Related papers: A free boundary problem driven by the biharmonic o…

We consider a one-phase free boundary problem of the minimizer of the energy \[ J_{\gamma}(u)=\frac{1}{2}\int_{(B_1^{n+1})^+}{y^{1-2s}|\nabla u(x,y)|^2dxdy}+\int_{B_1^{n}\times \{y=0\}}{u^{\gamma}dx}, \] with constants $0<s,\gamma<1$. It is…

Let $u$ be a weak solution of the free boundary problem $$\mathcal L u=\lambda_0 \mathcal H^1\lfloor\partial\{u>0\}, u\ge 0,$$ where $\mathcal L u={\text{div}}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and $g(\xi)$ is a given…

In this paper, we continue the 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 +…

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the…

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…

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $\Omega$ is obtained as the integral of a cost function $j(u,x)$…

We study minimizers of the functional $$ \int_{B_1^+}|\nabla u|^2 x_n^a\,d x +2\int_{B_1'} (\lambda_+ u^++\lambda_- u^-)\,d x', $$ for $a\in(-1,1)$. The problem arises in connection with heat flow with control on the boundary. It can also…

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: \[ \min \left\{f-|Du|^\gamma F(D^2u),u-\phi\right\} = 0 \quad\textrm{ in }\quad \Omega. \] We obtain existence of solutions…

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…

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\label{E} E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad…

We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\min\bigl\{-\Delta_p u,\,u-\varphi\bigr\}=0$ in $\Omega\subset\mathbb R^n$. Here, $\Delta_p u=\textrm{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)$, and…

In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_{\gamma}(u)=\int_{\Omega}\big(f(x,\nabla u)+\lambda_{+}…

The simplest genuinely multidimensional monopolist's problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)^2$. The geometry of the region of strict…

In this paper we present a survey concerning unconstrained free boundary problems of type $$ \left\{ \begin{array}{ll} F_1(D^2u,\nabla u,u,x)=0 & \text{in }B_1 \cap \Omega ,\\ F_2 (D^2 u,\nabla u,u,x)=0 & \text{in }B_1\setminus\Omega ,\\ u…

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove…

While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…

In this paper we study the following problem. For any $\ep>0$, take $u^{\ep}$ a solution of, $$ \L u^{\ep}:= {div}\Big(\di\frac {g(|\nabla \uep|)}{|\nabla \uep|}\nabla \uep\Big)=\beta_{\ep}(u^{\ep}),\quad u^{\ep}\geq 0. $$ A solution to…

In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_{\Omega}\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad…

In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_\Omega |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(\Omega)$.…

In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$ \Delta_\Phi u \coloneqq \mbox{div} \left(\phi(|\nabla u|)\dfrac{\nabla u}{|\nabla u|}\right), $$ where $\Phi(s)=e^{s^2}-1$…