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Discontinuous quasilinear elliptic problems at resonance

Nikolaos Kourogenis, Nikolaos 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.

Dynamic von Kármán equations involving nonlinear damping: Time-periodic solutions

Eduard Feireisl (1989)

Aplikace matematiky

In the paper, time-periodic solutions to dynamic von Kármán equations are investigated. Assuming that there is a damping term in the equations we are able to show the existence of at least one solution to the problem. The Faedo-Galerkin method is used together with some basic ideas concerning monotone operators on Orlicz spaces.

Efficient inexact Newton-like methods with application to problems of the deformation theory of plasticity

Radim Blaheta, Roman Kohut (1993)

Applications of Mathematics

Newton-like methods are considered with inexact correction computed by some inner iterative method. Composite iterative methods of this type are applied to the solution of nonlinear systems arising from the solution of nonlinear elliptic boundary value problems. Two main quastions are studied in this paper: the convergence of the inexact Newton-like methods and the efficient control of accuracy in computation of the inexact correction. Numerical experiments show the efficiency of the suggested composite...

Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations

Martin A. Grepl, Yvon Maday, Ngoc C. Nguyen, Anthony T. Patera (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function...

Elliptic problems with integral diffusion

Yannick Sire (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

In this paper, we review several recent results dealing with elliptic equations with non local diffusion. More precisely, we investigate several problems involving the fractional laplacian. Finally, we present a conformally covariant operator and the associated singular and regular Yamabe problem.

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : − Δ u ε = u ε ( n + 2 ) / ( n − 2 ) , u ε > 0 in A ε , u ε = 0 on ∂ A ε , where A ε is an annulus-shaped domain of ℝ n , n ≥ 3 , which becomes thinner as ε → 0 . We show that for every solution u ε , the energy ∫ A ε | ∇ u | 2 as well as the Morse index tend to infinity as ε → 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ℝ n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Energy quantization and mean value inequalities for nonlinear boundary value problems

Katrin Wehrheim (2005)

Journal of the European Mathematical Society

We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...

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