Related papers: Free boundary problems involving singular weights
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions…
For $\Omega\subset \mathbb{R}^2$ a smooth and bounded domain, we derive a sharp universal energy estimate for non-negative solutions of free boundary problems on $\Omega$ arising in plasma physics. As a consequence, we are able to deduce…
Minimizers in the least gradient problem with discontinuous boundary data need not be unique. However, all of them have a similar structure of level sets. Here, we give a full characterization of the set of minimizers in terms of any one of…
We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.
We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by…
We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are H\"older continuous and the free boundary has…
Let $\O$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincar\'e inequalities with weight function singular at the boundary of $\O$. In particular we give sufficient conditions so that the best…
We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of…
In this article we study a class of generalised linear systems of difference equations with given boundary conditions and assume that the boundary value problem is non-consistent, i.e. it has infinite many or no solutions. We take into…
In this manuscript we deal with regularity issues and the asymptotic behaviour (as $p \to \infty$) of solutions for elliptic free boundary problems of $p-$Laplacian type ($2 \leq p< \infty$): \begin{equation*} -\Delta_p u(x) +…
We explore singular second-order boundary value problems with mixed boundary conditions on a general time scale. Using the lower and upper solutions method combined with the Brouwer fixed point theorem we demonstrate the existence of a…
The main result of this paper concerns the behavior of a free boundary arising from a minimization problem, close to the fixed boundary in two dimensions.
In this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning $C^1$-extremals at which the second variation is strictly positive. The results are presented in the quasiconvex setting, in accordance…
This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a $C^1$ cost implies a locally Lipschitz free boundary. As an…
A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form $$\sup_{\int_D\theta\,dx=m}\ \inf_{u\in H^1_0(D)}\int_D\Big(\frac{1+\theta}{2}|\nabla…
In this article, we study a class of fully nonlinear double-divergence systems with free boundaries associated with a minimization problem. The variational structure of Hessian-dependent functional plays a fundamental role in proving the…
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$\Delta u = -\chi_{\{u>0\}} .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs…
The study of the regularity of the minimizer of the weighted anisotropic total variation with a general fidelity term is at the heart of this paper. We generalized some recent results on the inclusion of the discontinuities of the minimizer…
In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set…
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…