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 EDTWellposedness and smoothing effect for generalized nonlinear Schrödinger equationshttps://projecteuclid.org/euclid.apde/1523930419<strong>PierreYves Bienaimé</strong>, <strong>Abdesslam Boulkhemair</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 5, 12411284.</p><p><strong>Abstract:</strong><br/>
We improve the result obtained by one of the authors, Bienaimé (2014), and establish the wellposedness of the Cauchy problem for some nonlinear equations of Schrödinger type in the usual Sobolev space [math] for [math] instead of [math] . We also improve the smoothing effect of the solution and obtain the optimal exponent.
</p>projecteuclid.org/euclid.apde/1523930419_20180416220025Mon, 16 Apr 2018 22:00 EDTThe shape of low energy configurations of a thin elastic sheet with a single disclinationhttps://projecteuclid.org/euclid.apde/1523930420<strong>Heiner Olbermann</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 5, 12851302.</p><p><strong>Abstract:</strong><br/>
We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness [math] as a small parameter. We give an improvement of a recently proved energy scaling law, removing the nexttoleadingorder terms in the lower bound. Then we prove the convergence of (almost)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces [math] for every [math] , as the thickness [math] is sent to 0.
</p>projecteuclid.org/euclid.apde/1523930420_20180416220025Mon, 16 Apr 2018 22:00 EDTThe thinfilm equation close to selfsimilarityhttps://projecteuclid.org/euclid.apde/1523930421<strong>Christian Seis</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 5, 13031342.</p><p><strong>Abstract:</strong><br/>
We study wellposedness and regularity of the multidimensional thinfilm equation with linear mobility in a neighborhood of the selfsimilar Smyth–Hill solutions. To be more specific, we perform a von Mises change of dependent and independent variables that transforms the thinfilm free boundary problem into a parabolic equation on the unit ball. We show that the transformed equation is wellposed and that solutions are smooth and even analytic in time and angular direction. The latter gives the analyticity of level sets of the original equation, and thus, in particular, of the free boundary.
</p>projecteuclid.org/euclid.apde/1523930421_20180416220025Mon, 16 Apr 2018 22:00 EDTPropagation of chaos, Wasserstein gradient flows and toric Kähler–Einstein metricshttps://projecteuclid.org/euclid.apde/1527040814<strong>Robert J. Berman</strong>, <strong>Magnus Önnheim</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 6, 13431380.</p><p><strong>Abstract:</strong><br/>
Motivated by a probabilistic approach to Kähler–Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex Nparticle interaction energy. We show that a deterministic “macroscopic” evolution equation emerges in the large Nlimit of many particles. This is a strengthening of previous results which required a uniform twosided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at “negative temperature”, the corresponding limiting evolution equation yields a driftdiffusion equation, coupled to the Monge–Ampère operator, whose static solutions correspond to toric Kähler–Einstein metrics. This driftdiffusion equation is the gradient flow on the Wasserstein space of probability measures of the Kenergy functional in Kähler geometry and it can be seen as a fully nonlinear version of various extensively studied dissipative evolution equations and conservation laws, including the Keller–Segel equation and Burger’s equation. In a companion paper, applications to singular pair interactions in one dimension are given.
</p>projecteuclid.org/euclid.apde/1527040814_20180522220030Tue, 22 May 2018 22:00 EDTThe inverse problem for the DirichlettoNeumann map on Lorentzian manifoldshttps://projecteuclid.org/euclid.apde/1527040815<strong>Plamen Stefanov</strong>, <strong>Yang Yang</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 6, 13811414.</p><p><strong>Abstract:</strong><br/>
We consider the DirichlettoNeumann map [math] on a cylinderlike Lorentzian manifold related to the wave equation related to the metric [math] , the magnetic field [math] and the potential [math] . We show that we can recover the jet of [math] on the boundary from [math] up to a gauge transformation in a stable way. We also show that [math] recovers the following three invariants in a stable way: the lens relation of [math] , and the light ray transforms of [math] and [math] . Moreover, [math] is an FIO away from the diagonal with a canonical relation given by the lens relation. We present applications for recovery of [math] and [math] in a logarithmically stable way in the Minkowski case, and uniqueness with partial data.
