Nonlinear elliptic and parabolic equations involving measure data. (English) Zbl 0707.35060

Boccardo, Lucio; Gallouët, Thierry

J. Funct. Anal. 87, No. 1, 149-169 (1989).

Let \(\Omega\) be a nonempty bounded set in \({\mathbb{R}}^ N\). 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: \(\Omega\times {\mathbb{R}}^ N\to {\mathbb{R}}^ N\) 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 \(W_ 0^{1,p}(\Omega)\) for f in \(W^{-1},p'(\Omega)\) and then obtaining estimates on u which depend only on \(\Omega\), a and \(\| f\|_{L^ 1}\). Finally, f is approximated by a sequence in \(W^{-1/p'}(\Omega)\). 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, boundary and initial-boundary value problems for linear parabolic equations
35Dxx	Generalized solutions to partial differential equations
35B45	A priori estimates in context of PDEs

Keywords:

measure data; quasilinear

References:

[1]	Stampacchia, G, Equations elliptiques du second ordre à coefficients discontinus, () · Zbl 0151.15501
[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) · Zbl 0314.35077
[4]	{\scPh. Benilan, M. Crandall, and A. Pazy}, Evolution equations governed by accretive operators, in preparation. · Zbl 0635.34013
[5]	Gallouet, T; Morel, J.M, Resolution of a semilinear equation in L1, (), 275-288 · Zbl 0573.35030
[6]	Gallouet, T; Morel, J.M, On some semilinear problems in L1, Boll. un. mat. ital. A, 4, 121-131, (1985), (6) · Zbl 0585.35033
[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. 2, 161-169, (1988) · Zbl 0685.35041
[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, (), 53-73
[12]	Brezis, H; Veron, L, Removable singularities for some nonlinear elliptic equations, Arch. rational mech. anal., 75, 1-6, (1980) · Zbl 0459.35032
[13]	Lions, J.L, Quelques méthodes de rèsolution des problèmes aux limites non linéaires, (1968), Dunod Paris · Zbl 0189.40603
[14]	Simon, J, Compact sets in the space Lp(0, T; B), Ann. mat. pura appl., 196, 65-96, (1987) · Zbl 0629.46031
[15]	Temam, R, Navier-Stokes equations, (1977), North-Holland Amsterdam · Zbl 0335.35077
[16]	Talenti, G, Non-linear elliptic equations, rearrangements of functions and Orlicz sspaces, Ann. mat. pura appl., 120, 159-184, (1979), (4) · Zbl 0419.35041
[17]	Kichenassamy, S, Quasilinear problems with singularities, Manuscripta math., 57, 281-313, (1987) · Zbl 0595.35024