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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:
35J60Nonlinear elliptic equations
35K55Nonlinear parabolic equations
35B99Qualitative properties of solutions of PDE
WorldCat.org
Full Text: DOI
References:
[1] Boccardo, L.; Murat, F.; Puel, J. -P.: Existence de solutions non bornées pour certaines équations quasilinéaires. Portugaliae math. 41, 507-534 (1982) · Zbl 0524.35041
[2] Meyers, N. G.: An lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. scu. Norm. sup. Pisa 17, 189-206 (1963) · Zbl 0127.31904
[3] Murat, F.: L’injection du cône positif de H-1 dans W-1,q est compacte pour tout q 2. J. math. Pures appl. 60, 309-322 (1981) · Zbl 0471.46020
[4] Frehse, J.: A refinement of Rellich’s theorem. Rend. mat. 5, 229-242 (1985) · Zbl 0666.46041
[5] Landes, R.: On the existence of weak solutions of perturbed systems with critical growth. J. reine angew. Math. 393, 21-38 (1989) · Zbl 0664.35027
[6] Boccardo, L.; Murat, F.: Strongly nonlinear Cauchy problems with gradient dependent lower order nonlinearity. Pitman research notes in mathematics 208, 247-254 (1989) · Zbl 0703.35094
[7] Boccardo, L.; Gallouet, T.: Nonlinear elliptic and parabolic equations involving measure data. J. funct. Analysis 87, 149-169 (1989) · Zbl 0707.35060
[8] Boccardo, L.; Giachetti, D.; Murat, F.: A generalization of a theorem of H. Brezis & F.E. Browder and applications to some unilateral problems. Ann. inst. Poincaré, analyse non lin. 7, 367-384 (1990) · Zbl 0716.46032
[9] Del Vecchio, T.: Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math. 16 (1990) · Zbl 0714.35035
[10] Landes, R.: Solvability of perturbed elliptic equations with critical growth exponent for the gradient. J. math. Analysis applic. 139, 63-77 (1989) · Zbl 0691.35038
[11] Hedberg L. & Murat F., Compactness of intervals in some spaces of distributions (to appear).
[12] Leray, J.; Lions, J. -L.: Quelques résultats de visik sur LES problèmes elliptiques non linéaires par LES méthodes de minty-Browder. Bull. soc. Math. fr. 93, 97-107 (1965) · Zbl 0132.10502
[13] Lions, J. -L.: Quelques méthodes de résolution des problémes aux limites nonlinéaires. (1969)
[14] Cioranescu, D.; Murat, F.: Un terme étrange venu d’ailleurs. Research notes in mathematics 60 & 70, 98-138 (1982) · Zbl 0496.35030
[15] Bensoussan A., Boccardo L. & Murat F., H-convergence for quasilinear elliptic equations with quadratic growth (to appear). · Zbl 0795.35008
[16] Browder, F. E.: Existence theorems for nonlinear partial differential equations. Proc. symp. Pure math., 1-60 (1970) · Zbl 0211.17204
[17] Boccardo, L.; Murat, F.; Puel, J. -P.: Existence of bounded solutions for nonlinear elliptic unilateral problem. Ann. mat. Pura appl. 152, 183-196 (1988) · Zbl 0687.35042
[18] Boccardo, L.; Giachetti, D.: Strongly nonlinear unilateral problems. Appl. math. Opt. 9, 291-301 (1983) · Zbl 0535.49010
[19] Simon, J.: Compact sets in the space $Lp(0, T; B)$. Ann. mat. Pura appl. 146, 65-96 (1987)
[20] Boccardo, L.; Murat, F.; Puel, J. -P.: Existence results for some quasilinear parabolic equations. Nonlinear analysis 13, 373-392 (1989) · Zbl 0705.35066