Related papers: An epiperimetric inequality for the lower dimensio…

An inverse problem of finding an obstacle and the boundary condition on its surface from the fixed-energy scattering data is studied. A new method is developed for a proof of the uniqueness results. The method does not use the discreteness…

We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in $\Omega\times (0,T)$ -…

We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb{R}^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right hand side and the…

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…

We prove $C^{1, \alpha}$ regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the…

Inspired by a planar partitioning problem involving multiple improper chambers, this article investigates using classical techniques what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint…

The wave equation is time-reversal invariant. The enclosure method using a Neumann data generated by this invariance is introduced. The method yields the minimum ball that is centered at a given arbitrary point and encloses an unknown…

A theorem of W. Derrick ensures that the volume of any Riemannian cube $([0,1]^n,g)$ is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick's…

We provide a new and simple proof based on Harnack's inequality to the Lipschitz continuity of the solutions of a class of free boundary problems.

This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous…

We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is $C^\infty$ in space…

Now a final and maybe simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a {\it spacial domain with an outer boundary} over a finite time interval is fixed. The…

The aim of this paper is twofold: to prove, for L^1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability…

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we…

We extend the generic Bell inequalities suggested by Son, Lee, and Kim [Phys. Rev. Lett. 96, 060406 (2006)] to incorporate multiple observables for tripartite systems and introduce a geometric methodology for calculating classical upper…

In 2012, F.Leonetti and F.Siepe [1] considered solutions to boundary value problems of some anisotropic elliptic equations of the type $$ \left\{ \begin{array}{llll} \sum\limits _{i=1}\limits^{n} D_i (a_i(x,Du(x)))=0, &x\in \Omega,\\…

Given a simple closed plane curve $\Gamma$ of length $L$ enclosing a compact convex set $K$ of area $F$, Hurwitz found an upper bound for the isoperimetric deficit, namely $L^2-4\pi F\leq \pi |F_{e}|$, where $F_{e}$ is the algebraic area…

We prove the -- to the best knowledge of the authors -- first result on the fine asymptotic behavior of the regular part of the free boundary of the obstacle problem close to singularities. The result is motivated by our recent partial…

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by…

The classical Sobolev and Escobar inequalities are embedded into the same one-parameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known…