Hölder Regularity for Singular Parabolic Obstacle Problems of Porous Medium Type

Abstract

Abstract We study the regularity of weak solutions to parabolic obstacle problems related to equations of singular porous medium type that are modeled after the nonlinear equation $$\partial_{t} u - \Delta u^{m} = 0.$$For the range of exponents 0 < m < 1, we prove that locally bounded weak solutions are locally Hölder continuous, provided the obstacle function is. Moreover, in the case $\frac{(n-2)_{+}}{n+2} < m < 1$ we show that every weak solution is locally bounded and therefore Hölder continuous.