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Kari Astala

Publications and source records attributed to Kari Astala.

At least 19 recordsLinked to original sources
arXiv2024

Lower semicontinuity, Stoilow factorization and principal maps

We consider a strengthening of the usual quasiconvexity condition of Morrey in two dimensions, which allows us to prove lower semicontinuity for functionals which are unbounded as the determinant vanishes. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and \v{S}ver\'ak, plays a prominent role in our approach.

math.AP↗
arXiv2023

The local Burkholder functional, quasiconvexity and Geometric Function Theory

We show that the local Burkholder functional $\mathcal B_K$ is quasiconvex. In the limit of $p$ going to 2 we find a class of non-polyconvex functionals which are quasiconvex on the set of matrices with positive determinant. In order to prove the validity of lower semicontinuity arguments in this setting, we show that the Burkholder functionals satisfy a sharp extension of the classical function theoretic area formula. As a corollary, in addition to functionals in geometric function theory, one finds new classes of non-polyconvex functionals, degenerating as the determinant vanishes, for which there is existence of minimizers.

math.AP↗
arXiv2020

Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d., and $ϕ\in C^\infty_0$. We establish the almost sure and local uniform convergence as $j\to\infty$ of the maps $F_j$ to a deterministic quasiconformal limit $F_\infty$. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let $T_1,\ldots , T_{m}$ be translation and dilation invariant singular integrals on ${\bf R}^d, $ and consider a $d$-dimensional version of $μ_j$, e.g., as defined above or within a more general setting. We then prove that there is a deterministic function $f$ such that almost surely as $j\to\infty$, $$ μ_j T_{m}μ_j\ldots T_1μ_j\to f \quad \textrm{weakly in } L^p,\quad 1 < p < \infty\ . $$

math.CV↗
arXiv2020

Dimer Models and Conformal Structures

In this work we study the variational problem associated to dimer models, a class of models from integrable probability and statistical mechanics in dimension two which have been the focus of intense research efforts over the last decades. These models give rise to an infinite family of non-differentiable functionals on Lipschitz functions with gradient constraint, determined by solutions of the Dirichlet problem on compact convex polygons for a class of Monge-Amp\`ere equations. We settle a number or outstanding open questions for this infinite class functionals. In particular we prove a complete classification of the regularity of minimizers, also known as height functions, for all dimer models for a natural class of polygonal (simply or multiply connected) domains much studied in numerical simulations and elsewhere. Our classification in particular implies that the Pokrovsky-Talapov law holds for all dimer models at a generic point on the frozen boundary and in addition shows a very strong local rigidity of dimer models which can be interpreted as a geometric universality result. Furthermore, we give a complete classification of the regularity of the associated free boundary, also known in the literature as frozen boundary or arctic curves and prove that they are all algebraic curves. The lack of differentiability of the functionals is intimately connected to the boundary behaviour of the solutions to the Monge-Amp\`ere equations and we prove a complete classification for these, of independent interest.

math.AP↗
arXiv2019

Improved Hölder regularity for strongly elliptic PDEs

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of Hölder regularity, higher than what is given by the classical exponent $1/K$.

math.CV↗
arXiv2019

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We study quasiconformal mappings in planar domains $\Omega$ and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary $\partial \Omega$ and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below $1$, the same conditions give global regularity in $\Omega$ for the principal solutions with Beltrami coefficient supported in $\Omega$.

math.AP↗
arXiv2016

Manifolds of quasiconformal mappings and the nonlinear Beltrami equation

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold $\mathcal{F}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.

math.CV↗
arXiv2015

Asymptotic variance of the Beurling transform

We study the interplay between infinitesimal deformations of conformal mappings, quasiconformal distortion estimates and integral means spectra. By the work of McMullen, the second derivative of the Hausdorff dimension of the boundary of the image domain is naturally related to asymptotic variance of the Beurling transform. In view of a theorem of Smirnov which states that the dimension of a $k$-quasicircle is at most $1+k^2$, it is natural to expect that the maximum asymptotic variance $Σ^2 = 1$. In this paper, we prove $0.87913 \le Σ^2 \le 1$. For the lower bound, we give examples of polynomial Julia sets which are $k$-quasicircles with dimensions $1+ 0.87913 \, k^2$ for $k$ small, thereby showing that $Σ^2 \ge 0.87913$. The key ingredient in this construction is a good estimate for the distortion $k$, which is better than the one given by a straightforward use of the $λ$-lemma in the appropriate parameter space. Finally, we develop a new fractal approximation scheme for evaluating $Σ^2$ in terms of nearly circular polynomial Julia sets.

math.CV↗
arXiv2015

Rough Potential Recovery in the Plane

We reconstruct compactly supported potentials with only half a derivative in $L^2$ from the scattering amplitude at a fixed energy. For this we draw a connection between the recently introduced method of Bukhgeim, which uniquely determined the potential from the Dirichlet-to-Neumann map, and a question of Carleson regarding the convergence to initial data of solutions to time-dependent Schrödinger equations. We also provide examples of compactly supported potentials, with $s$ derivatives in $L^2$ for any $s<1/2$, which cannot be recovered by these means. Thus the recovery method has a different threshold in terms of regularity than the corresponding uniqueness result.

