A lower spatially Lipschitz bound for solutions to fully nonlinear parabolic equations and its optimality

Abstract

We derive a lower spatially Lipschitz bound for viscosity solutions to fully nonlinear parabolic partial differential equations when the initial datum belongs to the H(cid:127)older space. The resulting estimate depends on the initial H(cid:127)older exponent and the growth rates of the equation with respect to the (cid:12)rst and second order derivative terms. Our estimate is applicable to equations which are possibly singular at the initial time. Moreover, it gives the optimal rate of the regularizing effect for solutions, which occurs for some uniformly parabolic equations and (cid:12)rst order Hamilton{Jacobi equations. In the proof of our lower estimate, we construct a subsolution and a supersolution by optimally rescaling the solution of the heat equation and then compare them with the solution. For linear equations, the lower spatially Lipschitz bound for solutions can be obtained in a different way if the fundamental solution satis(cid:12)es the Aronson estimate. Examples include the heat convection equation whose convection term has singularities.