</p>projecteuclid.org/euclid.apde/1527040815_20180522220030Tue, 22 May 2018 22:00 EDTBlowup criteria for the Navier–Stokes equations in nonendpoint critical Besov spaceshttps://projecteuclid.org/euclid.apde/1527040816<strong>Dallas Albritton</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 6, 14151456.</p><p><strong>Abstract:</strong><br/>
We obtain an improved blowup criterion for solutions of the Navier–Stokes equations in critical Besov spaces. If a mild solution [math] has maximal existence time [math] , then the nonendpoint critical Besov norms must become infinite at the blowup time:
lim
t
↑
T
∗
∥
u
(
⋅
,
t
)
∥
Ḃ
p
,
q
−
1
+
3
∕
p
(
ℝ
3
)
=
∞
,
3
<
p
,
q
<
∞
.
In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.
</p>projecteuclid.org/euclid.apde/1527040816_20180522220030Tue, 22 May 2018 22:00 EDTDolgopyat's method and the fractal uncertainty principlehttps://projecteuclid.org/euclid.apde/1527040817<strong>Semyon Dyatlov</strong>, <strong>Long Jin</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 6, 14571485.</p><p><strong>Abstract:</strong><br/>
We show a fractal uncertainty principle with exponent [math] , [math] , for Ahlfors–David regular subsets of [math] of dimension [math] . This is an improvement over the volume bound [math] , and [math] is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Here the group invariance of the set is replaced by its fractal structure. As an application, we quantify the result of Naud on spectral gaps for convex cocompact hyperbolic surfaces and obtain a new spectral gap for open quantum baker maps.
</p>projecteuclid.org/euclid.apde/1527040817_20180522220030Tue, 22 May 2018 22:00 EDTDini and Schauder estimates for nonlocal fully nonlinear parabolic equations with driftshttps://projecteuclid.org/euclid.apde/1527040818<strong>Hongjie Dong</strong>, <strong>Tianling Jin</strong>, <strong>Hong Zhang</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 6, 14871534.</p><p><strong>Abstract:</strong><br/>
We obtain Dini and Schaudertype estimates for concave fully nonlinear nonlocal parabolic equations of order [math] with rough and nonsymmetric kernels and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dinitype estimates in the spacial variable. This is a continuation of work by the authors Dong and Zhang.
</p>projecteuclid.org/euclid.apde/1527040818_20180522220030Tue, 22 May 2018 22:00 EDTSquare function estimates for the Bochner–Riesz meanshttps://projecteuclid.org/euclid.apde/1527040819<strong>Sanghyuk Lee</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 11, Number 6, 15351586.</p><p><strong>Abstract:</strong><br/>
We consider the squarefunction (known as Stein’s square function) estimate associated with the Bochner–Riesz means. The previously known range of the sharp estimate is improved. Our results are based on vectorvalued extensions of Bennett, Carbery and Tao’s multilinear (adjoint) restriction estimate and an adaptation of an induction argument due to Bourgain and Guth. Unlike the previous work by Bourgain and Guth on [math] boundedness of the Bochner–Riesz means in which oscillatory operators associated to the kernel were studied, we take more direct approach by working on the Fourier transform side. This enables us to obtain the correct order of smoothing, which is essential for obtaining the sharp estimates for the square functions.
</p>projecteuclid.org/euclid.apde/1527040819_20180522220030Tue, 22 May 2018 22:00 EDTBoundary behavior of solutions to the parabolic $p$Laplace equationhttps://projecteuclid.org/euclid.apde/1534384913<strong>Benny Avelin</strong>, <strong>Tuomo Kuusi</strong>, <strong>Kaj Nyström</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 1, 142.</p><p><strong>Abstract:</strong><br/>
We establish boundary estimates for nonnegative solutions to the [math] parabolic equation in the degenerate range [math] . Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA domains together with sharp boundary decay estimates. If the underlying domain is [math] regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waitingtime phenomenon present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out wellknown examples of explicit solutions violating the boundary Harnack principle.
