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Nonlinear elliptic and parabolic equations involving measure data. (English) Zbl 0707.35060
Let $\Omega$ be a nonempty bounded set in ${\bbfR}\sp N$. The authors prove the existence of solutions to $$(E)\quad Au=f\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega,$$ where $Au=-div(a(x,Du))$ with a: $\Omega\times {\bbfR}\sp N\to {\bbfR}\sp N$ is subject to certain coerciveness and monotonicity conditions and f is a bounded measure. This is done by first showing that (E) has a unique weak solution u in $W\sb 0\sp{1,p}(\Omega)$ for f in $W\sp{-1},p'(\Omega)$ and then obtaining estimates on u which depend only on $\Omega$, a and $\Vert f\Vert\sb{L\sp 1}$. Finally, f is approximated by a sequence in $W\sp{-1/p'}(\Omega)$. Extension to the equation $$Au+g(x,u)=f\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega$$ and a parabolic analog of (E) is also given.
Reviewer: P.K.Wong

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35K60 Nonlinear initial value problems for linear parabolic equations 35Dxx Generalized solutions of PDE 35B45 A priori estimates for solutions of PDE
##### Keywords:
measure data; quasilinear
Full Text:
##### References:
 [1] Stampacchia, G.: Equations elliptiques du second ordre à coefficients discontinus. Séminaire de mathématiques supérieures (1965) · Zbl 0151.15401 [2] Brezis, H.; Strauss, W.: Seminilear elliptic equations in L1. J. math. Soc. Japan 25, 565-590 (1973) [3] Benilan, Ph; Brezis, H.; Crandall, M.: A semilinear equation in L1. Ann. scuola norm. Sup. Pisa 2, 523-555 (1975) [4] Ph. Benilan, M. Crandall, and A. Pazy, Evolution equations governed by accretive operators, in preparation. · Zbl 0895.47036 [5] Gallouet, T.; Morel, J. M.: Resolution of a semilinear equation in L1. Proc. roy. Soc. Edinburgh sect. A 96, 275-288 (1984) · Zbl 0573.35030 [6] Gallouet, T.; Morel, J. M.: On some semilinear problems in L1. Boll. un. Mat. ital. A 4, 121-131 (1985) [7] Gallouet, T.: Equations elliptiques semilinéaires avec, pour la non linéarité, une condition de signe et une dépendance sous-quadratique par rapport au gradient. Ann. fac. Sc. univ. Toulouse 9, No. No. 2, 161-169 (1988) [8] Baras, P.; Pierre, M.: Critère d’existence de solutions positives pour des équations semi-linéaires non monotones. Ann. inst. H. Poincaré 3, 185-212 (1985) · Zbl 0599.35073 [9] Leray, J.; Lions, J. L.: Quelques résultats de visik sur LES problèmes elliptiques non linéaires par LES méthodes de minty et Browder. Bull. soc. Math. France 93, 97-107 (1965) · Zbl 0132.10502 [10] Serrin, J.: Pathological solutions of elliptic differential equations. Ann. scuola norm. Sup. Pisa, 385-387 (1964) · Zbl 0142.37601 [11] Brezis, H.: Some variational problems of the Thomas Fermi type. Variational inequalities, 53-73 (1980) [12] Brezis, H.; Veron, L.: Removable singularities for some nonlinear elliptic equations. Arch. rational mech. Anal. 75, 1-6 (1980) [13] Lions, J. L.: Quelques méthodes de rèsolution des problèmes aux limites non linéaires. (1968) [14] Simon, J.: Compact sets in the space $Lp(0, T; B)$. Ann. mat. Pura appl. 196, 65-96 (1987) [15] Temam, R.: Navier-Stokes equations. (1977) · Zbl 0383.35057 [16] Talenti, G.: Non-linear elliptic equations, rearrangements of functions and Orlicz sspaces. Ann. mat. Pura appl. 120, 159-184 (1979) · Zbl 0419.35041 [17] Kichenassamy, S.: Quasilinear problems with singularities. Manuscripta math. 57, 281-313 (1987) · Zbl 0595.35024