Remarks on nonlinear uniformly parabolic equations
Michael G. Crandall, Ken Fok, Maciej Kocan, Andrzej Święch · Indiana University Mathematics Journal · 1998
This paper provides a numberofworking tools for the discussion of fully nonlinear parabolic equations. These include: a proof that the maximum principle which provides L 1 estimates of "strong" solutions of extremal equations by L n+1 norms of the forcing term over the "contact set" remains valid for viscosity solutions in an L n+1 sense, a gradient estimate in L p for p##n+ 1##n + 2# for solutions of extremal equations with forcing terms in L n+1 ,the use of this estimate in improving the range of p for which the maximum principle first alluded to holds #obtaining some p#n+ 1 - but without the contact set#, a proof of the strong solvability of Dirichlet problems for extremal equations with forcing terms in L p for some p#n+1,and the twice parabolic differentiability a.e. of W 2;1;p functions for #n +2#=2#p. 0. Introduction. In this work we provide a number of tools for the discussion of nonlinear parabolic equations under appropriate structure conditions. In particular, we...