Related papers: Stability of Bernstein type theorem for the minima…

In this paper, we are interested in shape optimization problems involving the ge ometry (normal, curvatures) of the surfaces. We consider a class of hypersurface s in $\mathbb{R}^{n}$ satisfying a uniform ball condition and we prove the…

The radial limits at a point ${\bf y}$ of the boundary of the domain $\Omega\subset {\bf R}^{2}$ of a bounded variational solution $f$ of Dirichlet or contact angle boundary value problems for a prescribed mean curvature equation are…

Let $\Omega $ be a bounded domain in $\mathbb{R} ^N $, and let $u\in C^1 (\overline{\Omega }) $ be a weak solution of the following overdetermined BVP: $-\nabla (g(|\nabla u|)|\nabla u|^{-1} \nabla u )=f(|x|,u)$, $ u>0 $ in $\Omega $ and…

In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish…

For $n\geq2,$ we obtain Liouville type theorems for minimal surface equations in half space $\mathbf R^n_+$ with affine Dirichlet boundary value or constant Neumann boundary value.

Let $\Omega$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $\phi_{\lambda}$ with eigenvalue $\lambda^2$. Our main result is a small-scale {\em non-concentration} estimate: We…

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on…

Let $\Omega \subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and $A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) = 0$ in…

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $u$ be the solution of $-\Delta u = 1$ vanishing on the boundary $\partial \Omega$. The estimate $$ \| \nabla u\|_{L^{\infty}(\Omega)} \leq c |\Omega|^{1/2}$$ is…

The minimal surfaces meeting in triples with equal angles along a common boundary naturally arise from soap films and other physical phenomenon. They are also the natural extension of the usual minimal surface. In this paper, we consider…

We study the semilinear indefinite elliptic problem \[ -\Delta u = Q_\Omega |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_\Omega = \chi_\Omega - \chi_{\mathbb{R}^N \setminus \Omega}$, $\Omega \subset \mathbb{R}^N$ is a bounded…

We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $\Omega \subset \mathbb{R}^n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz…

A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $\lambda$-hypersurfaces (\cite{…

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$. We show that the Bergman metric associated to $\Omega$ has the lower bound $\tilde g(\delta_\Omega(z)^{-1})$ where $\delta_\Omega(z)$…

In this paper we use the theory of mean-stable surfaces (stable minimal surfaces included) to explore the static Einstein-Maxwell space-time. We first prove that the zero set of the lapse function must be contained in the horizon boundary.…

We prove that the Robin ground state and the Robin torsion function are respectively log-concave and $\frac{1}{2}$-concave on an uniformly convex domain $\Omega\subset \mathbb{R}^N$ of class $\mathcal{C}^m$, with $[m -\frac{ N}{2}]\geq 4$,…

In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the $L^s$…

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $\Omega$. We prove existence, regularity and some structural properties of minimizers. In particular, when $\Omega$ is…

In this note we discuss graphs over a domain $\Omega\subset N^2$ in the product manifold $N^2\times \mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $\Omega$ has peice-wise smooth boundary. Let $\gamma \subset\partial\Omega$ be…

We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized…