Regularity and separation for Grušin-type p-Laplace operators

Abstract

We analyze p-Laplace type operators with degenerate elliptic coefficients. This investigation includes Grušin-type p-Laplace operators. We describe a separation phenomenon in elliptic and parabolic p-Laplace type equations, which provide an illuminating illustration of simple jump discontinuities of the corresponding weak solutions. Interestingly validity of an isoperimetric inequality for the considered setting does not imply the continuity of weak solutions to elliptic equations. On the other hand, we can establish global $L^1$-$L^{\infty }$-regularization and decay estimates of every mild solution of the parabolic Grušin-type p-Laplace equation.