## 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:

 [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 $$L^p$$-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, (Benilan, P.; Chipot, M.; Evans, L. C.; Pierre, M., Recent Advances in Nonlinear Elliptic and Parabolic Problems. Recent Advances in Nonlinear Elliptic and Parabolic Problems, Pitman Research Notes in Mathematics, Vol. 208 (1989), Longman: Longman Harlow), 247-254 · 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 [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), Dunod & Gauthier-Villars: Dunod & Gauthier-Villars Paris · Zbl 0189.40603 [14] Cioranescu, D.; Murat, F., Un terme étrange venu d’ailleurs, (Brezis, H.; Lions, J.-L., Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar. Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Research Notes in Mathematics, Vols 60 & 70 (1982), Pitman: Pitman London). (Brezis, H.; Lions, J.-L., Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar. Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Research Notes in Mathematics, Vols 60 & 70 (1982), Pitman: Pitman London), 154-178, Vols 2 & 3 · Zbl 0496.35030 [16] Browder, F. E., Existence theorems for nonlinear partial differential equations, (Chern, S. S.; Smale, S., Proc. Symp. Pure Math. (1970), American Mathematical Society: American Mathematical Society Providence, Rhode Island), 1-60 · Zbl 0212.27704 [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 $$L^p}(0, T; B)$$, Ann. Mat. pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 [20] Boccardo, L.; Murat, F.; Puel, J.-P., Existence results for some quasilinear parabolic equations, Nonlinear Analysis, 13, 373-392 (1989) · Zbl 0705.35066
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