Related papers: Free Boundary Regularity for Almost-Minimizers

We study the minimum problem for the functional $\int_{\Omega}\bigl( \vert \nabla \mathbf{u} \vert^{2} + Q^{2}\chi_{\{\vert \mathbf{u}\vert>0\}} \bigr)dx$ with the constraint $u_i\geq 0$ for $i=1,\cdots,m$ where…

In this paper, we consider the following semilinear vector-valued minimization problem $$\min\left\{\int_{D}({|\nabla\mathbf{u}|}^2 + F(|\mathbf{u}|))dx: \ \ \mathbf{u}\in W^{1,2}(D; \mathbb{R}^m) \ \text{and} \ \mathbf{u}=\mathbf{g}\…

We prove the partial H\"older continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \end{equation*} where $f$…

We investigate general semilinear (obstacle-like) problems of the form $\Delta u = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately…

Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and $$ \d\left( |\nabla u|^{p-2}\nabla u\right)=\chi_{\{u>0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$…

In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim \chi_{\{u > 0 \}}$. The…

In this paper, we study local minimizers of a degenerate version of the Alt-Caffarelli functional. Specifically, we consider local minimizers of the functional $J_{Q}(u, \Omega):= \int_{\Omega} |\nabla u|^2 + Q(x)^2\chi_{\{u>0\}}dx$ where…

We study a class of semilinear free boundary problems in which admissible functions $u$ have a topological constraint, or spanning condition, on their 1-level set. This constraint forces $\{u=1\}$, which is the free boundary, to behave like…

In this article we study functionals of the type considered in \cite{HS21}, i.e. $$ J(v):=\int_{B_1} A(x,u)|\nabla u|^2 +f(x,u)u+ Q(x)\lambda (u)\,dx $$ here $A(x,u)= A_+(x)\chi_{\{u>0\}}+A_-(x) \chi _{\{u<0\}}$, $f(x,u)=…

We study the regularity of the interface for a new free boundary problem introduced by Caffarelli and Kriventsov. We show that for minimizers of the functional \[ F_1(A,u) = \int_A |\nabla u|^2 d\mathcal{L}^n + \int_{\partial A} u^2 +…

For a fixed constant $\lambda > 0$ and a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type…

In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+\gamma}$ regularity…

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…

It is well known that minimizers of the Allen-Cahn-type functional \[ J_\epsilon(u):=\int_\Omega\frac{\epsilon|\nabla u|^2}{2}+\frac{W(u)}{\epsilon}, \] where $W$ is a double-well potential, resemble minimal surfaces in the sense that their…

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…

We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function strictly increasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. Our main result…

In this paper we consider a minimization problem for the functional $$ J(u)=\int_{B_1^+}|\nabla u|\sp 2+\lambda_{+}^2\chi_{\{u>0\}}+\lambda_{-}^2\chi_{\{u\leq0\}}, $$ in the upper half ball $B_1^+\subset\R^n, n\geq 2$ subject to a Lipschitz…

Let $N>2$, $p\in \left(\frac{2N}{N+2},+\infty\right)$, and $\Omega$ be an open bounded domain in $\mathbb{R}^N$. We consider the minimum problem $$ \mathcal{J} (u) := \displaystyle\int_{\Omega } \left(\frac{1}{p}| \nabla u|…

We study the regularity of quasi-minimal sets (in the sense of David and Semmes) with a boundary condition, which can be interpreted as quasi-minimizers of Plateau's problem in co-dimension one. For these Plateau-quasi-minimizers, we…

In this article we study functionals of the following type $$ \int_{\Omega} \Big ( \langle A(x,u)\nabla u, \nabla u\rangle + \Lambda (x,u) \Big )\,dx $$ here $A(x,u)= A_+(x)\chi_{\{u>0\}}+A_-(x) \chi _{\{u\leq 0\}}$ for some elliptic and…