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Multiple solutions for nonlinear discontinuous elliptic problems near resonance

Nikolaos KourogenisNikolaos Papageorgiou — 1999

Colloquium Mathematicae

We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when λ λ 1 from the left, λ 1 being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.

Discontinuous quasilinear elliptic problems at resonance

Nikolaos KourogenisNikolaos Papageorgiou — 1998

Colloquium Mathematicae

In this paper we study a quasilinear resonant problem with discontinuous right hand side. To develop an existence theory we pass to a multivalued version of the problem, by filling in the gaps at the discontinuity points. We prove the existence of a nontrivial solution using a variational approach based on the critical point theory of nonsmooth locally Lipschitz functionals.

Existence and multiplicity results for nonlinear eigenvalue problems with discontinuities

Nikolaos PapageorgiouFrancesca Papalini — 2000

Annales Polonici Mathematici

We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations. In order to study the existence of solutions we replace these two problems with their multivalued approximations and, for the first problem, we estabilish an existence result while for the second problem we prove the existence of multiple nontrivial solutions. The approach used is variational.

Quasilinear vector differential equations with maximal monotone terms and nonlinear boundary conditions

Ralf BaderNikolaos Papageorgiou — 2000

Annales Polonici Mathematici

We consider a quasilinear vector differential equation which involves the p-Laplacian and a maximal monotone map. The boundary conditions are nonlinear and are determined by a generally multivalued, maximal monotone map. We prove two existence theorems. The first assumes that the maximal monotone map involved is everywhere defined and in the second we drop this requirement at the expense of strengthening the growth hypothesis on the vector field. The proofs are based on the theory of operators of...

On nonlinear, nonconvex evolution inclusions

Nikolaos S. Papageorgiou — 1995

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider a nonlinear evolution inclusion driven by an m-accretive operator which generates an equicontinuous nonlinear semigroup of contractions. We establish the existence of extremal integral solutions and we show that they form a dense, G δ -subset of the solution set of the original Cauchy problem. As an application, we obtain “bang-bang”’ type theorems for two nonlinear parabolic distributed parameter control systems.

Topological properties of the solution set of a class of nonlinear evolutions inclusions

Nikolaos S. Papageorgiou — 1997

Czechoslovak Mathematical Journal

In the paper we study the topological structure of the solution set of a class of nonlinear evolution inclusions. First we show that it is nonempty and compact in certain function spaces and that it depends in an upper semicontinuous way on the initial condition. Then by strengthening the hypothesis on the orientor field F ( t , x ) , we are able to show that the solution set is in fact an R δ -set. Finally some applications to infinite dimensional control systems are also presented.

Minimax control of nonlinear evolution equations

Nikolaos S. Papageorgiou — 1995

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the minimax control of systems governed by a nonlinear evolution inclusion of the subdifferential type. Using some continuity and lower semicontinuity results for the solution map and the cost functional respectively, we are able to establish the existence of an optimal control. The abstract results are then applied to obstacle problems, semilinear systems with weakly varying coefficients (e.gȯscillating coefficients) and differential variational inequalities.

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