We prove some comparison results for Monge-Ampère type equations in dimension two. We consider also the case of eigenfunctions and we prove a kind of reverse inequalities.

A nonlinear elliptic partial differential equation with the Newton boundary conditions is examined. We prove that for greater data we get a greater weak solution. This is the so-called comparison principle. It is applied to a steady-state heat conduction problem in anisotropic magnetic cores of large transformers.

Given a bounded open set $\Omega$ in ${ℝ}^n$ (or in a Riemannian manifold) and a partition of $\Omega$ by $k$ open sets $D_j$, we consider the quantity ${\mathrm{𝚖𝚊𝚡}}_j\lambda \left(D_j\right)$ where $\lambda \left(D_j\right)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by ${ℒ}_k\left(\Omega \right)$ the infimum over all the $k$-partitions of ${\mathrm{𝚖𝚊𝚡}}_j\lambda \left(D_j\right)$, a minimal $k$-partition is then a partition which realizes the infimum. When $k=2$, we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$’s is non trivial and quite interesting. In this paper, we give...

We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

We show compactness of bounded sets of weak solutions to the isentropic compressible Navier-Stokes equations in three space dimensions under the hypothesis that the adiabatic constant $\gamma >3/2$.

We study convergence properties of ${\left\{v\left(\nabla u_k\right)\right\}}_{k\in \mathbb{N}}$ if $v\in C\left({ℝ}^{m\times n}\right)$, $\left|v\left(s\right)\right|\le C(1+|s{|}^p)$, $1<p<+\infty$, has a finite quasiconvex envelope, $u_k\to u$ weakly in $W^{1,p}(\Omega ;{ℝ}^m)$ and for some $g\in C\left(\Omega \right)$ it holds that ${\int }_{\Omega }g\left(x\right)v\left(\nabla u_k\left(x\right)\right)\mathrm{d}x\to {\int }_{\Omega }g\left(x\right)Qv\left(\nabla u\left(x\right)\right)\mathrm{d}x$ as $k\to \infty$. In particular, we give necessary and sufficient conditions for $L^1$-weak convergence of ${\{det\nabla u_k\}}_{k\in \mathbb{N}}$ to $det\nabla u$ if $m=n=p$.

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta +\frac{k}{x_n}\frac{\partial }{\partial x_n}$ on ${ℝ}^n$ is proved. In this note there is shown that in the cases $k\ne 0$, $k\ne 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.