Analysis & PDE Articles (Project Euclid)
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The latest articles from Analysis & PDE on Project Euclid, a site for mathematics and statistics resources.enusCopyright 2017 Cornell University LibraryEuclidL@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 12:57 EDTThu, 19 Oct 2017 12:57 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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The Fuglede conjecture holds in $\mathbb{Z}_p\times \mathbb{Z}_p$
https://projecteuclid.org/euclid.apde/1508432238
<strong>Alexander Iosevich</strong>, <strong>Azita Mayeli</strong>, <strong>Jonathan Pakianathan</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 10, Number 4, 757764.</p><p><strong>Abstract:</strong><br/>
In this paper we study subsets [math] of [math] such that any function [math] can be written as a linear combination of characters orthogonal with respect to [math] . We shall refer to such sets as spectral. In this context, we prove the Fuglede conjecture in [math] , which says in this context that [math] is spectral if and only if [math] tiles [math] by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof. The proof relies to a significant extent on the analysis of direction sets of Iosevich et al. ( Integers 11 (2011), art. id. A39) and the tiling results of Haessig et al. (2011).
</p>projecteuclid.org/euclid.apde/1508432238_20171019125725Thu, 19 Oct 2017 12:57 EDTFracture with healing: A first step towards a new view of cavitationhttps://projecteuclid.org/euclid.apde/1539050445<strong>Gilles Francfort</strong>, <strong>Alessandro Giacomini</strong>, <strong>Oscar LopezPamies</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 417447.</p><p><strong>Abstract:</strong><br/>
Recent experimental evidence on rubber has revealed that the internal cracks that arise out of the process, often referred to as cavitation, can actually heal.
We demonstrate that crack healing can be incorporated into the variational framework for quasistatic brittle fracture evolution that has been developed in the last twenty years. This will be achieved for twodimensional linearized elasticity in a topological setting, that is, when the putative cracks are closed sets with a preset maximum number of connected components.
Other important features of cavitation in rubber, such as near incompressibility and the evolution of the fracture toughness as a function of the cumulative history of fracture and healing, have yet to be addressed even in the proposed topological setting.
</p>projecteuclid.org/euclid.apde/1539050445_20181008220100Mon, 08 Oct 2018 22:01 EDTGeneral Clark model for finiterank perturbationshttps://projecteuclid.org/euclid.apde/1539050446<strong>Constanze Liaw</strong>, <strong>Sergei Treil</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 449492.</p><p><strong>Abstract:</strong><br/>
All unitary (contractive) perturbations of a given unitary operator [math] by finiterank [math] operators with fixed range can be parametrized by [math] unitary (contractive) matrices [math] ; this generalizes unitary rankone ( [math] ) perturbations, where the Aleksandrov–Clark family of unitary perturbations is parametrized by the scalars on the unit circle [math] .
For a strict contraction [math] the resulting perturbed operator [math] is (under the natural assumption about star cyclicity of the range) a completely nonunitary contraction, so it admits the functional model.
We investigate the Clark operator, i.e., a unitary operator that intertwines [math] (written in the spectral representation of the nonperturbed operator [math] ) and its model. We make no assumptions on the spectral type of the unitary operator [math] ; an absolutely continuous spectrum may be present.
We first find a universal representation of the adjoint Clark operator in the coordinatefree Nikolski–Vasyunin functional model; the word “universal” means that it is valid in any transcription of the model. This representation can be considered to be a special version of the vectorvalued Cauchy integral operator.
Combining the theory of singular integral operators with the theory of functional models, we derive from this abstract representation a concrete formula for the adjoint of the Clark operator in the Sz.Nagy–Foiaş transcription. As in the scalar case, the adjoint Clark operator is given by a sum of two terms: one is given by the boundary values of the vectorvalued Cauchy transform (postmultiplied by a matrixvalued function) and the second one is just the multiplication operator by a matrixvalued function.
Finally, we present formulas for the direct Clark operator in the Sz.Nagy–Foiaş transcription.
</p>projecteuclid.org/euclid.apde/1539050446_20181008220100Mon, 08 Oct 2018 22:01 EDTOn the maximal rank problem for the complex homogeneous Monge–Ampère equationhttps://projecteuclid.org/euclid.apde/1539050447<strong>Julius Ross</strong>, <strong>David Witt Nyström</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 493503.</p><p><strong>Abstract:</strong><br/>
We give examples of regular boundary data for the Dirichlet problem for the complex homogeneous Monge–Ampère equation over the unit disc, whose solution is completely degenerate on a nonempty open set and thus fails to have maximal rank.
</p>projecteuclid.org/euclid.apde/1539050447_20181008220100Mon, 08 Oct 2018 22:01 EDTA viscosity approach to the Dirichlet problem for degenerate complex Hessiantype equationshttps://projecteuclid.org/euclid.apde/1539050448<strong>Sławomir Dinew</strong>, <strong>HoangSon Do</strong>, <strong>Tat Dat Tô</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 505535.</p><p><strong>Abstract:</strong><br/>
A viscosity approach is introduced for the Dirichlet problem associated to complex Hessiantype equations on domains in [math] . The arguments are modeled on the theory of viscosity solutions for real Hessiantype equations developed by Trudinger (1990). As a consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.
</p>projecteuclid.org/euclid.apde/1539050448_20181008220100Mon, 08 Oct 2018 22:01 EDTResolvent estimates for spacetimes bounded by Killing horizonshttps://projecteuclid.org/euclid.apde/1539050449<strong>Oran Gannot</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 537560.</p><p><strong>Abstract:</strong><br/>
We show that the resolvent grows at most exponentially with frequency for the wave equation on a class of stationary spacetimes which are bounded by nondegenerate Killing horizons, without any assumptions on the trapped set. Correspondingly, there exists an exponentially small resonancefree region, and solutions of the Cauchy problem exhibit logarithmic energy decay.
</p>projecteuclid.org/euclid.apde/1539050449_20181008220100Mon, 08 Oct 2018 22:01 EDTInterpolation by conformal minimal surfaces and directed holomorphic curveshttps://projecteuclid.org/euclid.apde/1539050450<strong>Antonio Alarcón</strong>, <strong>Ildefonso CastroInfantes</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 561604.</p><p><strong>Abstract:</strong><br/>
Let [math] be an open Riemann surface and [math] be an integer. We prove that on any closed discrete subset of [math] one can prescribe the values of a conformal minimal immersion [math] . Our result also ensures jetinterpolation of given finite order, and hence, in particular, one may in addition prescribe the values of the generalized Gauss map. Furthermore, the interpolating immersions can be chosen to be complete, proper into [math] if the prescription of values is proper, and injective if [math] and the prescription of values is injective. We may also prescribe the flux map of the examples.