</p>projecteuclid.org/euclid.apde/1534384913_20180815220215Wed, 15 Aug 2018 22:02 EDTOn asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbationhttps://projecteuclid.org/euclid.apde/1534384914<strong>Yang Lan</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 1, 43112.</p><p><strong>Abstract:</strong><br/>
We consider the [math] critical gKdV equation with a saturated perturbation: [math] , where [math] and [math] . For any initial data [math] , the corresponding solution is always global and bounded in [math] . This equation has a family of solutions, and our goal is to classify the dynamics near solitons. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave whose [math] norm is of size [math] as [math] ; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at [math] ; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves.
This extends the classification of the rigidity dynamics near the ground state for the unperturbed [math] critical gKdV (corresponding to [math] ) by Martel, Merle and Raphaël. However, the blowdown behavior (ii) is completely new, and the dynamics of the saturated equation cannot be viewed as a perturbation of the [math] critical dynamics of the unperturbed equation. This is the first example of classification of the dynamics near the ground state for a saturated equation in this context. The cases of [math] critical NLS and [math] supercritical gKdV, where similar classification results are expected, are completely open.
</p>projecteuclid.org/euclid.apde/1534384914_20180815220215Wed, 15 Aug 2018 22:02 EDTOn the stability of type II blowup for the 1corotational energysupercritical harmonic heat flowhttps://projecteuclid.org/euclid.apde/1534384915<strong>Tejeddine Ghoul</strong>, <strong>Slim Ibrahim</strong>, <strong>Van Tien Nguyen</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 1, 113187.</p><p><strong>Abstract:</strong><br/>
We consider the energysupercritical harmonic heat flow from [math] into the [math] sphere [math] with [math] . Under an additional assumption of 1corotational symmetry, the problem reduces to the onedimensional semilinear heat equation
∂
t
u
=
∂
r
2
u
+
(
d
−
1
)
r
∂
r
u
−
(
d
−
1
)
2
r
2
sin
(
2
u
)
.
We construct for this equation a family of [math] solutions which blow up in finite time via concentration of the universal profile
u
(
r
,
t
)
∼
Q
(
r
λ
(
t
)
)
,
where [math] is the stationary solution of the equation and the speed is given by the quantized rates
λ
(
t
)
∼
c
u
(
T
−
t
)
ℓ
γ
,
ℓ
∈
ℕ
∗
,
2
ℓ
>
γ
=
γ
(
d
)
∈
(
1
,
2
]
.
The construction relies on two arguments: the reduction of the problem to a finitedimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski ( Camb. J. Math. 3 :4 (2015), 439–617) for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer ( Anal. PDE 7 :8 (2014), 1713–1805) for the energy critical harmonic heat flow. Then we proceed by contradiction to solve the finitedimensional problem and conclude using the Brouwer fixedpoint theorem. Moreover, our constructed solutions are in fact [math] codimension stable under perturbations of the initial data. As a consequence, the case [math] corresponds to a stable type II blowup regime.
</p>projecteuclid.org/euclid.apde/1534384915_20180815220215Wed, 15 Aug 2018 22:02 EDTOn propagation of higher space regularity for nonlinear Vlasov equationshttps://projecteuclid.org/euclid.apde/1534384916<strong>Daniel HanKwan</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 1, 189244.</p><p><strong>Abstract:</strong><br/>
This work is concerned with the broad question of propagation of regularity for smooth solutions to nonlinear Vlasov equations. For a class of equations (that includes Vlasov–Poisson and relativistic Vlasov–Maxwell systems), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced by D. HanKwan and F. Rousset ( Ann. Sci. É cole Norm. Sup. 49 :6 (2016) 1445–1495) in the context of the quasineutral limit of the Vlasov–Poisson system.
