Related papers: Regularity for one-phase Bernoulli problems with d…
We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase…
We consider the Bernoulli one-phase free boundary problem in a domain $\Omega$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial \Omega$. We achieve this by relating the behavior of…
In this paper, we consider a vector-valued one-phase Bernoulli-type free boundary problem on a metric measure space $(X,d,\mu)$ with Riemannian curvature-dimension condition $RCD(K,N)$. We first prove the existence and the local Lipschitz…
In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a "regular" part and a "singular" part, as Alt and Caffarelli have shown in their pioneer work (J. Reine Angew. Math., 325, 105-144, 1981) that…
We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,\alpha}$…
In this paper we study the regularity of the free boundary for a vector-valued Bernoulli problem, with no sign assumptions on the boundary data. More precisely, given an open, smooth set of finite measure $D\subset \mathbb{R}^d$,…
We study the regularity of minimizers of a multiphase vectorial Bernoulli free boundary problem. This problem consists in a minimization problem for the Bernoulli functional over families of Sobolev functions with disjoint supports and non…
In this paper we prove a $C^{1,\alpha}$ regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As…
In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$ J(u):=\int_{\Omega}|\nabla u|^p +\lambda_+^p\,\chi_{\{u>0\}} +\lambda_-^p\,\chi_{\{u\le 0\}}, \quad 1<p<\infty. $$…
In this article, we show that for one-phase free boundary problems in noncollapsed limits of $n$-dimensional manifolds with two-sided Ricci curvature bounds, the Hausdorff dimension of the singular set of the free boundary can be bounded by…
We prove new boundary regularity results for minimizers to the one-phase Alt-Caffarelli functional (also known as Bernoulli free boundary problem) in the case of continuous and H\"older-continuous boundary data. As an application, we use…
For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,\alpha}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free…
In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers \cite{C77,C98}, regular points consist of smooth hypersurfaces, while singular points…
We investigate the regularity of the free boundary for the Signorini problem in $\mathbb{R}^{n+1}$. It is known that regular points are $(n-1)$-dimensional and $C^\infty$. However, even for $C^\infty$ obstacles $\varphi$, the set of…
We consider a one-phase Bernoulli free boundary problem in a container $D$ - a smooth open subset of $\mathbb{R}^d$ - under the condition that on the fixed boundary $\partial D$ the normal derivative of the solutions is prescribed. We study…
We consider minimizers of the one-phase Bernoulli free boundary problem in domains with analytic fixed boundary. In any dimension $d$, we prove that the branching set at the boundary has Hausdorff dimension at most $d-2$. As a consequence,…
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples…
We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta \varphi\leq 0$ near the contact region. Our main result establishes that…
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…
Consider the parabolic free boundary problem $$ \Delta u - \partial_t u = 0 \textrm{in} \{u>0\}, |\nabla u|=1 \textrm{on} \partial\{u>0\} . $$ For a realistic class of solutions, containing for example {\em all} limits of the singular…