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Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. (English) Zbl 0783.35020

Let ${\Omega }$ be a bounded open set of ${ℝ}^{N}$, $1, ${p}^{\text{'}}<\infty$, $1/p+1/{p}^{\text{'}}=1$. Consider the nonlinear elliptic equations

$-\text{div}a\left(x,{u}_{n},D{u}_{n}\right)={f}_{n}+{g}_{n}\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{𝒟}^{\text{'}}\left({\Omega }\right)\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $a:{\Omega }×ℝ×{ℝ}^{N}\to {ℝ}^{N}$ is a Carathéodory function satisfying the classical Leray-Lions hypotheses. Assume that ${u}_{n}⇀u$ weakly in ${W}^{1,p}\left({\Omega }\right)$, strongly in ${L}_{\text{loc}}^{p}\left({\Omega }\right)$ and a.e. in ${\Omega }$, and ${f}_{n}\to f$ strongly in ${W}^{-1,{p}^{\text{'}}}\left({\Omega }\right)$. Moreover, assume that ${g}_{n}\in {W}^{-1,{p}^{\text{'}}}\left({\Omega }\right)$ is bounded in the space $ℳ\left({\Omega }\right)$ of Radon measures.

In the present paper, the authors prove that $D{u}_{n}\to Du$ strongly in ${\left({L}^{q}\left({\Omega }\right)\right)}^{N}$ for any $q. This implies that, for a suitable subsequence ${n}^{\text{'}}$, $D{u}_{{n}^{\text{'}}}\to Du$ a.e. in ${\Omega }$ (cf. the title of the paper) and, moreover, that it is allowed to pass to the limit in (1) such that $-\text{div}a\left(x,u,Du\right)=f+g$ in ${𝒟}^{\text{'}}\left({\Omega }\right)$.

Besides, under the stronger hypotheses $a\left(x,s,\zeta \right)\zeta \ge {\alpha |\zeta |}^{p}$ for some $\alpha >0$ (a.e. $x\in {\Omega }$, and $s\in ℝ$, $\zeta \in {ℝ}^{N}$ arbitrary) and ${g}_{n}⇀g$ weakly in ${L}^{1}\left({\Omega }\right)$, they show that, for any fixed $k>0$, the truncation ${T}_{k}$ of ${u}_{n}$ at height $k$ satisfies $D{T}_{k}\left({u}_{n}\right)\to D{T}_{k}\left(u\right)$ strongly in ${\left({L}_{\text{log}}^{p}\left({\Omega }\right)\right)}^{N}$. Under suitably modified assumptions, corresponding results are obtained also in the parabolic case, i.e., when (1) is replaced by

$\partial {u}_{n}/\partial t-\text{div}a\left(x,t,{u}_{n},D{u}_{n}\right)={f}_{n}+{g}_{n}\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{𝒟}^{\text{'}}\left({\Omega }×\left(0,T\right)\right)\phantom{\rule{4pt}{0ex}}\left(T>0\phantom{\rule{4.pt}{0ex}}\text{fixed}\right)·$

##### MSC:
 35J60 Nonlinear elliptic equations 35K55 Nonlinear parabolic equations 35B99 Qualitative properties of solutions of PDE