We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space $C^{-1,\alpha }(\mathrm{\Omega })$ consinsting of all derivatives of hölder-continuous functions in $\mathrm{\Omega }$ where $\mathrm{\Omega }$ is a domain in ${ℝ}^n$ not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space $C^{-1,\alpha }(\mathrm{\Omega })$. We prove also that the spaces $C^{-1,\alpha }(\mathrm{\Omega })$ can be considered as extrapolation spaces relative to suitable non-variational operators....

We study the asymptotic behaviour of a sequence of strongly degenerate parabolic equations ${\partial }_t\left(r_{h}u\right)-\mathrm{div}(a_h·Du)$ with $r_h(x,t)\ge 0$, $r_h\in L^{\infty }(\Omega \times (0,T))$. The main problem is the lack of compactness, by-passed via a regularity result. As particular cases, we obtain G-convergence for elliptic operators $(r_h\equiv 0)$, G-convergence for parabolic operators $(r_h\equiv 1)$, singular perturbations of an elliptic operator $(a_h\equiv a$ and $r_h\to r$, possibly $r\equiv 0)$.

We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.

Let $ℒ=$ -div $\left(A\right(x\left)\nabla \right)$ be a second order elliptic operator with real, symmetric, bounded measurable coefficients on ${ℝ}^n$ or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed $p\>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform $\nabla {\left(ℒ\right)}^{-1/2}$ on the $L^p$ space. As an application, for $1\<p\<3+ϵ$, we establish the $L^p$ boundedness of Riesz transforms on Lipschitz domains for operators with $VMO$ coefficients. The range of $p$ is sharp. The closely related boundedness of ...

Caccioppoli estimates are instrumental in virtually all analytic aspects of the theory of partial differential equations, linear and nonlinear. And there is always something new to add to these estimates. We emphasize the fundamental role of the natural domain of definition of a given differential operator and the associated weak solutions. However, we depart from this usual setting (energy estimates) and move into the realm of the so-called very weak solutions where important new applications lie....

Dans cet article, on considère les opérateurs différentiels $T=b\left(x\right)D\left(a\right(x\left)D\right)$, où $a\left(x\right)$ et $b\left(x\right)$ sont deux fonctions mesurables, bornées et accrétives, et $D=-i\frac{d}{dx}$. Les résultats principaux portent sur les propriétés fonctionnelles de $T$, de sa racine carrée, avec applications à l’équation elliptique ${\partial }_t^{2}u-Tu=0$ sur $ℝ\times [0,+\infty [$. On démontre que $T^{1/2}{D}^{-1}$ est un opérateur de Calderón-Zygmund qui dépend analytiquement du couple $(a,b)$. Les estimations ponctuelles optimales sur le noyau du semi-groupe $\exp (-t{L}^{1/2})$ et le calcul fonctionnel permettent de développer une théorie...

We consider linear elliptic problems with variable coefficients, which may sharply change values and have a complex behavior in the domain. For these problems, a new combined discretization-modeling strategy is suggested and studied. It uses a sequence of simplified models, approximating the original one with increasing accuracy. Boundary value problems generated by these simplified models are solved numerically, and the approximation and modeling errors are estimated by a posteriori estimates of...