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A global differentiability result for solutions of nonlinear elliptic problems with controlled growths

Luisa Fattorusso (2008)

Czechoslovak Mathematical Journal

Let Ω be a bounded open subset of n , n > 2 . In Ω we deduce the global differentiability result u H 2 ( Ω , N ) for the solutions u H 1 ( Ω , n ) of the Dirichlet problem u - g H 0 1 ( Ω , N ) , - i D i a i ( x , u , D u ) = B 0 ( x , u , D u ) with controlled growth and nonlinearity q = 2 . The result was obtained by first extending the interior differentiability result near the boundary and then proving the global differentiability result making use of a covering procedure.

A Hölder infinity Laplacian

Antonin Chambolle, Erik Lindgren, Régis Monneau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

A Hölder infinity Laplacian

Antonin Chambolle, Erik Lindgren, Régis Monneau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study the limit as p → ∞ of minimizers of the fractional Ws,p-norms. In particular, we prove that the limit satisfies a non-local and non-linear equation. We also prove the existence and uniqueness of solutions of the equation. Furthermore, we prove the existence of solutions in general for the corresponding inhomogeneous equation. By making strong use of the barriers in this construction, we obtain some regularity results.

A la recherche du spectre perdu: An invitation to nonlinear spectral theory

Appell, Jürgen (2003)

Nonlinear Analysis, Function Spaces and Applications

We give a survey on spectra for various classes of nonlinear operators, with a particular emphasis on a comparison of their advantages and drawbacks. Here the most useful spectra are the asymptotic spectrum by M. Furi, M. Martelli and A. Vignoli (1978), the global spectrum by W. Feng (1997), and the local spectrum (called “phantom”) by P. Santucci and M. Väth (2000). In the last part we discuss these spectra for homogeneous operators (of any degree), and derive a discreteness result and a nonlinear...

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