Related papers: Viscosity solutions to complex first eigenvalue eq…

This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and…

We study the Dirichlet problem for the non-local diffusion equation $u_t=\int\{u(x+z,t)-u(x,t)\}\dmu(z)$, where $\mu$ is a $L^1$ function and $``u=\phi$ on $\partial\Omega\times(0,\infty)$'' has to be understood in a non-classical sense. We…

In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a…

We consider the problem of finding a real number lambda and a function u satisfying the PDE max{lambda -\Delta u -f,|Du|-1}=0, for all x in R^n. Here f is a convex, superlinear function. We prove that there is a unique lambda* such that the…

We consider the quasi-linear eigenvalue problem $-\Delta_p u = \lambda g(u)$ subject to Dirichlet boundary conditions on a bounded open set $\Omega$, where $g$ is a locally Lipschitz continuous functions. Imposing no further conditions on…

We examine the equation \[\Delta^2 u = \lambda f(u) \qquad \Omega, \] with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for…

We study positive solutions of the Dirichlet problem $-\Delta u = u^p$ in a uniformly convex domain $\Omega \subset \mathbb S^2$, $u= 0$ on $\partial\Omega.$ For $p=1$, we assume that the right-hand side is replaced by $\lambda_1 u$, where…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

In this paper we study some relationships between the first Dirichlet eigenvalue $\Lambda(\Omega)$ and the torsional rigidity $T(\Omega)$ of a domain $\Omega$. We consider the problem of optimizing the product $\Lambda(\Omega)T(\Omega)$…

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain $\Omega\subset\mathbb R^2$, for a vectorial elliptic operator $-\nabla\cdot A^\epsilon(\cdot)\nabla$ with $\epsilon$-periodic…

For a bounded domain $\Omega$ with a piecewise smooth boundary in an $n$-dimensional Euclidean space $\mathbf{R}^{n}$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. First we give a general inequality for…

We consider minimizers of \[ F(\lambda_1(\Omega),\ldots,\lambda_N(\Omega)) + |\Omega|, \] where $F$ is a function nondecreasing in each parameter, and $\lambda_k(\Omega)$ is the $k$-th Dirichlet eigenvalue of $\Omega$. This includes, in…

Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $\Delta_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of…

Let $m$ be a bounded function and $\alpha$ a nonnegative parameter. This article is concerned with the first eigenvalue $\lambda\_\alpha(m)$ of the drifted Laplacian type operator $\mathcal L\_m$ given by $\mathcal L\_m(u)=…

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by…

In this paper we are interested on the well-posedness of Dirichlet problems associated to integro-differential elliptic operators of order $\alpha < 1$ in a bounded smooth domain $\Omega$ . The main difficulty arises because of losses of…

We consider the singularly perturbed problem $F_\varepsilon (u,\Omega):=\int_\Omega \varepsilon |\nabla^2u| + \varepsilon^{-1}|1-|\nabla u|^2|^2$ on bounded domains $\Omega \subset\mathbb{R}^2$. Under appropriate boundary conditions, we…

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…

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…

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among…