相关论文: Bernstein-type techniques for 2D free boundary gra…
We prove that, given~$p>\max\left\{\frac{2n}{n+2},1\right\}$, the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz…
We investigate existence and regularity properties of one-phase free boundary graphs, in connection with the question of whether there exists a complete non-planar free boundary graph in high dimensions.
On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be…
Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -{\epsilon}^2\Delta_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partial{\nu}=0 on \partial M…
We consider the elliptic equation $-\Delta u+ u=0$ in a bounded, smooth domain $\Omega\subset\mathbb R^{2}$ subject to the nonlinear Neumann boundary condition $\partial u/\partial\nu = |u|^{p-1}u$ on $\partial\Omega$ and study the…
This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in $\mathbb {R}^n$. Among other results we prove the following one. Let $U(x',x_n)$ be a real-valued bounded harmonic function on the upper…
Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\Delta$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$ \left \{…
We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for…
The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove…
The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds for stream functions as well as free-surface profiles and the total head are obtained under the…
We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free…
We study the singular Lane-Emden-Fowler equation \begin{equation} -\Delta u=f(X)\cdot u^{-\gamma} \end{equation} in a bounded Lipschitz domain $\Omega$, with the Dirichlet boundary condition and a positive, bounded function $f(X)$. A…
In this paper, we consider the following free boundary problem $$ (P)\left\{\begin{array}{ll} \Delta u = \lambda \phi(x)\Sum_{i=1}^n H(u-\mu_i )& \quad \mbox{ in }\ \Omega=\Omega_2\setminus \overline{\Omega}_1, \\[0.3cm]u =0 &\quad \mbox{…
We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $ {\rm div} G(\nabla u)=0$ in $B_1\subset\mathbb{R}^2 $, for any continuous strictly monotone vector field $G \colon \mathbb{R}^2 \to \mathbb{R}^2$. We show…
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 $\Omega$…
Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any entire solution of the fourth order…
We classify global Lipschitz solutions to two-phase free boundary problems governed by concave fully nonlinear equations, as either two-plane solutions or solutions to a one-phase problem.
In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} \beta(u))=0\tag{$\sharp$} \end{equation} for prescribed constant velocity $\bf v$ and enthalpy $\beta(u)$ with…
In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has…
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…