Title: Remarks on nonlinear uniformly parabolic equations

Authors: M. G. Crandall, K. Fok, M. Kocan and A. Swiech

Issue: Volume 47 (1998), Issue 4, 1293-1326

Abstract: This paper provides a number of working tools for the discussion of fully nonlinear parabolic equations. These include: a full proof that the maximum principle which provides $L^{\infty}$ 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, and merely measurable forcing, 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 $(n + 2)/2 < p$.