We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form
The function $ A = A(xi, X, Y, t): mathbb R^mtimes mathbb R^mtimes mathbb R^mtimes mathbb Rto mathbb R^m $ is assumed to be continuous with respect to $ xi $, and measurable with respect to $ X, Y $ and $ t $. $ A = A(xi, X, Y, t) $ is allowed to be nonlinear but with linear growth. We establish higher integrability and local boundedness of weak sub-solutions, weak Harnack and Harnack inequalities, and Hölder continuity with quantitative estimates. In addition we establish existence and uniqueness of weak solutions to a Dirichlet problem in certain bounded $ X $, $ Y $ and $ t $ dependent domains.
In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $ Omegasubsetmathbb{R}^d $ with time-dependent Dirichlet boundary condition for the temperature $ vartheta = vartheta(x, t) $, $ vartheta = g $ on $ Omega^ctimes]0, T[$, and initial condition $ eta_0 $ for the enthalpy $ eta = eta(x, t) $, given in $ Omegatimes]0, T[$ by
where $ mathcal{L}_A^s $ is an anisotropic fractional operator defined in the distributional sense by
$ beta $ is a maximal monotone graph, $ A(x) $ is a symmetric, strictly elliptic and uniformly bounded matrix, and $ D^s $ is the distributional Riesz fractional gradient for $ 0 < s < 1 $. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $ snearrow 1 $ towards the classical local problem, the asymptotic behaviour as $ ttoinfty $, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $ beta $.
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