Subsolution-supersolution method in variational inequalities. (English) Zbl 1040.49008

Le, Vy Khoi

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 45, No. 6, 775-800 (2001).

The subsolution-supersolution method for equations is extended to a class of elliptic variational inequalities of the type \[ \int_\Omega A(x,\nabla u) \cdot(\nabla v-\nabla u)\;dx\geq \int_\Omega F(x,u)(v-u)\;dx \] \(\forall v\in K, \;K\subset W^{1,p}(\Omega),\) closed convex. Under additional assumptions on \(K\), the author proves the existence of a maximal and a minimal solution. The proofs rely mainly on the lattice structure of the Sobolev space \(W^{1,p}(\Omega)\). The obtained results are also of interest in view of the fact that the solution set of variational inequalities is in general more complicated than the solution set of equations.

Reviewer: Erich Miersemann (Leipzig)

MSC:

49J40	Variational inequalities
35J85	Unilateral problems; variational inequalities (elliptic type) (MSC2000)
49K20	Optimality conditions for problems involving partial differential equations

Keywords:

subsolution; supersolution; variational inequalities

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