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Differentiability for minimizers of anisotropic integrals

Paola Cavaliere, Anna D'Ottavio, Francesco Leonetti, Maria Longobardi (1998)

Commentationes Mathematicae Universitatis Carolinae

We consider a function u : Ω N , Ω n , minimizing the integral Ω ( | D 1 u | 2 + + | D n - 1 u | 2 + | D n u | p ) d x , 2 ( n + 1 ) / ( n + 3 ) p < 2 , where D i u = u / x i , or some more general functional with the same behaviour; we prove the existence of second weak derivatives D ( D 1 u ) , , D ( D n - 1 u ) L 2 and D ( D n u ) L p .

Diffusion and cross-diffusion in pattern formation

Wei-Ming Ni (2004)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We discuss the stability and instability properties of steady state solutions to single equations, shadow systems, as well as 2 × 2 systems. Our basic observation is that the more complicated the pattern are, the more unstable they tend to be.

Diffusion and propagation problems in some ramified domains with a fractal boundary

Yves Achdou, Christophe Sabot, Nicoletta Tchou (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of 2 with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous...

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