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Editorials' note

Miroslav Fiedler, Pankaj Jain, Lars-Erik Persson (2009)

Czechoslovak Mathematical Journal

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 in generalized Orlicz-Musielak spaces

Piotr Gwiazda, Piotr Minakowski, Aneta Wróblewska-Kamińska (2012)

Open Mathematics

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.

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...

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