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

##### MSC:
 35J60 Nonlinear elliptic equations 35K55 Nonlinear parabolic equations 35B99 Qualitative properties of solutions to partial differential equations
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##### References:
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