We also show analogous results for a large family of directed holomorphic immersions [math] , including null curves.
</p>projecteuclid.org/euclid.apde/1539050450_20181008220100Mon, 08 Oct 2018 22:01 EDTThe BMODirichlet problem for elliptic systems in the upper halfspace and quantitative characterizations of VMOhttps://projecteuclid.org/euclid.apde/1540432867<strong>José María Martell</strong>, <strong>Dorina Mitrea</strong>, <strong>Irina Mitrea</strong>, <strong>Marius Mitrea</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 3, 605720.</p><p><strong>Abstract:</strong><br/>
We prove that for any homogeneous, secondorder, constant complex coefficient elliptic system [math] in [math] , the Dirichlet problem in [math] with boundary data in [math] is wellposed in the class of functions [math] for which the Littlewood–Paley measure associated with [math] , namely
d
μ
u
(
x
′
,
t
)
:
=

∇
u
(
x
′
,
t
)

2
t
d
x
′
d
t
,
is a Carleson measure in [math] .
In the process we establish a Fatoutype theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth nullsolutions [math] of such systems satisfying the said Carleson measure condition. In concert, these results imply that the space [math] can be characterized as the collection of nontangential pointwise traces of smooth nullsolutions [math] to the elliptic system [math] with the property that [math] is a Carleson measure in [math] .
We also establish a regularity result for the BMODirichlet problem in the upper halfspace, to the effect that the nontangential pointwise trace on the boundary of [math] of any given smooth nullsolutions [math] of [math] in [math] satisfying the above Carleson measure condition actually belongs to Sarason’s space [math] if and only if [math] as [math] , uniformly with respect to the location of the cube [math] (where [math] is the Carleson box associated with [math] , and [math] denotes the Euclidean volume of [math] ).
Moreover, we are able to establish the wellposedness of the Dirichlet problem in [math] for a system [math] as above in the case when the boundary data are prescribed in Morrey–Campanato spaces in [math] . In such a scenario, the solution [math] is required to satisfy a vanishing Carleson measure condition of fractional order.
By relying on these wellposedness and regularity results we succeed in producing characterizations of the space [math] as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified (such as the class of smooth functions satisfying a Hölder or Lipschitz condition). This improves on Sarason’s classical result describing [math] as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. In turn, this allows us to show that any Calderón–Zygmund operator [math] satisfying [math] extends as a linear and bounded mapping from [math] (modulo constants) into itself. In turn, this is used to describe algebras of singular integral operators on [math] , and to characterize the membership to [math] via the action of various classes of singular integral operators.
</p>projecteuclid.org/euclid.apde/1540432867_20181024220121Wed, 24 Oct 2018 22:01 EDTConvergence of the Kähler–Ricci iterationhttps://projecteuclid.org/euclid.apde/1540432868<strong>Tamás Darvas</strong>, <strong>Yanir A. Rubinstein</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 3, 721735.</p><p><strong>Abstract:</strong><br/>
The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kähler–Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.
</p>projecteuclid.org/euclid.apde/1540432868_20181024220121Wed, 24 Oct 2018 22:01 EDTConcentration of ground states in stationary meanfield games systemshttps://projecteuclid.org/euclid.apde/1540432869<strong>Annalisa Cesaroni</strong>, <strong>Marco Cirant</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 3, 737787.</p><p><strong>Abstract:</strong><br/>
We provide the existence of classical solutions to stationary meanfield game systems in the whole space [math] , with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the nonconvex energy associated to the system. Finally, we show that in the vanishing viscosity limit, mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e., classical solutions to meanfield game systems in the whole space without potential, and with aggregating coupling.
</p>projecteuclid.org/euclid.apde/1540432869_20181024220121Wed, 24 Oct 2018 22:01 EDTGeneralized crystalline evolutions as limits of flows with smooth anisotropieshttps://projecteuclid.org/euclid.apde/1540432870<strong>Antonin Chambolle</strong>, <strong>Massimiliano Morini</strong>, <strong>Matteo Novaga</strong>, <strong>Marcello Ponsiglione</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 3, 789813.</p><p><strong>Abstract:</strong><br/>
We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as a limit of the viscosity solutions to flows with smooth anisotropies.
</p>projecteuclid.org/euclid.apde/1540432870_20181024220121Wed, 24 Oct 2018 22:01 EDTGlobal weak solutions of the Teichmüller harmonic map flow into general targetshttps://projecteuclid.org/euclid.apde/1540432871<strong>Melanie Rupflin</strong>, <strong>Peter M. Topping</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 3, 815842.</p><p><strong>Abstract:</strong><br/>
We analyse finitetime singularities of the Teichmüller harmonic map flow — a natural gradient flow of the harmonic map energy — and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a nolossoftopology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves.
</p>projecteuclid.org/euclid.apde/1540432871_20181024220121Wed, 24 Oct 2018 22:01 EDTA rigorous derivation from the kinetic Cucker–Smale model to the pressureless Euler system with nonlocal alignmenthttps://projecteuclid.org/euclid.apde/1540432872<strong>Alessio Figalli</strong>, <strong>MoonJin Kang</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 3, 843866.</p><p><strong>Abstract:</strong><br/>
We consider the kinetic Cucker–Smale model with local alignment as a mesoscopic description for the flocking dynamics. The local alignment was first proposed by Karper, Mellet and Trivisa (2014), as a singular limit of a normalized nonsymmetric alignment introduced by Motsch and Tadmor (2011). The existence of weak solutions to this model was obtained by Karper, Mellet and Trivisa (2014), and in the same paper they showed the timeasymptotic flocking behavior. Our main contribution is to provide a rigorous derivation from a mesoscopic to a macroscopic description for the Cucker–Smale flocking models. More precisely, we prove the hydrodynamic limit of the kinetic Cucker–Smale model with local alignment towards the pressureless Euler system with nonlocal alignment, under a regime of strong local alignment. Based on the relative entropy method, a main difficulty in our analysis comes from the fact that the entropy of the limit system has no strict convexity in terms of density variable. To overcome this, we combine relative entropy quantities with the 2Wasserstein distance.
</p>projecteuclid.org/euclid.apde/1540432872_20181024220121Wed, 24 Oct 2018 22:01 EDTQuantum dynamical bounds for ergodic potentials with underlying dynamics of zero topological entropyhttps://projecteuclid.org/euclid.apde/1540864855<strong>Rui Han</strong>, <strong>Svetlana Jitomirskaya</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 867902.</p><p><strong>Abstract:</strong><br/>
We show that positive Lyapunov exponents imply upper quantum dynamical bounds for Schrödinger operators [math] , where [math] is a piecewise Hölder function on a compact Riemannian manifold [math] , and [math] is a uniquely ergodic volumepreserving map with zero topological entropy. As corollaries we also obtain localizationtype statements for shifts and skewshifts on higherdimensional tori with arithmetic conditions on the parameters. These are the first localizationtype results with precise arithmetic conditions for multifrequency quasiperiodic and skewshift potentials.
