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

Let Ω be a bounded open set of N , 1<p, p ' <, 1/p+1/p ' =1. Consider the nonlinear elliptic equations

-diva(x,u n ,Du n )=f n +g n in𝒟 ' (Ω)(1)

where a:Ω×× N N is a Carathéodory function satisfying the classical Leray-Lions hypotheses. Assume that u n u weakly in W 1,p (Ω), strongly in L loc p (Ω) and a.e. in Ω, and f n f strongly in W -1,p ' (Ω). Moreover, assume that g n W -1,p ' (Ω) is bounded in the space (Ω) of Radon measures.

In the present paper, the authors prove that Du n Du strongly in L q (Ω) N for any q<p. This implies that, for a suitable subsequence n ' , Du n ' Du a.e. in Ω (cf. the title of the paper) and, moreover, that it is allowed to pass to the limit in (1) such that -diva(x,u,Du)=f+g in 𝒟 ' (Ω).

Besides, under the stronger hypotheses a(x,s,ζ)ζα|ζ| p for some α>0 (a.e. xΩ, and s, ζ N arbitrary) and g n g weakly in L 1 (Ω), they show that, for any fixed k>0, the truncation T k of u n at height k satisfies DT k (u n )DT k (u) strongly in L log p (Ω) N . Under suitably modified assumptions, corresponding results are obtained also in the parabolic case, i.e., when (1) is replaced by

u n /t-diva(x,t,u n ,Du n )=f n +g n in𝒟 ' Ω × ( 0 , T )(T>0fixed)·

MSC:
35J60Nonlinear elliptic equations
35K55Nonlinear parabolic equations
35B99Qualitative properties of solutions of PDE