Existence of localizable solutions to nonlinear parabolic problems with irregular obstacles
In this work, the author treats parabolic obstacle problems of the form \[ \partial_{t} u - \operatorname{div} (\cdot, Du) = f - \operatorname{div}\bigl( a(| F |^{p-2} F)\bigr) \] on \(\Omega_T = \Omega \times (0,T)\), where \(\Omega \subset \mathbb R^n\) is a bounded domain of dimension \(n \geq 2\), \(u \mapsto \operatorname{div}\left(a(\cdot, Du)\right)\) is of \(p\)-Laplace-type for some \(p>\frac{2n}{n+2}\) and the inhomogeneity is given by functions \(f \in L^{p'}(\Omega_T)\) and \(F \in L^p(\Omega_T, \mathbb{R}^n)\), where \(p'\) denotes the dual Hölder index. Further, the obstacle is assumed to be given by a function \(\psi : \Omega_T \to \mathbb R\) whose distributional derivative with respect to time is assumed to lie in \(L^{p'}(0,T; W^{-1,p'}(\Omega))\). For such equations, the author introduces the new concept of localizable solutions and then shows that for sufficiently regular domains there exists a unique localizable solution of the above obstacle problem which agrees with the solutions obtained from the weak formulation of the problem. Further weaker results are also obtained for arbitrary bounded domains. As an application, Calderón-Zygmund estimates for the spatial gradient of localizable solutions are established.
nonlinear parabolic variational inequalities
scientific article; zbMATH DE number 6387827
scientific article