Related papers: Obstacle problems and free boundaries: an overview
We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was…
While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains unresolved. For two…
We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and…
In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory…
This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp $C^{1,1/3}$--regularity at the free…
We study the regularity of free boundaries in the multiple elastic membrane problem in the plane. We prove the uniqueness of blow-ups, and that the free boundaries are $C^{1,\log}$-curves near a regular intersection point.
We consider the free boundary problem arising from an energy functional which is the sum of a Dirichlet energy and a nonlinear function of either the classical or the fractional perimeter. The main difference with the existing literature is…
We examine boundary regularity for a fully nonlinear free transmission problem. We argue using approximation methods, comparing the operators driving the problem with a limiting profile. Working natural conditions on the data of the…
We study the existence and multiplicity of solutions and the global solution curve of the following free boundary value problem, arising in plasma physics, see R. Temam [18], and H. Berestycki and H. Brezis [3]: find a function $u(x)$ and a…
Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of…
We study a family of variants of Erd\H os' unit distance problem, concerning distances and dot products between pairs of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, we look for…
We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis Caffarelli on free boundaries and by…
The problem of finding boundary conditions for fastening of a ring membrane, which are inaccessible for direct observation from the natural frequencies of its flexural oscillations, is considered. Two theorems on the uniqueness of this…
In this paper, we consider a free boundary problem of a semilinear nonhomogeneous elliptic equation with Bernoulli's type free boundary. The existence and regularity of the solution to the free boundary problem are established by use of the…
We consider an oscillatory obstacle problem where the coincidence set and free boundary are also highly oscillatory. We establish a rate of convergence for a regularized notion of free boundary to the free boundary of a corresponding…
A free boundary value problem related to the genesis of multispecies granular biofilms is presented. The granular biofilm is modelled as a spherical free boundary domain with radial symmetry. The proposed model is conceived in the framework…
In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we…
We give a definition for Obstacle Problems with measure data and general obstacles. For such problems we prove existence and uniqueness of solutions and consistency with the classical theory of Variational Inequalities. Continuous…
We present a series of recent results on some new classes of free boundary problems. Differently from the classical literature, the problems considered have either a "nonlocal" feature (e.g., the interaction or/and the interfacial energy…
Modelling diffusion processes in heterogeneous media requires addressing inherent discontinuities across interfaces, where specific conditions are to be met. These challenges fall under the purview of Mathematical Analysis as…