</p>projecteuclid.org/euclid.apde/1540864855_20181029220123Mon, 29 Oct 2018 22:01 EDTTwodimensional gravity water waves with constant vorticity, I: Cubic lifespanhttps://projecteuclid.org/euclid.apde/1540864856<strong>Mihaela Ifrim</strong>, <strong>Daniel Tataru</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 903967.</p><p><strong>Abstract:</strong><br/>
This article is concerned with the incompressible, infinitedepth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in positionvelocity potential holomorphic coordinates, and prove local wellposedness for large data, as well as cubic lifespan bounds for smalldata solutions.
</p>projecteuclid.org/euclid.apde/1540864856_20181029220123Mon, 29 Oct 2018 22:01 EDTAbsolute continuity and $\alpha$numbers on the real linehttps://projecteuclid.org/euclid.apde/1540864857<strong>Tuomas Orponen</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 969996.</p><p><strong>Abstract:</strong><br/>
Let [math] , [math] be Radon measures on [math] , with [math] nonatomic and [math] doubling, and write [math] for the Lebesgue decomposition of [math] relative to [math] . For an interval [math] , define [math] , the Wasserstein distance of normalised blowups of [math] and [math] restricted to [math] . Let [math] be the square function
S
ν
2
(
μ
)
=
∑
I
∈
D
α
μ
,
ν
2
(
I
)
χ
I
,
where [math] is the family of dyadic intervals of sidelength at most 1. I prove that [math] is finite [math] almost everywhere and infinite [math] almost everywhere. I also prove a version of the result for a nondyadic variant of the square function [math] . The results answer the simplest “ [math] ” case of a problem of J. Azzam, G. David and T. Toro.
</p>projecteuclid.org/euclid.apde/1540864857_20181029220123Mon, 29 Oct 2018 22:01 EDTGlobal wellposedness for the twodimensional Muskat problem with slope less than 1https://projecteuclid.org/euclid.apde/1540864858<strong>Stephen Cameron</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 9971022.</p><p><strong>Abstract:</strong><br/>
We prove the existence of global, smooth solutions to the twodimensional Muskat problem in the stable regime whenever the product of the maximal and minimal slope is less than 1. The curvature of these solutions decays to 0 as [math] goes to infinity, and they are unique when the initial data is [math] . We do this by getting a priori estimates using a nonlinear maximum principle first introduced in a paper by Kiselev, Nazarov, and Volberg (2007), where the authors proved global wellposedness for the surface quasigeostraphic equation.
</p>projecteuclid.org/euclid.apde/1540864858_20181029220123Mon, 29 Oct 2018 22:01 EDTGlobal wellposedness and scattering for the radial, defocusing, cubic wave equation with initial data in a critical Besov spacehttps://projecteuclid.org/euclid.apde/1540864859<strong>Benjamin Dodson</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 10231048.</p><p><strong>Abstract:</strong><br/>
We prove that the cubic wave equation is globally wellposed and scattering for radial initial data lying in [math] . This space of functions is a scaleinvariant subspace of [math] .
</p>projecteuclid.org/euclid.apde/1540864859_20181029220123Mon, 29 Oct 2018 22:01 EDTNonexistence of Wente's $L^\infty$ estimate for the Neumann problemhttps://projecteuclid.org/euclid.apde/1540864860<strong>Jonas Hirsch</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 10491063.</p><p><strong>Abstract:</strong><br/>
We provide a counterexample of Wente’s inequality in the context of Neumann boundary conditions. We will also show that Wente’s estimate fails for general boundary conditions of Robin type.
</p>projecteuclid.org/euclid.apde/1540864860_20181029220123Mon, 29 Oct 2018 22:01 EDTGlobal geometry and $C^1$ convex extensions of 1jetshttps://projecteuclid.org/euclid.apde/1540864861<strong>Daniel Azagra</strong>, <strong>Carlos Mudarra</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 10651099.</p><p><strong>Abstract:</strong><br/>
Let [math] be an arbitrary subset of [math] (not necessarily bounded) and [math] , [math] be functions. We provide necessary and sufficient conditions for the [math] jet [math] to have an extension [math] with [math] convex and [math] . Additionally, if [math] is bounded we can take [math] so that [math] . As an application we also solve a similar problem about finding convex hypersurfaces of class [math] with prescribed normals at the points of an arbitrary subset of [math] .
</p>projecteuclid.org/euclid.apde/1540864861_20181029220123Mon, 29 Oct 2018 22:01 EDTClassification of positive singular solutions to a nonlinear biharmonic equation with critical exponenthttps://projecteuclid.org/euclid.apde/1540864862<strong>Rupert L. Frank</strong>, <strong>Tobias König</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 11011113.</p><p><strong>Abstract:</strong><br/>
For [math] , we consider positive solutions [math] of the biharmonic equation
Δ
2
u
=
u
(
n
+
4
)
∕
(
n
−
4
)
on
ℝ
n
∖
{
0
}
,
with a nonremovable singularity at the origin. We show that [math] is a periodic function of [math] and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior of local solutions near singularities and for the [math] curvature problem in conformal geometry.
</p>projecteuclid.org/euclid.apde/1540864862_20181029220123Mon, 29 Oct 2018 22:01 EDTOptimal multilinear restriction estimates for a class of hypersurfaces with curvaturehttps://projecteuclid.org/euclid.apde/1540864863<strong>Ioan Bejenaru</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 4, 11151148.</p><p><strong>Abstract:</strong><br/>
Bennett, Carbery and Tao (2006) considered the [math] linear restriction estimate in [math] and established the near optimal [math] estimate under transversality assumptions only. In 2017, we showed that the trilinear restriction estimate improves its range of exponents under some curvature assumptions. In this paper we establish almost sharp multilinear estimates for a class of hypersurfaces with curvature for [math] . Together with previous results in the literature, this shows that curvature improves the range of exponents in the multilinear restriction estimate at all levels of lower multilinearity, that is, when [math] .
</p>projecteuclid.org/euclid.apde/1540864863_20181029220123Mon, 29 Oct 2018 22:01 EDTOn the Luzin $N$property and the uncertainty principle for Sobolev mappingshttps://projecteuclid.org/euclid.apde/1546657228<strong>Adele Ferone</strong>, <strong>Mikhail V. Korobkov</strong>, <strong>Alba Roviello</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 11491175.</p><p><strong>Abstract:</strong><br/>
We say that a mapping [math] satisfies the [math]  [math] property if [math] whenever [math] , where [math] means the Hausdorff measure. We prove that every mapping [math] of Sobolev class [math] with [math] satisfies the [math]  [math] property for every [math] with
[math]
We prove also that for [math] and for the critical value [math] the corresponding [math]  [math] property fails in general. Nevertheless, this [math]  [math] property holds for [math] if we assume in addition that the highest derivatives [math] belong to the Lorentz space [math] instead of [math] .
