We present families of scalar nonconforming finite elements of arbitrary order $r\ge 1$ with optimal approximation properties on quadrilaterals and hexahedra. Their vector-valued versions together with a discontinuous pressure approximation of order $r-1$ form inf-sup stable finite element pairs of order r for the Stokes problem. The well-known elements by Rannacher and Turek are recovered in the case r=1. A numerical comparison between conforming and nonconforming discretisations will be given. Since higher order...

On démontre dans cet article des versions probabilistes des injections de Sobolev sur une variété riemannienne compacte, $(M,g)$. Plus précisément on démontre que pour des mesures de probabilité naturelles sur l’espace $L^2\left(M\right)$, presque toute fonction appartient à tous les espaces $L^p\left(M\right)$, $p\<+\infty$. On donne ensuite des applications à l’étude des harmoniques sphériques sur la sphère ${𝕊}^d$ : on démontre (encore pour des mesures de probabilité naturelles) que presque toute base hilbertienne de $L^2\left({𝕊}^d\right)$ formée d’harmoniques sphériques...

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact sequence framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem for the discrete function spaces. This...

In this paper we prove an estimate for the measure of superlevel sets of weak solutions to quasilinear elliptic systems in divergence form. In some special cases, such an estimate allows us to improve on the integrability of the solution.

We prove that the higher integrability of the data $f,f_0$ improves on the integrability of minimizers $u$ of functionals $ℱ$, whose model is $${\int }_{\Omega }\left[{\left|Du\right|}^p+{(det\left(Du\right))}^2-\langle f,Du\rangle +\langle f_0,u\rangle \right]dx,$$ where $u:\Omega \subset {ℝ}^n\to {ℝ}^n$ and $p\ge 2$.

The paper deals with very weak solutions $u\in \theta +W_0^{1,r}\left(\Omega \right)$, $max\{1,p-1\}<r<p<n$, to boundary value problems of the $p$-harmonic equation $$\left\{\begin{array}{cc}-\text{div}\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=0,\hfill & x\in \Omega ,\hfill \\ u\left(x\right)=\theta \left(x\right),\hfill & x\in \partial \Omega .\hfill \end{array}\right.(*)$$ We show that, under the assumption $\theta \in W^{1,q}\left(\Omega \right)$, $q>r$, any very weak solution $u$ to the boundary value problem ($*$) is integrable with $$u\in \left\{\begin{array}{cc}\theta +L_{\mathrm{weak}}^{q^{*}}\left(\Omega \right)\hfill & \text{for}q<n,\hfill \\ \theta +L_{\mathrm{weak}}^{\tau }\left(\Omega \right)\hfill & \text{for}q=n\text{and}\text{any}\tau <\infty ,\hfill \\ \theta +L^{\infty }\left(\Omega \right)\hfill & \text{for}q>n,\hfill \end{array}\right.$$ provided that $r$ is sufficiently close to $p$.

The aim of this note is to indicate how inequalities concerning the integral of ${\left|\nabla u\right|}^2$ on the subsets where |u(x)| is greater than k ($k\in I{R}^{+}$) can be used in order to prove summability properties of u (joint work with Daniela Giachetti). This method was introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems. In some joint works with Thierry Gallouet, inequalities concerning the integral of ${\left|\nabla u\right|}^2$ on the subsets where |u(x)| is less than k ($k\in I{R}^{+}$) or...