</p>projecteuclid.org/euclid.apde/1534384916_20180815220215Wed, 15 Aug 2018 22:02 EDTOn a boundary value problem for conically deformed thin elastic sheetshttps://projecteuclid.org/euclid.apde/1534384918<strong>Heiner Olbermann</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 1, 245258.</p><p><strong>Abstract:</strong><br/>
We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. These are the boundary conditions of a socalled “dcone”. We define the free elastic energy as a variation of the von Kármán energy, which penalizes bending energy in [math] with [math] (instead of, as usual, [math] ). We prove ansatzfree upper and lower bounds for the elastic energy that scale like [math] , where [math] is the thickness of the sheet.
</p>projecteuclid.org/euclid.apde/1534384918_20180815220215Wed, 15 Aug 2018 22:02 EDTA unified flow approach to smooth, even $L_p$Minkowski problemshttps://projecteuclid.org/euclid.apde/1539050438<strong>Paul Bryan</strong>, <strong>Mohammad N. Ivaki</strong>, <strong>Julian Scheuer</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 259280.</p><p><strong>Abstract:</strong><br/>
We study longtime existence and asymptotic behavior for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even [math] Minkowski problems in [math] for [math] .
</p>projecteuclid.org/euclid.apde/1539050438_20181008220100Mon, 08 Oct 2018 22:01 EDTThe Muskat problem in two dimensions: equivalence of formulations, wellposedness, and regularity resultshttps://projecteuclid.org/euclid.apde/1539050439<strong>BogdanVasile Matioc</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 281332.</p><p><strong>Abstract:</strong><br/>
We consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal twodimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functionalanalytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local wellposedness of the problem for arbitrary large initial data and show that the solutions become instantly realanalytic in time and space. Our method allows us to choose the initial data in the class [math] , [math] , when neglecting surface tension, respectively in [math] , [math] , when surfacetension effects are included. Besides, we provide new criteria for the global existence of solutions.
</p>projecteuclid.org/euclid.apde/1539050439_20181008220100Mon, 08 Oct 2018 22:01 EDTMaximal gain of regularity in velocity averaging lemmashttps://projecteuclid.org/euclid.apde/1539050441<strong>Diogo Arsénio</strong>, <strong>Nader Masmoudi</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 333388.</p><p><strong>Abstract:</strong><br/>
We investigate new settings of velocity averaging lemmas in kinetic theory where a maximal gain of half a derivative is obtained. Specifically, we show that if the densities [math] and [math] in the transport equation [math] belong to [math] , where [math] and [math] is the dimension, then the velocity averages belong to [math] .
We further explore the setting where the densities belong to [math] and show, by completing the work initiated by PierreEmmanuel Jabin and Luis Vega on the subject, that velocity averages almost belong to [math] in this case, in any dimension [math] , which strongly indicates that velocity averages should almost belong to [math] whenever the densities belong to [math] .
These results and their proofs bear a strong resemblance to the famous and notoriously difficult problems of boundedness of Bochner–Riesz multipliers and Fourier restriction operators, and to smoothing conjectures for Schrödinger and wave equations, which suggests interesting links between kinetic theory, dispersive equations and harmonic analysis.
</p>projecteuclid.org/euclid.apde/1539050441_20181008220100Mon, 08 Oct 2018 22:01 EDTOn the existence and stability of blowup for wave maps into a negatively curved targethttps://projecteuclid.org/euclid.apde/1539050442<strong>Roland Donninger</strong>, <strong>Irfan Glogić</strong>. <p><strong>Source: </strong>Analysis & PDE, Volume 12, Number 2, 389416.</p><p><strong>Abstract:</strong><br/>
We consider wave maps on [math] dimensional Minkowski space. For each dimension [math] we construct a negatively curved, [math] dimensional target manifold that allows for the existence of a selfsimilar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem.
</p>projecteuclid.org/euclid.apde/1539050442_20181008220100Mon, 08 Oct 2018 22:01 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 EDT