We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubinitype theorems for [math] Nproperties and discuss their applications to the Morse–Sard theorem and its recent extensions.
</p>projecteuclid.org/euclid.apde/1546657228_20190104220040Fri, 04 Jan 2019 22:00 ESTUnstable normalized standing waves for the space periodic NLShttps://projecteuclid.org/euclid.apde/1546657229<strong>Nils Ackermann</strong>, <strong>Tobias Weth</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 11771213.</p><p><strong>Abstract:</strong><br/>
For the stationary nonlinear Schrödinger equation [math] with periodic potential [math] we study the existence and stability properties of multibump solutions with prescribed [math] norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow us to match the [math] constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated [math] sphere, and we show their orbital instability with respect to the Schrödinger flow. Our results apply in both, the masssubcritical and the masssupercritical regime.
</p>projecteuclid.org/euclid.apde/1546657229_20190104220040Fri, 04 Jan 2019 22:00 ESTScaleinvariant Fourier restriction to a hyperbolic surfacehttps://projecteuclid.org/euclid.apde/1546657230<strong>Betsy Stovall</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 12151224.</p><p><strong>Abstract:</strong><br/>
This result sharpens the bilineartolinear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid in [math] to the sharp line, leading to the first scaleinvariant restriction estimates, beyond the Stein–Tomas range, for a hypersurface on which the principal curvatures have different signs.
</p>projecteuclid.org/euclid.apde/1546657230_20190104220040Fri, 04 Jan 2019 22:00 ESTSteady threedimensional rotational flows: an approach via two stream functions and Nash–Moser iterationhttps://projecteuclid.org/euclid.apde/1546657231<strong>Boris Buffoni</strong>, <strong>Erik Wahlén</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 12251258.</p><p><strong>Abstract:</strong><br/>
We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region [math] . We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary [math] . The Bernoulli equation states that the “Bernoulli function” [math] (where [math] is the velocity field and [math] the pressure) is constant along stream lines, that is, each particle is associated with a particular value of [math] . We also prescribe the value of [math] on [math] . The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form [math] and deriving a degenerate nonlinear elliptic system for [math] and [math] . This system is solved using the Nash–Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see, e.g., the book by Q. Han and J.X. Hong (2006). Since we can allow [math] to be nonconstant on [math] , our theory includes threedimensional flows with nonvanishing vorticity.
</p>projecteuclid.org/euclid.apde/1546657231_20190104220040Fri, 04 Jan 2019 22:00 ESTSparse bounds for the discrete cubic Hilbert transformhttps://projecteuclid.org/euclid.apde/1546657232<strong>Amalia Culiuc</strong>, <strong>Robert Kesler</strong>, <strong>Michael T. Lacey</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 12591272.</p><p><strong>Abstract:</strong><br/>
Consider the discrete cubic Hilbert transform defined on finitely supported functions [math] on [math] by
[math]
We prove that there exists [math] and universal constant [math] such that for all finitely supported [math] on [math] there exists an [math] sparse form [math] for which
[math]
This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.
</p>projecteuclid.org/euclid.apde/1546657232_20190104220040Fri, 04 Jan 2019 22:00 ESTOn the dimension and smoothness of radial projectionshttps://projecteuclid.org/euclid.apde/1546657233<strong>Tuomas Orponen</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 12731294.</p><p><strong>Abstract:</strong><br/>
This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces.
To introduce the first one, assume that [math] are nonempty Borel sets with [math] . Does the radial projection of [math] to some point in [math] have positive dimension? Not necessarily: [math] can be zerodimensional, or [math] and [math] can lie on a common line. I prove that these are the only obstructions: if [math] , and [math] does not lie on a line, then there exists a point in [math] such that the radial projection [math] has Hausdorff dimension at least [math] . Applying the result with [math] gives the following corollary: if [math] is a Borel set which does not lie on a line, then the set of directions spanned by [math] has Hausdorff dimension at least [math] .
For the second result, let [math] and [math] . Let [math] be a compactly supported Radon measure in [math] with finite [math] energy. I prove that the radial projections of [math] are absolutely continuous with respect to [math] for every centre in [math] , outside an exceptional set of dimension at most [math] . In fact, for [math] outside an exceptional set as above, the proof shows that [math] for some [math] . The dimension bound on the exceptional set is sharp.
</p>projecteuclid.org/euclid.apde/1546657233_20190104220040Fri, 04 Jan 2019 22:00 ESTCartan subalgebras of tensor products of free quantum group factors with arbitrary factorshttps://projecteuclid.org/euclid.apde/1546657234<strong>Yusuke Isono</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 12951324.</p><p><strong>Abstract:</strong><br/>
Let [math] be a free (unitary or orthogonal) quantum group. We prove that for any nonamenable subfactor [math] which is an image of a faithful normal conditional expectation, and for any [math] finite factor [math] , the tensor product [math] has no Cartan subalgebras. This generalizes our previous work that provides the same result when [math] is finite. In the proof, we establish Ozawa–Popa and Popa–Vaes’s weakly compact action on the continuous core of [math] as the one relative to [math] , by using an operatorvalued weight to [math] and the central weak amenability of [math] .
</p>projecteuclid.org/euclid.apde/1546657234_20190104220040Fri, 04 Jan 2019 22:00 ESTCommutators of multiparameter flag singular integrals and applicationshttps://projecteuclid.org/euclid.apde/1546657235<strong>Xuan Thinh Duong</strong>, <strong>Ji Li</strong>, <strong>Yumeng Ou</strong>, <strong>Jill Pipher</strong>, <strong>Brett D. Wick</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 13251355.</p><p><strong>Abstract:</strong><br/>
We introduce the iterated commutator for the Riesz transforms in the multiparameter flag setting, and prove the upper bound of this commutator with respect to the symbol [math] in the flag BMO space. Our methods require the techniques of semigroups, harmonic functions and multiparameter flag Littlewood–Paley analysis. We also introduce the big commutator in this multiparameter flag setting and prove the upper bound with symbol [math] in the flag little bmo space by establishing the “exponentiallogarithmic” bridge between this flag little bmo space and the Muckenhoupt [math] weights with flag structure. As an application, we establish the divcurl lemmas with respect to the appropriate Hardy spaces in the multiparameter flag setting.
