zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Existence of bounded solutions for non linear elliptic unilateral problems. (English) Zbl 0687.35042
Let $\Omega$ be a bounded domain in ${\bbfR}\sp n$, $\psi$ a measurable function on $\Omega$, $p>1$, and $A(u)=div a(x,u,Du)+a\sb 0(x,u,Du)$ an elliptic quasilinear differential operator whose coefficients a, $a\sb 0$ satisfy natural regularity and growth conditions which in particular guarantee that A is a continuous, pseudomonotone operator from the Sobolev space $W\sb 0\sp{1,p}(\Omega)$ into its dual. The authors prove the existence of a solution $u\in W\sb 0\sp{1,p}(\Omega)\cap L\sp{\infty}(\Omega)$ of the variational inequality $u\ge \psi$, $<A(u),v-u>+\int H(x,u,Du)(v-u)dx\ge 0$ for all $v\in W\sb 0\sp{1,p}(\Omega)\cap L\sp{\infty}(\Omega)$ such that $v\ge \psi$. Here it is important to notice that the inhomogeneous term H is allowed to grow like $\vert Du\vert\sp p$. The proof is carried out by an approximation of H by bounded functions $H\sb{\epsilon}$ for which a solution $u\sb{\epsilon}$ of the corresponding problem is known to exist. Then it is shown that the family $u\sb{\epsilon}$ is compact in $W\sb 0\sp{1,p}(\Omega)$.
Reviewer: F.Tomi

35J85Unilateral problems; variational inequalities (elliptic type) (MSC2000)
47H05Monotone operators (with respect to duality) and generalizations
35J65Nonlinear boundary value problems for linear elliptic equations
35D05Existence of generalized solutions of PDE (MSC2000)
49J40Variational methods including variational inequalities
Full Text: DOI
[1] H. Amann -M. G. Crandall,On some existence theorems for semilinear elliptic equa- tions, Indiana Univ. Math. J.,27 (1978), pp. 779--790. · Zbl 0391.35030 · doi:10.1512/iumj.1978.27.27050
[2] L. Boccardo,An L s-estimate for the gradient of solutions of some nonlinear unilateral problems, Ann. Mat. Pura Appl.,141 (1985), pp. 277--287. · Zbl 0599.49009 · doi:10.1007/BF01763177
[3] L. Boccardo -F. Murat -J. P. Puel,Existence de solutions faibles pour des équations elliptiques quasilinéaires à croissance quadratique, in Konlinear partial differential equations and their applications, Collège de France Seminar, Vol. IV, ed. byH. Brezis andJ. L. Lions, Research Notes in Mathematics,84 Pitman, London, (1983), pp. 19--73. · Zbl 0588.35041
[4] L. Boccardo -F. Murat -J. P. Puel,Résultats d’existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa,11 (1984), pp. 213--235.
[5] A, Bensoussan -J. Frehse,Nonlinear elliptic systems in stochastic game theory, J. reine ang. Math.,350 (1984), pp. 23--67. · Zbl 0531.93052
[6] A. Bensoussan -J. Frehse -U. Mosco,A stochastic impulse control problem with quadratic growth Mamiltonian and the corresponding quasi-variational inequality, J. reine ang. Math.,331 (1982), pp. 124--145. · Zbl 0474.49013
[7] H. Brezis,Equations et inéquations non linéaires dans les espaces en dualité, Ann. Inst. Fourier,18 (1968), pp. 115--175.
[8] H. Brezis,Problèmes unilatéraux, J. Math. Pures et Appl.,51 (1972), pp. 1--168.
[9] F. E. Browder,Existence theorems for nonlinear partial differential equations, in Proceedings of Symposia in Pure Mathematics, Vol. XVI, ed. byS. S. Chern andS. Smale, American Mathematical Society, Providence, (1970), pp. 1--60. · Zbl 0211.17204
[10] P. Donato -D. Giachetti,Quasilinear elliptic equations with quadratic growth in unbounded domains, Nonlinear Anal. T.M.A.,10 (8) (1986), pp. 791--804. · Zbl 0602.35036 · doi:10.1016/0362-546X(86)90038-6
[11] P. Donato -D. Giachetti,Unilateral problems with quadratic growth in unbounded domains, Boll. Un. Mat. Ital., (6)5 A (1986), pp. 361--369. · Zbl 0602.49008
[12] D. Kinderlehrer -G. Stampacchia,An introduction to variational inequalities and their applications, Academic Press, New York, 1980. · Zbl 0457.35001
[13] P. Hartman -G. Stampacchia,On some nonlinear elliptic differential functional equations, Acta Math.,115 (1966), pp. 153--188. · Zbl 0142.38102 · doi:10.1007/BF02392210
[14] J. Leray -J. L. Lions,Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France,93 (1965), pp. 97--107. · Zbl 0132.10502
[15] J. L. Lions,Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier Villars, Paris, 1969.
[16] P. L. Lions,Résolution de problèmes elliptiques quasilinéaires, Arch. Rat. Mech. Anal.,74 (1980), pp. 336--353. · Zbl 0449.35036 · doi:10.1007/BF00249679
[17] P. L. Lions,Quelques remarques sur les problèmes elliptiques quasilinéaires du second ordre, J. Anal. Math.,45 (1985), pp. 234--254. · Zbl 0614.35034 · doi:10.1007/BF02792551
[18] J. M.Rakotoson - R.Temam,Relative rearrangement in quasilinear variational inequalities, to appear.