We prove the existence of solutions to $-\text{div}a\left(x,\text{grad}u\right)=f$, together with appropriate boundary conditions, whenever $a\left(x,e\right)$ is a maximal monotone graph in $e$, for every fixed $x$. We propose an adequate setting for this problem, in particular as far as measurability is concerned. It consists in looking at the graph after a ${45}^{\circ }$ rotation, for every fixed $x$; in other words, the graph $d\in a\left(x,e\right)$ is defined through $d-e=\phi \left(x,d+e\right)$, where $\phi$ is a Carathéodory contraction in ${\mathbb{R}}^N$. This definition is shown to be equivalent to the fact that $a(x,\cdot )$ is pointwise monotone...

We present some monotonicity and symmetry results for positive solutions of the equation $-\text{div}\left({\left|Du\right|}^{p-2}Du\right)=f\left(u\right)$ satisfying an homogeneous Dirichlet boundary condition in a bounded domain $\mathrm{\Omega }$. We assume 1 < p < 2 and $f$ locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if $\mathrm{\Omega }$ is a ball then the solutions are radially symmetric and strictly radially decreasing.

In this Note we consider the following problem $$\left\{\begin{array}{cc}-\mathrm{△}u=N\left(N-2\right)u^{p_{ϵ}}-\lambda u\hfill & \text{in }\mathrm{\Omega }\hfill \\ u>0\hfill & \text{in }\mathrm{\Omega }\hfill \\ u=0\hfill & \text{on }\partial \mathrm{\Omega }.\hfill \end{array}\right.$$ where $\mathrm{\Omega }$ is a bounded smooth starshaped domain in ${\mathbb{R}}^N$, $N\ge 3$, $p_{ϵ}=\frac{N+2}{N-2}-ϵ$, $ϵ>0$, and $\lambda \ge 0$. We prove that if $u_{ϵ}$ is a solution of Morse index $m>0$ than $u_{ϵ}$ cannot have more than $m$ maximum points in $\mathrm{\Omega }$ for $ϵ$ sufficiently small. Moreover if $\mathrm{\Omega }$ is convex we prove that any solution of index one has only one critical point and the level sets are starshaped for $ϵ$ sufficiently small.

This paper deals with the mortar spectral element discretization of two equivalent problems, the Laplace equation and the Darcy system, in a domain which corresponds to a nonhomogeneous anisotropic medium. The numerical analysis of the discretization leads to optimal error estimates and the numerical experiments that we present enable us to verify its efficiency.

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. We reduce the original problem by a Fourier expansion in the angular variable to a countable family of two-dimensional problems. We decompose the meridian domain, assumed polygonal, in a finite number of rectangles and we discretize by a spectral method. Then we describe the main features of the mortar method and use the algorithm Strang Fix to improve the accuracy of our discretization.

We consider the Laplace equation posed in a three-dimensional axisymmetric domain. We reduce the original problem by a Fourier expansion in the angular variable to a countable family of two-dimensional problems. We decompose the meridian domain, assumed polygonal, in a finite number of rectangles and we discretize by a spectral method. Then we describe the main features of the mortar method and use the algorithm Strang Fix to improve the accuracy...

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.