</p>projecteuclid.org/euclid.apde/1546657235_20190104220040Fri, 04 Jan 2019 22:00 ESTRokhlin dimension: absorption of model actionshttps://projecteuclid.org/euclid.apde/1546657236<strong>Gábor Szabó</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 5, 13571396.</p><p><strong>Abstract:</strong><br/>
We establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly selfabsorbing [math] algebras. Namely, as to be made precise in the paper, let [math] be a wellbehaved locally compact group. If [math] is a strongly selfabsorbing [math] algebra and [math] is an action on a separable, [math] absorbing [math] algebra that has finite Rokhlin dimension with commuting towers, then [math] tensorially absorbs every semistrongly selfabsorbing [math] action on [math] . In particular, this is the case when [math] satisfies any version of what is called the Rokhlin property, such as for [math] or [math] . This contains several existing results of similar nature as special cases. We will in fact prove a more general version of this theorem, which is intended for use in subsequent work. We will then discuss some nontrivial applications. Most notably it is shown that for any [math] and on any strongly selfabsorbing Kirchberg algebra, there exists a unique [math] action having finite Rokhlin dimension with commuting towers up to (very strong) cocycle conjugacy.
</p>projecteuclid.org/euclid.apde/1546657236_20190104220040Fri, 04 Jan 2019 22:00 ESTLong time behavior of the master equation in mean field game theoryhttps://projecteuclid.org/euclid.apde/1552356123<strong>Pierre Cardaliaguet</strong>, <strong>Alessio Porretta</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 6, 13971453.</p><p><strong>Abstract:</strong><br/>
Mean field game (MFG) systems describe equilibrium configurations in games with infinitely many interacting controllers. We are interested in the behavior of this system as the horizon becomes large, or as the discount factor tends to [math] . We show that, in these two cases, the asymptotic behavior of the mean field game system is strongly related to the long time behavior of the socalled master equation and to the vanishing discount limit of the discounted master equation, respectively. Both equations are nonlinear transport equations in the space of measures. We prove the existence of a solution to an ergodic master equation, towards which the timedependent master equation converges as the horizon becomes large, and towards which the discounted master equation converges as the discount factor tends to [math] . The whole analysis is based on new estimates for the exponential rates of convergence of the timedependent and the discounted MFG systems, respectively.
</p>projecteuclid.org/euclid.apde/1552356123_20190311220222Mon, 11 Mar 2019 22:02 EDTOn the cost of observability in small times for the onedimensional heat equationhttps://projecteuclid.org/euclid.apde/1552356127<strong>Jérémi Dardé</strong>, <strong>Sylvain Ervedoza</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 6, 14551488.</p><p><strong>Abstract:</strong><br/>
We aim at presenting a new estimate on the cost of observability in small times of the onedimensional heat equation, which also provides a new proof of observability for the onedimensional heat equation. Our proof combines several tools. First, it uses a Carlemantype estimate borrowed from our previous work ( SIAM J. Control Optim. 56 :3 (2018), 1692–1715), in which the weight function is derived from the heat kernel and which is therefore particularly easy. We also use explicit computations in the Fourier domain to compute the highfrequency part of the solution in terms of the observations. Finally, we use the Phragmén–Lindelöf principle to estimate the lowfrequency part of the solution. This last step is done carefully with precise estimations coming from conformal mappings.
</p>projecteuclid.org/euclid.apde/1552356127_20190311220222Mon, 11 Mar 2019 22:02 EDTZeros of repeated derivatives of random polynomialshttps://projecteuclid.org/euclid.apde/1552356128<strong>Renjie Feng</strong>, <strong>Dong Yao</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 6, 14891512.</p><p><strong>Abstract:</strong><br/>
It has been shown that zeros of Kac polynomials [math] of degree [math] cluster asymptotically near the unit circle as [math] under some assumptions. This property remains unchanged for the [math] th derivative of the Kac polynomials [math] for any fixed order [math] . So it’s natural to study the situation when the number of the derivatives we take depends on [math] , i.e., [math] . We will show that the limiting behavior of zeros of [math] depends on the limit of the ratio [math] . In particular, we prove that when the limit of the ratio is strictly positive, the property of the uniform clustering around the unit circle fails; when the ratio is close to 1, the zeros have some rescaling phenomenon. Then we study such problem for random polynomials with more general coefficients. But things, especially the rescaling phenomenon, become very complicated for the general case when [math] , where we compute the case of the random elliptic polynomials to illustrate this.
</p>projecteuclid.org/euclid.apde/1552356128_20190311220222Mon, 11 Mar 2019 22:02 EDTGross–Pitaevskii dynamics for Bose–Einstein condensateshttps://projecteuclid.org/euclid.apde/1552356129<strong>Christian Brennecke</strong>, <strong>Benjamin Schlein</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 6, 15131596.</p><p><strong>Abstract:</strong><br/>
We study the timeevolution of initially trapped Bose–Einstein condensates in the Gross–Pitaevskii regime. We show that condensation is preserved by the manybody evolution and that the dynamics of the condensate wave function can be described by the timedependent Gross–Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e., on the number of excitations of the Bose–Einstein condensate). To reach this goal, we combine the method of Lewin, Nam and Schlein (2015), who analyzed fluctuations around the Hartree dynamics for [math] particle initial data in the meanfield regime, with ideas of Benedikter, de Oliveira and Schlein (2015), who considered the evolution of Fockspace initial data in the Gross–Pitaevskii regime.
</p>projecteuclid.org/euclid.apde/1552356129_20190311220222Mon, 11 Mar 2019 22:02 EDTDimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphshttps://projecteuclid.org/euclid.apde/1552356130<strong>Riccardo Adami</strong>, <strong>Simone Dovetta</strong>, <strong>Enrico Serra</strong>, <strong>Paolo Tilli</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 6, 15971612.</p><p><strong>Abstract:</strong><br/>
We investigate the existence of ground states for the focusing nonlinear Schrödinger equation on a prototypical doubly periodic metric graph. When the nonlinearity power is below 4, ground states exist for every value of the mass, while, for every nonlinearity power between 4 (included) and 6 (excluded), a mark of [math] criticality arises, as ground states exist if and only if the mass exceeds a threshold value that depends on the power. This phenomenon can be interpreted as a continuous transition from a twodimensional regime, for which the only critical power is 4, to a onedimensional behavior, in which criticality corresponds to the power 6. We show that such a dimensional crossover is rooted in the coexistence of onedimensional and twodimensional Sobolev inequalities, leading to a new family of Gagliardo–Nirenberg inequalities that account for this continuum of critical exponents.
