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

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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.