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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 $\bbfR\sp N$, $1<p$, $p'<\infty$, $1/p+1/p'=1$. Consider the nonlinear elliptic equations $$-\text{div} a(x,u\sb n,Du\sb n)=f\sb n+ g\sb n \quad \text{in } {\cal D}'(\Omega) \tag 1$$ where $a:\Omega \times \bbfR \times \bbfR\sp N \to \bbfR\sp N$ is a Carathéodory function satisfying the classical Leray-Lions hypotheses. Assume that $u\sb n\rightharpoonup u$ weakly in $W\sp{1,p}(\Omega)$, strongly in $L\sp p\sb{\text{loc}}(\Omega)$ and a.e. in $\Omega$, and $f\sb n \to f$ strongly in $W\sp{-1,p'} (\Omega)$. Moreover, assume that $g\sb n \in W\sp{-1,p'}(\Omega)$ is bounded in the space ${\cal M}(\Omega)$ of Radon measures. In the present paper, the authors prove that $Du\sb n \to Du$ strongly in $\bigl( L\sp q(\Omega) \bigr)\sp N$ for any $q<p$. This implies that, for a suitable subsequence $n'$, $Du\sb{n'} \to Du$ 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 $-\text{div} a(x,u,Du) = f+g$ in ${\cal D}'(\Omega)$. Besides, under the stronger hypotheses $a(x,s,\zeta) \zeta \ge \alpha \vert \zeta \vert\sp p$ for some $\alpha>0$ (a.e. $x \in \Omega$, and $s \in \bbfR$, $\zeta \in \bbfR\sp N$ arbitrary) and $g\sb n \rightharpoonup g$ weakly in $L\sp 1(\Omega)$, they show that, for any fixed $k>0$, the truncation $T\sb k$ of $u\sb n$ at height $k$ satisfies $DT\sb k(u\sb n) \to DT\sb k(u)$ strongly in $\bigl( L\sp p\sb{\text{log}} (\Omega) \bigr)\sp N$. Under suitably modified assumptions, corresponding results are obtained also in the parabolic case, i.e., when (1) is replaced by $$\partial u\sb n/ \partial t-\text{div} a(x,t,u\sb n,Du\sb n)=f\sb n+g\sb n \quad \text{in } {\cal D}'\bigl( \Omega \times(0,T)\bigr)\ (T>0 \text{ fixed}).$$

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