</p>projecteuclid.org/euclid.apde/1552356130_20190311220222Mon, 11 Mar 2019 22:02 EDTAlexandrov's theorem revisitedhttps://projecteuclid.org/euclid.apde/1552356131<strong>Matias Gonzalo Delgadino</strong>, <strong>Francesco Maggi</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 6, 16131642.</p><p><strong>Abstract:</strong><br/>
We show that among sets of finite perimeter balls are the only volumeconstrained critical points of the perimeter functional.
</p>projecteuclid.org/euclid.apde/1552356131_20190311220222Mon, 11 Mar 2019 22:02 EDTGeneralized $q$gaussian von Neumann algebras with coefficients, I: Relative strong solidityhttps://projecteuclid.org/euclid.apde/1564538421<strong>Marius Junge</strong>, <strong>Bogdan Udrea</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 7, 16431709.</p><p><strong>Abstract:</strong><br/>
We define [math] , the generalized [math] gaussian von Neumann algebras associated to a sequence of symmetric independent copies [math] and to a subset [math] and, under certain assumptions, prove their strong solidity relative to [math] . We provide many examples of strongly solid generalized [math] gaussian von Neumann algebras. We also obtain nonisomorphism and nonembedability results about some of these von Neumann algebras.
</p>projecteuclid.org/euclid.apde/1564538421_20190730220039Tue, 30 Jul 2019 22:00 EDTComplex interpolation and Calderón–Mityagin couples of Morrey spaceshttps://projecteuclid.org/euclid.apde/1564538422<strong>Mieczysław Mastyło</strong>, <strong>Yoshihiro Sawano</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 7, 17111740.</p><p><strong>Abstract:</strong><br/>
We study interpolation spaces between global Morrey spaces and between local Morrey spaces. We prove that for a wide class of couples of these spaces the upper complex Calderón spaces are not described by the [math] method of interpolation. A byproduct of our results is that couples of Morrey spaces belonging to this class are not Calderón–Mityagin couples. A Banach couple [math] is said to have the universal [math] property if all relative interpolation spaces from any Banach couple to [math] are relatively [math] monotone. A couple of local Morrey spaces is proved to have the universal [math] property once it is a Calderón–Mityagin couple.
</p>projecteuclid.org/euclid.apde/1564538422_20190730220039Tue, 30 Jul 2019 22:00 EDTMonotonicity and local uniqueness for the Helmholtz equationhttps://projecteuclid.org/euclid.apde/1564538423<strong>Bastian Harrach</strong>, <strong>Valter Pohjola</strong>, <strong>Mikko Salo</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 7, 17411771.</p><p><strong>Abstract:</strong><br/>
This work extends monotonicitybased methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation [math] in a bounded domain for fixed nonresonance frequency [math] and realvalued scattering coefficient function [math] . We show a monotonicity relation between the scattering coefficient [math] and the local NeumanntoDirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicitybased characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions [math] and [math] can be distinguished by partial boundary data if there is a neighborhood of the boundary part where [math] and [math] .
</p>projecteuclid.org/euclid.apde/1564538423_20190730220039Tue, 30 Jul 2019 22:00 EDTSolutions of the 4species quadratic reactiondiffusion system are bounded and $C^\infty$smooth, in any space dimensionhttps://projecteuclid.org/euclid.apde/1564538424<strong>M. Cristina Caputo</strong>, <strong>Thierry Goudon</strong>, <strong>Alexis F. Vasseur</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 7, 17731804.</p><p><strong>Abstract:</strong><br/>
We establish the boundedness of solutions of reactiondiffusion systems with quadratic (in fact slightly superquadratic) reaction terms that satisfy a natural entropy dissipation property, in any space dimension [math] . This bound implies the [math] regularity of the solutions. This result extends the theory which was restricted to the twodimensional case. The proof heavily uses De Giorgi’s iteration scheme, which allows us to obtain local estimates. The arguments rely on duality reasoning in order to obtain new estimates on the total mass of the system, both in the [math] norm and in a suitable weak norm. The latter uses [math] regularization properties for parabolic equations.
</p>projecteuclid.org/euclid.apde/1564538424_20190730220039Tue, 30 Jul 2019 22:00 EDTSpacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski spacehttps://projecteuclid.org/euclid.apde/1564538425<strong>Denis Bonheure</strong>, <strong>Alessandro Iacopetti</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 7, 18051842.</p><p><strong>Abstract:</strong><br/>
We investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz–Minkowski space [math] , for [math] , spanning a given boundary datum lying on the hyperbolic space [math] .
</p>projecteuclid.org/euclid.apde/1564538425_20190730220039Tue, 30 Jul 2019 22:00 EDTSquare function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lowerdimensional setshttps://projecteuclid.org/euclid.apde/1564538428<strong>Svitlana Mayboroda</strong>, <strong>Zihui Zhao</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 7, 18431890.</p><p><strong>Abstract:</strong><br/>
In the recent work G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lowerdimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary.
The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, [math] , in terms of solvability of boundary value problems. The authors establish, in particular, square function estimates and solvability of the Dirichlet problem in BMO for domains with lowerdimensional boundaries under the underlying assumption [math] . More generally, it is proved that in all domains with Ahlfors regular boundaries the BMO solvability of the Dirichlet problem is necessary and sufficient for the absolute continuity of the harmonic measure.
</p>projecteuclid.org/euclid.apde/1564538428_20190730220039Tue, 30 Jul 2019 22:00 EDTTangent measures of elliptic measure and applicationshttps://projecteuclid.org/euclid.apde/1576292432<strong>Jonas Azzam</strong>, <strong>Mihalis Mourgoglou</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 8, 18911941.</p><p><strong>Abstract:</strong><br/>
Tangent measure and blowup methods are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly nonsymmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains [math] , [math] :
We extend the results of Kenig, Preiss, and Toro ( J. Amer. Math. Soc. 22 :3
(2009), 771–796) by showing mutual absolute continuity of interior and exterior elliptic
measures for any domains implies the tangent measures are a.e. flat and the elliptic
measures have dimension
n
.
We generalize the work of Kenig and Toro ( J. Reine Agnew. Math. 596 (2006),
1–44) and show that
VMO
equivalence of doubling interior and exterior elliptic measures for general domains
implies the tangent measures are always supported on the zero sets of elliptic polynomials.
In a uniform domain that satisfies the capacity density condition and whose boundary is
locally finite and has a.e. positive lower
n
Hausdorff density, we show that if the elliptic measure is absolutely continuous with
respect to
n
Hausdorff measure then the boundary is rectifiable. This generalizes the work of
Akman, Badger, Hofmann, and Martell ( Trans. Amer. Math. Soc. 369:8 (2017),
5711–5745).
