Related papers: Remarks on a mean field equation on $\mathbb{S}^{2…

In this paper, we prove uniqueness of solutions of mean field equations with general boundary conditions for the critical and subcritical total mass regime, extending the earlier results for null Dirichlet boundary condition. The proof is…

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are…

We study the local H\"older regularity of strong solutions $u$ of second-order uniformly elliptic equations having a gradient term with superquadratic growth $\gamma > 2$, and right-hand side in a Lebesgue space $L^q$. When $q >…

We study existence and uniqueness of solutions of (E 1) --$\Delta$u + $\mu$ |x| ^{-2} u + g(u) = $\nu$ in $\Omega$, u = $\lambda$ on $\partial$$\Omega$, where $\Omega$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$…

In Euclidean space, we investigate surfaces whose mean curvature $H$ satisfies the equation $H=\alpha\langle N,\mathbf{x}\rangle+\lambda$, where $N$ is the Gauss map, $\mathbf{x}$ is the position vector and $\alpha$ and $\lambda$ are two…

We study stable compact constant mean curvature surfaces in the product spaces S2 X R and H2 X R and in some other Riemannian 3-manifolds.

In this paper we obtain rigidity results for a bounded non-constant entire solution $u$ of the Allen-Cahn equation in $\mathbb{R}^n$, whose level set $\{u=0\}$ is contained in a half-space. If $n\leq 3$ we prove that the solution must be…

We examine the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=(1+|x|^2)^{\frac{\alpha}{2}} v,\\ -\Delta v=(1+|x|^2)^{\frac{\alpha}{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$,…

We consider the mean curvature rigidity problem of an equatorial zone on a sphere which is symmetric about the equator with width $2w$. There are two different notions on rigidity, i.e. strong rigidity and local rigidity. We prove that for…

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta…

Consider the solutions $u$ to the elliptic equation $\mathcal{L}(u) = \partial_i(a^{ij}(x) \partial_j u) + b^i(x) \partial_i u + c(x) u= 0$ with $a^{ij}$ assumed only to be H\"older continuous. In this paper we prove an explicit bound for…

In this paper, we study the following problem $$ \{{ll} \Delta_{H^n} u-u+u^p=0 & in H^n u>0& in H^n u(x)\to 0 &\rho(x)\to\infty}. $$ where $1<p < \frac{Q+2}{Q-2}$, Q is the homogeneous dimension of Heisenberg group $H^n$. Our main result is…

We obtain and study static, spherically symmetric solutions for the Einstein - generalized Maxwell field system in 2n dimensions, with possible inclusion of a massless scalar field. The generalization preserves the conformal invariance of…

We consider the regularity of the extremal solution of the nonlinear eigenvalue problem (S)_\lambda \qquad {rcr} -\Delta u + c(x) \cdot \nabla u &=& \frac{\lambda}{(1-u)^2} \qquad {in $ \Omega$}, u &=& 0 \qquad {on $ \pOm$}, where $ \Omega…

We give an interpretation of the hemisphere rigidity theorem of Hang-Wang in the framework of Gelfand problem. More precisely, Hang-Wang showed that for a metric $g$ conformal to the standard metric $g_0$ on $S^{n}_{+}$ with $R\geq n(n-1)$…

In this paper we prove the monotonicity of positive solutions to $ -\Delta_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$.…

The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to $- L u = f(u)$, where $L$ is a linear uniformly elliptic operator and $f$ is…

We propose using an equation-of-motion approach as an impurity solver for dynamical mean field theory. As an illustration of this technique, we consider a finite-$U$ Hubbard model defined on the Bethe lattice with infinite connectivity at…