We prove the Hölder continuity of the homogeneous gradient of the weak solutions $u\in W_{loc}^{1,p}$ of the p-Laplacian on the Heisenberg group ${\mathscr{H}}^n$, for $1+\frac{1}{\sqrt{5}}<p\le 2$.

We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $\Omega$. We prove the global $C^{1,\alpha }$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1,\alpha }$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.

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...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures $\mu$ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures $\mu$ for which the Sobolev space $W^{2,p}$ can be imbedded into $L^p\left(\mu \right)$. The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...

La métrique attachée de façon naturelle à des champs de vecteurs $C^{\infty }$ est susceptible de plusieurs définitions voisines ; on montre que, suivant la définition adoptée, elle peut avoir, ou ne pas avoir, un caractère localement lipschitzien qui a pour conséquence l’existence de points $L$-réguliers, pour certains opérateurs différentiels $L$, sur les frontières des boules pour la métrique.

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

We establish a Carleman type inequality for the subelliptic operator $ℒ={\Delta }_z+|x{|}^2{\partial }_t^2$ in ${ℝ}^{n+1}$, $n\ge 2$, where $z\in {ℝ}^n$, $t\in ℝ$. As a consequence, we show that $-ℒ+V$ has the strong unique continuation property at points of the degeneracy manifold $\left\{\right(0,t)\in {ℝ}^{n+1}|t\in ℝ\}$ if the potential $V$ is locally in certain $L^p$ spaces.

In this work we introduce a new class of lowest order methods for diffusive problems on general meshes with only one unknown per element. The underlying idea is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. To do so we borrow ideas from multi-point finite volume methods, although we use them in a rather different context. The incomplete polynomial space replaces classical complete polynomial spaces in discrete...