</p>projecteuclid.org/euclid.apde/1576292432_20191213220105Fri, 13 Dec 2019 22:01 ESTDiscretely selfsimilar solutions to the Navier–Stokes equations with data in $L^2_{\mathrm{loc}}$ satisfying the local energy inequalityhttps://projecteuclid.org/euclid.apde/1576292433<strong>Zachary Bradshaw</strong>, <strong>TaiPeng Tsai</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 8, 19431962.</p><p><strong>Abstract:</strong><br/>
Chae and Wolf recently constructed discretely selfsimilar solutions to the Navier–Stokes equations for any discretely selfsimilar data in [math] . Their solutions are in the class of local Leray solutions with projected pressure and satisfy the “local energy inequality with projected pressure”. In this note, for the same class of initial data, we construct discretely selfsimilar suitable weak solutions to the Navier–Stokes equations that satisfy the classical local energy inequality of Scheffer and Caffarelli–Kohn–Nirenberg. We also obtain an explicit formula for the pressure in terms of the velocity. Our argument involves a new purely local energy estimate for discretely selfsimilar solutions with data in [math] and an approximation of divergencefree, discretely selfsimilar vector fields in [math] by divergencefree, discretely selfsimilar elements of [math] .
</p>projecteuclid.org/euclid.apde/1576292433_20191213220105Fri, 13 Dec 2019 22:01 ESTContinuity properties for divergence form boundary data homogenization problemshttps://projecteuclid.org/euclid.apde/1576292436<strong>William M. Feldman</strong>, <strong>Yuming Paul Zhang</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 8, 19632002.</p><p><strong>Abstract:</strong><br/>
We study the asymptotic behavior at rational directions of the effective boundary condition in periodic homogenization of oscillating Dirichlet data. We establish a characterization for the directional limits at a rational direction in terms of a relatively simple twodimensional boundary layer problem for the homogenized operator. Using this characterization we show continuity of the effective boundary condition for divergence form linear systems, and for divergence form nonlinear equations we give an example of discontinuity.
</p>projecteuclid.org/euclid.apde/1576292436_20191213220105Fri, 13 Dec 2019 22:01 ESTDynamics of onefold symmetric patches for the aggregation equation and collapse to singular measurehttps://projecteuclid.org/euclid.apde/1576292437<strong>Taoufik Hmidi</strong>, <strong>Dong Li</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 8, 20032065.</p><p><strong>Abstract:</strong><br/>
We are concerned with the dynamics of onefold symmetric patches for the twodimensional aggregation equation associated to the Newtonian potential. We reformulate a suitable graph model and prove a local wellposedness result in subcritical and critical spaces. The global existence is obtained only for small initial data using a weak damping property hidden in the velocity terms. This allows us to analyze the concentration phenomenon of the aggregation patches near the blowup time. In particular, we prove that the patch collapses to a collection of disjoint segments and we provide a description of the singular measure through a careful study of the asymptotic behavior of the graph.
</p>projecteuclid.org/euclid.apde/1576292437_20191213220105Fri, 13 Dec 2019 22:01 ESTCoupled Kähler–Ricci solitons on toric Fano manifoldshttps://projecteuclid.org/euclid.apde/1576292438<strong>Jakob Hultgren</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 8, 20672094.</p><p><strong>Abstract:</strong><br/>
We prove a necessary and sufficient condition in terms of the barycenters of a collection of polytopes for existence of coupled Kähler–Einstein metrics on toric Fano manifolds. This confirms the toric case of a coupled version of the Yau–Tian–Donaldson conjecture and as a corollary we obtain an example of a coupled Kähler–Einstein metric on a manifold which does not admit Kähler–Einstein metrics. We also obtain a necessary and sufficient condition for existence of torusinvariant solutions to a system of solitontype equations on toric Fano manifolds.
</p>projecteuclid.org/euclid.apde/1576292438_20191213220105Fri, 13 Dec 2019 22:01 ESTCarleson measure estimates and the Dirichlet problem for degenerate elliptic equationshttps://projecteuclid.org/euclid.apde/1576292441<strong>Steve Hofmann</strong>, <strong>Phi Le</strong>, <strong>Andrew J. Morris</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 8, 20952146.</p><p><strong>Abstract:</strong><br/>
We prove that the Dirichlet problem for degenerate elliptic equations [math] in the upper halfspace [math] is solvable when [math] and the boundary data is in [math] for some [math] . The coefficient matrix [math] is only assumed to be measurable, realvalued and [math] independent with a degenerate bound and ellipticity controlled by an [math] weight [math] . It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is in [math] with respect to the [math] weighted Lebesgue measure on [math] . The Carleson measure estimate allows us to avoid applying the method of [math] approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.
</p>projecteuclid.org/euclid.apde/1576292441_20191213220105Fri, 13 Dec 2019 22:01 ESTAbsence of Cartan subalgebras for rightangled Hecke von Neumann algebrashttps://projecteuclid.org/euclid.apde/1579143666<strong>Martijn Caspers</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 128.</p><p><strong>Abstract:</strong><br/>
For a rightangled Coxeter system [math] and [math] , let [math] be the associated Hecke von Neumann algebra, which is generated by selfadjoint operators [math] , [math] , satisfying the Hecke relation [math] , as well as suitable commutation relations. Under the assumption that [math] is irreducible and [math] it was proved by Garncarek ( J. Funct. Anal. 270 :3 (2016), 1202–1219) that [math] is a factor (of type II [math] ) for a range [math] and otherwise [math] is the direct sum of a II [math] factor and [math] .
In this paper we prove (under the same natural conditions as Garncarek) that [math] is noninjective, that it has the weak [math] completely contractive approximation property and that it has the Haagerup property. In the hyperbolic factorial case [math] is a strongly solid algebra and consequently [math] cannot have a Cartan subalgebra. In the general case [math] need not be strongly solid. However, we give examples of nonhyperbolic rightangled Coxeter groups such that [math] does not possess a Cartan subalgebra.
</p>projecteuclid.org/euclid.apde/1579143666_20200115220114Wed, 15 Jan 2020 22:01 ESTA vector field method for radiating black hole spacetimeshttps://projecteuclid.org/euclid.apde/1579143667<strong>Jesús Oliver</strong>, <strong>Jacob Sterbenz</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 2992.</p><p><strong>Abstract:</strong><br/>
We develop a commuting vector field method for a general class of radiating spacetimes. The metrics we consider are modeled on those constructed from global nonlinear stability problems in general relativity, and our method provides sharp peeling estimates for solutions to both linear and (null form) nonlinear scalar fields.
