In questo articolo studiamo problemi di Dirichlet singolari, lineari e semilineari, della forma ${\left|x\right|}^2\mathrm{\Delta }u=f\left(u\right)$ in $\mathrm{\Omega }$, $u=0$ su $\partial \mathrm{\Omega }$, dove $\mathrm{\Omega }$ è un dominio in ${\mathbb{R}}^2$ e $f\left(u\right)=\lambda u$ o $f\left(u\right)=\lambda u+{\left|u\right|}^{p-2}u$ con $p>2$ (o nonlinearità più generali). In tali problemi bidimensionali emergono alcune difficoltà a causa della non validità della disuguaglianza di Hardy in ${\mathbb{R}}^2$ e a causa delle invarianze dell'equazione $-{\left|x\right|}^2\mathrm{\Delta }u=f\left(u\right)$. Pertanto opportune condizioni su $\lambda$ e $\mathrm{\Omega }$ sono necessarie al fine di garantire l'esistenza di una soluzione positiva. Per esempio, se ${\mathrm{\Gamma }}_0$ è una curva non costante...

Consider the boundary value problem (L.P):$(h-A)u=f$ in $D$, $(v-\Gamma )u=g$ on $\partial D$ where $A$ is written as $A=1/2{\sum }_{i=1}^m{Y}_i^2+Y_0$, and $\Gamma$ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_i$ ($0\le i\le m$), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A,\Gamma )$-diffusion process...

Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form $$-div\left(a\left(T_1\left(\frac{x}{{\epsilon }_1}\right){\omega }_1,T_2\left(\frac{x}{{\epsilon }_2}\right){\omega }_2,\nabla u_{\epsilon }^{\omega }\right)\right)={\lambda }_{\epsilon }^{\omega }𝒞\left(u_{\epsilon }^{\omega }\right).$$ It is shown, under certain structure assumptions on the random map $a({\omega }_1,{\omega }_2,\xi )$, that the sequence $\{{\lambda }_{\epsilon }^{\omega ,k},u_{\epsilon }^{\omega ,k}\}$ of $k$th eigenpairs converges to the $k$th eigenpair $\{{\lambda }^k,u^k\}$ of the homogenized eigenvalue problem $$-\mathrm{div}\left(b\left(\nabla u\right)\right)=\lambda \overline{𝒞}\left(u\right).$$ For the case of $p$-Laplacian type maps we characterize $b$ explicitly.

Strong convergence estimates for pseudospectral methods applied to ordinary boundary value problems are derived. The results are also used for a convergence analysis of the Schwarz algorithm (a special domain decomposition technique). Different types of nodes (Chebyshev, Legendre nodes) are examined and compared.

We consider the Neumann Laplacian with constant magnetic field on a regular domain in ${ℝ}^2$. Let $B$ be the strength of the magnetic field and let ${\lambda }_1\left(B\right)$ be the first eigenvalue of this Laplacian. It is proved that $B\mapsto {\lambda }_1\left(B\right)$ is monotone increasing for large $B$. Together with previous results of the authors, this implies the coincidence of all the “third” critical fields for strongly type 2 superconductors.

The Steklov postprocessing operator for the linear finite element method is studied. Superconvergence of order 𝓞(h²) is proved for a class of second order differential equations with zero Dirichlet boundary conditions for arbitrary space dimensions. Relations to other postprocessing and averaging schemes are discussed.