Minimal solutions of nonlinear parabolic problems with unilateral constraints
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Papalini, F
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Vercillo, S
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Papageorgiou, NS
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In this paper we consider a class of nonlinear parabolic variational inequalities, we assume that it has an upper solution and we look for the minimal solution bounded above by the given upper soltuion. Our approach uses truncation and penalization techniques, which lead us to an auxiliary closely related problem. This is transformed to an equivalent abstract subdifferential evolution equation, which we solve. We then show that these solutions also solve the original variational inequality and they are bounded from above by the upper solution. Finally, an application of Zorn's lemma gives us the desired minimal solution.
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