</p>projecteuclid.org/euclid.apde/1579143667_20200115220114Wed, 15 Jan 2020 22:01 ESTStable ODEtype blowup for some quasilinear wave equations with derivativequadratic nonlinearitieshttps://projecteuclid.org/euclid.apde/1579143668<strong>Jared Speck</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 93146.</p><p><strong>Abstract:</strong><br/>
We prove a constructive stable ODEtype blowupresult for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions. The blowup is driven by a Riccatitype derivativequadratic semilinear term, and the singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowuptime. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the “elliptic part” of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or large in [math] in an absolute sense, but we assume that their spatial derivatives satisfy a nonlinear smallness condition relative to the size of the time derivative. The first main idea of the proof is to construct a quasilinear integrating factor, which allows us to reformulate the wave equation as a firstorder system whose solutions remain regular, all the way up to the singularity. Using the integrating factor, we construct quasilinear vector fields adapted to the nonlinear flow. The second main idea is to exploit some crucial monotonic terms in various estimates, especially the energy estimates, that feature the integrating factor. The availability of the monotonicity is tied to our size assumptions on the initial data and on the structure of the nonlinear terms. The third main idea is to propagate the relative smallness of the spatial derivatives all the way up to the singularity so that the solution behaves, in many ways, like an ODE solution. As a corollary of our main results, we show that there are quasilinear wave equations that exhibit two distinct kinds of blowup: the formation of shocks for one nontrivial set of data, and ODEtype blowup for another nontrivial set.
</p>projecteuclid.org/euclid.apde/1579143668_20200115220114Wed, 15 Jan 2020 22:01 ESTAsymptotic expansions of fundamental solutions in parabolic homogenizationhttps://projecteuclid.org/euclid.apde/1579143669<strong>Jun Geng</strong>, <strong>Zhongwei Shen</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 147170.</p><p><strong>Abstract:</strong><br/>
For a family of secondorder parabolic systems with rapidly oscillating and timedependent periodic coefficients, we investigate the asymptotic behavior of fundamental solutions and establish sharp estimates for the remainders.
</p>projecteuclid.org/euclid.apde/1579143669_20200115220114Wed, 15 Jan 2020 22:01 ESTCapillary surfaces arising in singular perturbation problemshttps://projecteuclid.org/euclid.apde/1579143670<strong>Aram L. Karakhanyan</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 171200.</p><p><strong>Abstract:</strong><br/>
We prove some Bernsteintype theorems for a class of stationary points of the Alt–Caffarelli functional in [math] and [math] arising as limits of the singular perturbation problem
△
u
ε
(
x
)
=
β
ε
(
u
ε
)
in
B
1
,

u
ε

≤
1
in
B
1
,
in the unit ball [math] as [math] . Here [math] , [math] , [math] , is an approximation of the Dirac measure and [math] . The limit functions [math] of uniformly converging sequences [math] solve a Bernoullitype free boundary problem in some weak sense. Our approach has two novelties: First we develop a hybrid method for stratification of the free boundary [math] of blowup solutions which combines some ideas and techniques of viscosity and variational theory. An important tool we use is a new monotonicity formula for the solutions [math] based on a computation of J. Spruck. It implies that any blowup [math] of [math] either vanishes identically or is a homogeneous function of degree 1, that is, [math] , [math] , in spherical coordinates [math] . In particular, this implies that in two dimensions the singular set is empty at the nondegenerate points, and in three dimensions the singular set of [math] is at most a singleton. Second, we show that the spherical part [math] is the support function (in Minkowski’s sense) of some capillary surface contained in the sphere of radius [math] . In particular, we show that [math] is an almost conformal and minimal immersion and the singular Alt–Caffarelli example corresponds to a piece of catenoid which is a unique ringtype stationary minimal surface determined by the support function [math] .
</p>projecteuclid.org/euclid.apde/1579143670_20200115220114Wed, 15 Jan 2020 22:01 ESTA spiral interface with positive Alt–Caffarelli–Friedman limit at the originhttps://projecteuclid.org/euclid.apde/1579143671<strong>Mark Allen</strong>, <strong>Dennis Kriventsov</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 201214.</p><p><strong>Abstract:</strong><br/>
We give an example of a pair of nonnegative subharmonic functions with disjoint support for which the Alt–Caffarelli–Friedman monotonicity formula has strictly positive limit at the origin, and yet the interface between their supports lacks a (unique) tangent there. This clarifies a remark of Caffarelli and Salsa ( A geometric approach to free boundary problems , 2005) that the positivity of the limit of the ACF formula implies unique tangents; this is true under some additional assumptions, but false in general. In our example, blowups converge to the expected piecewise linear twoplane function along subsequences, but the limiting function depends on the subsequence due to the spiraling nature of the interface.
</p>projecteuclid.org/euclid.apde/1579143671_20200115220114Wed, 15 Jan 2020 22:01 ESTInfinitetime blowup for the 3dimensional energycritical heat equationhttps://projecteuclid.org/euclid.apde/1579143672<strong>Manuel del Pino</strong>, <strong>Monica Musso</strong>, <strong>Juncheng Wei</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 215274.</p><p><strong>Abstract:</strong><br/>
We construct globally defined in time, unbounded positive solutions to the energycritical heat equation in dimension 3
u
t
=
Δ
u
+
u
5
in
ℝ
3
×
(
0
,
∞
)
,
u
(
x
,
0
)
=
u
0
(
x
)
in
ℝ
3
.
For each [math] we find initial data (not necessarily radially symmetric) with [math] such that as [math]
∥
u
(
⋅
,
t
)
∥
∞
∼
t
γ
−
1
2
if
1
<
γ
<
2
,
∥
u
(
⋅
,
t
)
∥
∞
∼
t
if
γ
>
2
,
∥
u
(
⋅
,
t
)
∥
∞
∼
t
(
ln
t
)
−
1
if
γ
=
2
.
Furthermore we show that this infinitetime blowup is codimensional1 stable. The existence of such solutions was conjectured by Fila and King ( Netw. Heterog. Media 7 :4 (2012), 661–671).
</p>projecteuclid.org/euclid.apde/1579143672_20200115220114Wed, 15 Jan 2020 22:01 ESTA wellposedness result for viscous compressible fluids with only bounded densityhttps://projecteuclid.org/euclid.apde/1579143673<strong>Raphaël Danchin</strong>, <strong>Francesco Fanelli</strong>, <strong>Marius Paicu</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 13, Number 1, 275316.</p><p><strong>Abstract:</strong><br/>
We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier–Stokes equations. Assuming that the initial velocity has slightly subcritical regularity and that the initial density is a small perturbation (in the [math] norm) of a positive constant, we prove the existence of localintime solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector fields), we get uniqueness. This latter result supplements the work by D. Hoff ( Comm. Pure Appl. Math. 55 :11 (2002), 1365–1407) with a uniqueness statement, and is valid in any dimension [math] and for general pressure laws.
</p>projecteuclid.org/euclid.apde/1579143673_20200115220114Wed, 15 Jan 2020 22:01 EST