Newton-like methods are considered with inexact correction computed by some inner iterative method. Composite iterative methods of this type are applied to the solution of nonlinear systems arising from the solution of nonlinear elliptic boundary value problems. Two main quastions are studied in this paper: the convergence of the inexact Newton-like methods and the efficient control of accuracy in computation of the inexact correction. Numerical experiments show the efficiency of the suggested composite...

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function...

In this paper, we review several recent results dealing with elliptic equations with non local diffusion. More precisely, we investigate several problems involving the fractional laplacian. Finally, we present a conformally covariant operator and the associated singular and regular Yamabe problem.

We consider the Yamabe type family of problems $\left(P_{\epsilon }\right):-\Delta u_{\epsilon }=u_{\epsilon }^{(n+2)/\left(n-2\right)}$, $u_{\epsilon }>0$ in $A_{\epsilon }$, $u_{\epsilon }=0$ on $\partial A_{\epsilon }$, where $A_{\epsilon }$ is an annulus-shaped domain of ${ℝ}^n$, $n\ge 3$, which becomes thinner as $\epsilon \to 0$. We show that for every solution $u_{\epsilon }$, the energy ${\int }_{A_{\epsilon }}|\nabla u_{|}^2$ as well as the Morse index tend to infinity as $\epsilon \to 0$. This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ${ℝ}^n$, a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...

Dans l'article, on a défini une équation d'operateur équivalent à la formulation variationnelle du problème. Les solutions de cette équation sont des points critiques de la fonctionnelle qu'elle porte le nom d'énergie totale de déformation. La fonctionnelle est coercive et faiblement séquentiellement semi-continue inférieure. Par le théorème de l'analyse fonctionnelle, on a obtenu le résultat d'existence pour le problème.

Dans l'article, on a donné quelques conditions suffisantes pour l'unicité locale et globale de la solution du problème. On a construit une solution variationnelle du problème par la méthode de Newton-Kantorovitch et la méthode du prolongement continu avec ces conditions suffisantes pour l'unicité.

On considère le problème :$$\left\{\begin{array}{c}-\mathrm{div}a(x,Du)=f\mathrm{dans}\Omega ,\hfill \\ u=0\mathrm{sur}\partial \Omega ,\hfill \end{array}\right.$$où $\Omega$ est un ouvert borné de ${𝐑}^N$, où $a(x,\xi )$ est une fonction de Carathéodory, monotone en $\xi$, coercive, qui définit un opérateur dans $W_0^{1,p}\left(\Omega \right)$ (avec $1\<p\le N$), et où $f$ appartient à $L^1\left(\Omega \right)$ ou est une mesure bornée sur $\Omega$. On introduit une nouvelle définition de la solution de ce problème, la notion de solution renormalisée (ou entropique), et on montre l’existence d’une telle solution et sa continuité par rapport à $f$. Quand $f$ appartient à $L^1\left(\Omega \right)$, on montre en outre que cette solution est unique.