Fundamental solutions for the anisotropic Laplacian: existence and a priori estimates

Date/heure
30 mai 2014
14:00 - 15:00

Oratrice ou orateur
Florica Cirstea

Catégorie d'évènement
Séminaire EDP, Analyse et Applications (Metz)

Résumé

Let $v a r O m e g a$ be a domain in ${m a t h b b R}^{n}$ with $n g e q 2$ and $0 i n v a r O m e g a$ . We study anisotropic elliptic equations such as $- s u m_{i = 1}^{n}, p a r t i a l_{x_{i}} (| p a r t i a l_{x_{i}} u |^{p_{i} - 2} p a r t i a l_{x_{i}} u) = d e l t a_{0}$ in $v a r O m e g a$ (with Dirac mass $d e l t a_{0}$ at zero), subject to $u = 0$ on $p a r t i a l v a r O m e g a$ . We assume that all $p_{i}$ are in $(1, i n f t y)$ with their harmonic mean $p$ satisfying either Case 1: $p < n$ and $m a x_{1 l e q i l e q n} p_{i} < f r a c p (n - 1) n - p$ or Case 2: $p = n$ and $v a r O m e g a$ is bounded. We introduce a suitable notion of fundamental solution and establish its existence, together with sharp pointwise upper bound estimates near the origin for the solution and its derivatives. The latter is based on a Moser-type iteration scheme specific to each case, which is intricate due to our anisotropic analogue of the reverse H"older inequality. This is joint work with Jérôme Vétois (University of Nice)."