math.CA↗
arXiv2015

On Plancherel's identity for a two-dimensional scattering transform

We consider the $\overline{\partial}$-Dirac system that Ablowitz and Fokas used to transform the defocussing Davey-Stewartson system to a linear evolution equation. The nonlinear Plancherel identity for the associated scattering transform was established by Beals and Coifman for Schwartz functions. Sung extended the validity of the identity to functions belonging to $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$ and Brown to $L^2(\mathbb{R}^2)$-functions with sufficiently small norm. More recently, Perry extended to the weighted Sobolev space $H^{1,1}(\mathbb{R}^2)$ and here we extend to $H^{s,s}(\mathbb{R}^2)$ with $s\in(0,1)$.

math.CA↗
arXiv2014

Pavlovic's theorem in space

We study higher dimensional counterparts to the well-known theorem of Pavlovic \cite{pa3}, that every harmonic quasiconformal mapping of the disk is bi-Lipschitz.

math.CV↗
arXiv2014

Nonlinear Fourier analysis for discontinuous conductivities: computational results

Two reconstruction methods of Electrical Impedance Tomography (EIT) are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics (CGO) solutions to D-bar equations involving the global uniqueness proofs for Calderón problem exposed in [Nachman; Annals of Mathematics 143, 1996] and [Astala and Päivärinta; Annals of Mathematics 163, 2006]: the Astala-Päivärinta theory-based "low-pass transport matrix method" implemented in [Astala et al.; Inverse Problems and Imaging 5, 2011] and the "shortcut method" which considers ingredients of both theories. The latter method is formally similar to the Nachman theory-based regularized EIT reconstruction algorithm studied in [Knudsen, Lassas, Mueller and Siltanen; Inverse Problems and Imaging 3, 2009] and several references from there. New numerical results are presented using parallel computation with size parameters larger than ever, leading mainly to two conclusions as follows. First, both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases, and there seems to be a Gibbs-like phenomenon producing ringing artifacts. Second, the transport matrix method loses accuracy away from a (freely chosen) pivot point located outside of the object to be studied, whereas the shortcut method produces reconstructions with more uniform quality.

math.AP↗
arXiv2014

A hunt for sharp $L ^p$-estimates and rank-one convex variational integrals

Learning how to figure out sharp $L^p$-estimates of nonlinear differential expressions, to prove and use them, is a fundamental part of the development of PDEs and Geometric Function Theory (GFT). Our survey presents, among what is known to date, some notable recent efforts and novelties made in this direction. We focus attention here on the historic Morrey's Conjecture and Burkholder's martingale inequalities for stochastic integrals. Some of these topics have already been discussed by the present authors [5] and by Rodrigo Bañuelos [10]. Nevertheless, there is always something new to add.

math.CV↗
arXiv2013

Bilipschitz and quasiconformal rotation, stretching and multifractal spectra

We establish sharp bounds for simultaneous local rotation and Hölder-distortion of planar quasiconformal maps. In addition, we give sharp estimates for the corresponding joint quasiconformal multifractal spectrum, based on new estimates for Burkholder functionals with complex parameters. As a consequence, we obtain optimal rotation estimates also for bi-Lipschitz maps.

math.CV↗
arXiv2011

The borderlines of the invisibility and visibility for Calderon's inverse problem

We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than $L^\infty$. Also, we give new counterexamples for the uniqueness of the inverse conductivity problem. We say that a conductivity is visible if the inverse problem is solvable so that the inside of the domain can be uniquely determined, up to a change of coordinates, using the boundary measurements. The present counterexamples for the inverse problem have been related to the invisibility cloaking. This means that there are conductivities for which a part of the domain is shielded from detection via boundary measurements. Such conductivities are called invisibility cloaks. In the present paper we identify the borderline of the visible conductivities and the borderline of invisibility cloaking conductivities. Surprisingly, these borderlines are not the same. We show that between the visible and the cloaking conductivities there are the electric holograms, conductivities which create an illusion of a non-existing body. The electric holograms give counterexamples for the uniqueness of the inverse problem which are less degenerate than the previously known ones.

math.AP↗
arXiv2010

Burkholder integrals, Morrey's problem and quasiconformal mappings

Inspired by Morrey's Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals $B_p$, $p \ge 2$, are quasiconcave, when tested on deformations of identity $f\in Id + C^\infty_0(Ω)$ with $B_p(Df(x)) \ge 0$ pointwise, or equivalently, deformations such that $|Df|^2 \leq \frac{p}{p-2} J_f$. In particular, this holds in explicit neighbourhoods of the identity map. Among the many immediate consequences, this gives the strongest possible $L^p$- estimates for the gradient of a principal solution to the Beltrami equation $\f_{\bar{z}} = μ(z) f_z$, for any $p$ in the critical interval $2 \leq p \leq 1+1/\|μ_f\|_\infty$. Examples of local maxima lacking symmetry manifest the intricate nature of the problem.

math.CA↗