Related papers: An Elliptic-Parabolic Free Boundary Problem with D…
We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its…
We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity…
We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are $C^{1,\gamma}$. In particular, viscosity solutions…
We prove existence of strong solutions to a family of some semilinear parabolic free boundary problems by means of elliptic regularization. Existence of solutions is obtained in two steps: we first show some uniform energy estimates and…
We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain mountain pass solutions of critical and…
We consider fully nonlinear obstacle-type problems of the form \begin{equation*} \begin{cases} F(D^{2}u,x)=f(x) & \text{a.e. in}B_{1}\cap\Omega,|D^{2}u|\le K & \text{a.e. in}B_{1}\backslash\Omega, \end{cases} \end{equation*} where $\Omega$…
In this work, we show how to obtain a free boundary problem as the limit of a fully non linear elliptic system of equations that models population segregation (Gause-Lotka-Volterra type). We study the regularity of the solutions. In…
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the total head are obtained under the…
We study a class of two-phase inhomogeneous free boundary problems governed by elliptic equations in divergence form. In particular we prove that Lipschitz or flat free boundaries are $C^{1,\gamma}$. Our results apply to the classical…
We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the…
A free boundary problem for the dynamics of a glasslike binary fluid naturally leads to a singular perturbation problem for a strongly degenerate parabolic partial differential equation in 1D. We present a conjecture for an asymptotic…
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…
We provide rigorous asymptotic models for the free boundary Darcy and Forchheimer problem under the assumption of weak nonlinear interaction, in a regime in which the steepness parameter of the interface is considered to be very small. The…
In this paper we consider a quasilinear elliptic PDE, $\text{div} (A(x,u) \nabla u) =0$, where the underlying physical problem gives rise to a jump for the conductivity $A(x,u)$, across a level surface for $u$. Our analysis concerns…
We discuss an evolution free boundary problem of mixed type with two free boundaries modeling an idealized electrostatically actuated MEMS device. While the electric potential is the solution of an elliptic equation, the dynamics of the…
In this work we demonstrate that a class of some one and two phase free boundary problems can be recast as nonlocal parabolic equations on a submanifold. The canonical examples would be one-phase Hele Shaw flow, as well as its two-phase…
We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show the $C^{1,\alpha}$ regularity of the free boundary, when the solution is directionally monotone in space…
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…
In this work we investigate the following isoperimetric problem: to find the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the area of the part of the boundary contained in…
In this paper we study the following parabolic system \begin{equation*} \Delta \u -\partial_t \u =|\u|^{q-1}\u\,\chi_{\{ |\u|>0 \}}, \qquad \u = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|\u | >0\}$. For $0\leq…