Optimal regularity estimates for general nonlinear parabolic equations
We develop a global Calderón–Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution u and its spatial gradient Du in a nonsmooth domain. The nonlinearity behaves as the parabolic p-Laplacian in Du, its discontinuity with respect to the independent variables is measured in terms of small-BMO, while only Hölder continuity is required with respect to u and the underlying domain is assumed to be \(\delta \)-Reifenberg flat. We introduce and employ essentially a new concept of the intrinsic parabolic maximal function in order to overcome the main difficulties stemming from both the parabolic scaling deficiency and the nonlinearity of u-variable of such a very general parabolic operator, obtaining optimal \(L^q\)-estimates for the spatial gradient under a minimal geometric condition on the domain.
Mathematics Subject ClassificationPrimary: 35K59 35B65 Secondary: 35R05 42B37
S.-S. Byun was supported by NRF-2017R1A2B003877. D.K. Palagachev is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). P. Shin was supported by NRF-2015R1A4A1041675.