Gamma limits in some Bernoulli free boundary problem

摘要

We study the limit cases p → ∞ and p → 1 of the functionals (1) $$ E_p (u): = intlimits_{mathbb{R}^n } {left{ {left. {frac{1} {p}left( {frac{{|nabla u|}} {a}} right)^p + frac{{p - 1}} {p}chi { u > 0} } right}} right.dx,} $$ where u ≡ 1 on a given compact set $K subset mathbb{R}^n ,$ and a > 0 is also given. Minimizers u p of these functionals have uniformly bounded support Ω p := {u p > 0} and satisfy (2) $$ - Delta _p u_p = 0{text{ in }}Omega _p ,,u_p equiv 1{text{ on }}K,;|nabla u_p |{kern 1pt} = {kern 1pt} a{text{ on }}partial Omega _p . $$