Behaviour in the Limit, as $p \to \infty $, of Minimizers of Functionals Involving p -Dirichlet Integrals
Ulf Janfalk · SIAM Journal on Mathematical Analysis · 1996
The purpose of this paper is to study the behaviour, as $p \to \infty $, of minimizers of functionals involving p-Dirichlet integrals in a bounded Lipschitz domain, $\Omega \subset {\bf R}^n $. In the case where $\Omega $ is a convex ring it is proved that the minimizers converge monotonically and uniformly. In the paper by T. Bhattacharya, E. DiBenedetto, and J. Manfredi [Limits as $p \to \infty $ of $\Delta _p u_p = f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino, (1989), pp. 15–68], the problem of torsional creep is studied. Here the situation is generalized by introducing a more general functional and relaxing the boundary conditions. Various aspects of the Green function of the p-laplacian are considered and it is proved that the Green function is not symmetric if p is sufficiently large. Finally, it is proved that the extremals to the dual problem tend to zero in the mean as $p \to \infty $, outside a well-specified subset of $\Omega $.