Generalizations of Stampacchia Lemma and applications to quasilinear elliptic systems

Abstract

Abstract We present two generalizations of the classical Stampacchia Lemma. As applications, we consider Dirichlet problem for elliptic systems of the form − ∑ i = 1 n ∂ ∂ x i ∑ β = 1 N ∑ j = 1 n a i , j α , β ( x , u ( x ) ) ∂ u β ( x ) ∂ x j = f α , in\xa0 Ω , u ( x ) = 0 , on\xa0 ∂ Ω . Under ellipticity and degenerate ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive some global regularity results.