Showing papers by "Angkana Rüland published in 2017"

Exponential instability in the fractional Calder\'on problem

Abstract: In this note we prove the exponential instability of the fractional Calder\'on problem and thus prove the optimality of the logarithmic stability estimate from \cite{RS17}. In order to infer this result, we follow the strategy introduced by Mandache in \cite{M01} for the standard Calder\'on problem. Here we exploit a close relation between the fractional Calder\'on problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in \cite{RS17}. Finally, in one dimension, we show a close relation between the fractional Calder\'on problem and the truncated Hilbert transform.

Quantitative Approximation Properties for the Fractional Heat Equation

Abstract: In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal} pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain \emph{qualitative} approximation results from \cite{DSV16}. Using propagation of smallness arguments, we then provide bounds on the \emph{cost} of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss generalizations of these results to a larger class of operators involving both local and nonlocal contributions.

Surface energies emerging in a microscopic, two-dimensional two-well problem

Abstract: In this article we are interested in the microscopic modeling of a two-dimensional two-well problem which arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyze the Hamiltonian which was introduced in \cite{KLR14}. It turns out that this class of Hamiltonians allows for a direct control of the discrete second order gradients and for a one-sided comparison with a two-dimensonal spin system. Using this and relying on the ideas of Conti and Schweizer \cite{CS06}, \cite{CS06a}, \cite{CS06c}, which were developed for a continuous analogue of the model under consideration, we derive a (first order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.

Optimal regularity for the thin obstacle problem with $$C^{0,\alpha }$$ C 0 , α coefficients

Abstract: In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson (Invent Math 204(1):1–82, 2016. doi: 10.1007/s00222-015-0608-6 ) and the epiperimetric inequality from Focardi and Spadaro (Adv Differ Equ 21(1–2):153–200, 2016), Garofalo, Petrosyan and Smit Vega Garcia (J Math Pures Appl 105(6):745–787, 2016. doi: 10.1016/j.matpur.2015.11.013 ), we prove the optimal $$C^{1,\min \{\alpha ,1/2\}}$$ regularity of solutions in the presence of $$C^{0,\alpha }$$ coefficients $$a^{ij}$$ and $$C^{1,\alpha }$$ obstacles $$\phi $$ . Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $$C^{1,\gamma }$$ manifold for some $$\gamma \in (0,1)$$ .

Quantitative Invertibility and Approximation for the Truncated Hilbert and Riesz Transforms

Abstract: In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. In this context we then invoke quantitative propagation of smallness estimates in combination with qualitative Runge approximation results. These results can be viewed as quantifications of the approximation properties which have recently gained prominence in the context of nonlocal operators, c.f. [DSV14], [DSV16].

The Variable Coefficient Thin Obstacle Problem: Higher Regularity

Abstract: In this article, we continue our investigation of the variable coefficients thin obstacle problem which was initiated in [20], [21]. Using a partial Hodograph-Legendre transform and the implicit function theorem, we prove the higher order Holder regularity for the regular free boundary, if the associated coefficients are of the corresponding regularity. For the zero obstacle, this yields an improvement of a full derivative for the free boundary regularity compared to the regularity of the coefficients. In the presence of inhomogeneities, we gain three halves of a derivative for the free boundary regularity with respect to the regularity of the inhomogeneity. Further, we show analyticity of the regular free boundary for analytic coefficients. We also discuss the set-up of $W^{1,p}$ coefficients with $p>n+1$ and $L^p$ inhomogeneities. Key ingredients in our analysis are the introduction of generalized Holder spaces, which allow to interpret the transformed fully nonlinear, degenerate (sub)elliptic equation as a perturbation of the Baouendi-Grushin operator, various uses of intrinsic geometries associated with appropriate operators, the application of the implicit function theorem to deduce (higher) regularity.

A Compactness and Structure Result for a Discrete Multi-Well Problem with $SO(n)$ Symmetry in Arbitrary Dimension

Abstract: In this note we combine the "spin-argument" from [KLR15] and the $n$-dimensional incompatible, one-well rigidity result from [LL16], in order to infer a new proof for the compactness of discrete multi-well energies associated with the modelling of surface energies in certain phase transitions. Mathematically, a main novelty here is the reduction of the problem to an incompatible one-well problem. The presented argument is very robust and applies to a number of different physically interesting models, including for instance phase transformations in shape-memory materials but also anti-ferromagnetic transformations or related transitions with an "internal" microstructure on smaller scales.

Quantitative Runge Approximation and Inverse Problems

Abstract: In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application we provide a new proof of the result from \cite{F07}, \cite{AK12} on stability for the Calder\'on problem with local data.

Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in $\text{int}(K^{lc})$

Abstract: In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev regularity for Dirichlet problems with affine data in $\text{int}(K^{lc})$. This allows us to simultaneously deal with linear and nonlinear differential inclusion problems. We show that the derived higher integrability and differentiability exponent has a lower bound, which is independent of the position of the Dirichlet boundary data in $\text{int}(K^{lc})$. As applications we discuss the regularity of weak isometric immersions in two and three dimensions as well as the differential inclusion problem for the geometrically linear hexagonal-to-rhombic and the cubic-to-orthorhombic phase transformations occurring in shape memory alloys.