On montre que le faisceau $ℱ$ des sursolutions locales dans $W_{\mathrm{loc}}^2$ d’un certain opérateur elliptique $L$ est maximal pour un principe du minimum adapté aux espaces de Sobolev. La continuité de la réduite variationnelle des éléments continus permet alors d’étudier des représentants s.c.i.

Soit $\mathbf{F}$ le faisceau des sursolutions variationnelles d’un opérateur différentiel elliptique du second ordre à coefficients ${\mathbf{L}}_{\mathrm{loc}}^{\infty }$. Soit $\widehat{\mathbf{F}}$ le faisceau des régularitées essentielles inférieures des éléments de $\mathbf{F}$. On démontre que $\widehat{\mathbf{F}}$ est contenu dans un seul préfaisceau ${\mathbf{F}}^{*}$ maximal de cônes convexes de fonctions s.c.i. $\>-\infty$ vérifiant le principe du minimum sur une base d’ouverts suffisamment petits. On démontre que ${\mathbf{F}}^{*}$ possède toutes les bonnes propriétés d’une théorie locale du potentiel.

We consider a general elliptic Robin boundary value problem. Using orthogonal Coifman wavelets (Coiflets) as basis functions in the Galerkin method, we prove that the rate of convergence of an approximate solution to the exact one is O(2−nN ) in the H 1 norm, where n is the level of approximation and N is the Coiflet degree. The Galerkin method needs to evaluate a lot of complicated integrals. We present a structured approach for fast and effective evaluation of these integrals via trivariate connection...

In this paper a black-box solver based on combining the unknowns aggregation with smoothing is suggested. Convergence is improved by overcorrection. Numerical experiments demonstrate the efficiency.

In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if $a$ and $b$ are symbols in $S_{1,0}^2$ such that $\left|a\right|\le b$, then for all $ϵ\>0$ we have the estimate $\parallel a^{w}u{\parallel }_s^2\le C_{ϵ}(\parallel b^{w}u{\parallel }_s^2+\parallel u{\parallel }_{s+ϵ}^2)$ for all $u$ in the Schwartz space, where $\parallel {\parallel }_t$ is the usual $H_t$ norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.

Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that $E={\mathbf{R}}^{m+1}$ is given the presheaf of solutions of$$\sum _{i=1}^m\frac{\partial u}{\partial x_i}=\frac{\partial u}{\partial x_{m+1}}$$and $B$ is a subset of ${\mathbf{R}}^m\times [-\infty ,0]$ satisfying$$\left\{\right(\alpha x,{\alpha }^{2}t):(x,t)\in B,x\in {\mathbf{R}}^m,t\in \mathbf{R}\}\subset B$$for $\alpha \>0$ arbitrarily small. Then $B$ is thin at 0 if and only if $B$ is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several approximation theorems on reducing and balayage...

This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Wentzell boundary conditions in the characteristic case. Our results may be stated as follows: We can construct Feller semigroups corresponding to a diffusion phenomenon including absorption, reflection, viscosity, diffusion along the boundary and jump at each point of the boundary.

It is shown that the $(1,p)$-fine topology defined via a Wiener criterion is the coarsest topology making all supersolutions to the $p$-Laplace equation$$\mathrm{div}\left(\right|\nabla u{|}^{p-2}\nabla u)=0$$continuous. Fine limits of quasiregular and BLD mappings are also studied.