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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 \[ \partial u_ n/ \partial t-\text{div} a(x,t,u_ n,Du_ n)=f_ n+g_ n \quad \text{in } {\mathcal D}'\bigl( \Omega \times(0,T)\bigr)\;(T>0 \text{ fixed}). \]

35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
35B99 Qualitative properties of solutions to partial differential equations
Full Text: DOI
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