Assume that $M$ is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian ${\mathrm{\Delta }}_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from $M$ when the above data are given with a small error. We give an explicit $\log$-$\log$-type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gelfand inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric boundary control method.

This is the first part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions.

Consider the ${\mathbb{𝕊}}^2$-valued equivariant energy critical wave maps equation on ${ℝ}^{1+2}$, with equivariance class $k\ge 4$. It is known that every topologically trivial wave map with energy less than twice that of the unique $k$-equivariant harmonic map $Q_k$ scatters in both time directions. We study maps with precisely the threshold energy $\mathcal{ℰ}=2\mathcal{ℰ}(Q_k)$.

In this paper, we give a refined construction of a wave map with threshold energy that converges to a superposition of two harmonic maps (bubbles), asymptotically decoupling in scale. We show that this two-bubble solution possesses $H^2$ regularity. We give a precise dynamical description of the modulation parameters as well as an expansion of the map into profiles. In the next paper in the series, we show that this solution is unique (up to the natural invariances of the equation) relying crucially on the detailed properties of the solution constructed here. Combined with our earlier work (in 2018), we can now give an exact description of every threshold wave map.

We show a couple of typicality results for weak solutions $v\in C^{𝜃}$ of the Euler equations, in the case . It is known that convex integration schemes produce wild weak solutions that exhibit anomalous dissipation of the kinetic energy $e_v$. We show that those solutions are typical in the Baire category sense. From work of Isett (2013, arXiv:1307.0565), it is know that the kinetic energy $e_v$ of a $𝜃$-Hölder continuous weak solution $v$ of the Euler equations satisfies $e_v\in C^{2𝜃∕(1-𝜃)}$. As a first result we prove that solutions with that behavior are a residual set in suitable complete metric space $X_{𝜃}$ that is contained in the space of all $C^{𝜃}$ weak solutions, whose choice is discussed at the end of the paper. More precisely we show that the set of solutions $v\in X_{𝜃}$, with $e_v\in C^{2𝜃∕(1-𝜃)}$ but for any open $I\subset [0,T]$, are a residual set in $X_{𝜃}$. This, in particular, partially solves Conjecture 1 of Isett and Oh (Arch. Ration. Mech. Anal. 221:2 (2016), 725–804). We also show that smooth solutions form a nowhere dense set in the space of all the $C^{𝜃}$ weak solutions. The technique is the same and what really distinguishes the two cases is that in the latter there is no need to introduce a different complete metric space with respect to the natural one.

We extend the results of Córdoba, Faraco and Gancedo (Arch. Ration. Mech. Anal. 200:3 (2011), 725–746) and Székelyhidi (Ann. Sci. Éc. Norm. Supér. $(4)$ 45:3 (2012), 491–509) on the 2-dimensional incompressible porous media system with constant viscosity (Atwood number $A_{\mu }=0$) to the case of viscosity jump (). We prove an h-principle whereby (infinitely many) weak solutions in $C_t{L}_{w^{\ast }}^{\infty }$ are recovered via convex integration whenever a subsolution is provided. As a first example, nontrivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Székelyhidi (2012) for $A_{\mu }=0$, between the subsolution and the Lagrangian relaxed solution of Otto (Comm. Pure Appl. Math. 52:7 (1999), 873–915) holds for too. For different viscosities, we show how a pinch singularity in the relaxation prevents the two fluids from mixing wherever there is neither Rayleigh–Taylor nor vorticity at the interface.

Distinguished algebraic varieties in ${ℂ}^2$ have been the focus of much research in recent years for good reasons. This note gives a different perspective.

At the root of our work is the Berger–Coburn–Lebow theorem characterizing a commuting tuple of isometries.

We investigate gaps of $n$-term arithmetic progressions $x$, $x+y$, …, $x+(n-1)y$ inside a positive-measure subset $A$ of the unit cube ${[0,1]}^d$. If lengths of their gaps $y$ are evaluated in the ${\ell }^p$-norm for any $p$ other than $1$, $2$, …, $n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\parallel y{\parallel }_{{\ell }^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemerédi’s theorem on the integers, which are used as a black box. A key ingredient of the proof is power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$-dimensional cubes with vertices lying in a positive-measure subset of ${({[0,1]}^2)}^n$.

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10 and in some unbounded domains of dimension at most 11.