Convergence of Rothe's method for fully nonlinear parabolic equations

TL;DR

The paper analyzes the convergence of Rothe's method (backward-Euler in time) for fully nonlinear parabolic equations in the viscosity-solution framework: $u_t + F(D^2u, Du, u, x, t)=0$ with uniform ellipticity. By solving elliptic problems at each time step and forming a piecewise-linear in time interpolant, the authors prove the Rothe limit exists, is locally Lipschitz in time, Hölder continuous in space, and satisfies the equation in the viscosity sense, under a structural continuity condition on $F$ expressed via Pucci operators. The key contributions include establishing a priori bounds through Pucci inequalities, employing sup-/inf-convolutions and Aleksandrov–Jensen theory to manage regularity, and a viscosity-solution stability argument to pass to the limit. The results yield Lipschitz-in-time regularity for viscosity solutions of the fully nonlinear parabolic problem and provide a rigorous justifications for Rothe-type discretizations in this nonlinear, non-divergence form setting, with potential implications for numerical analysis and approximation schemes.

Abstract

Convergence of Rothe's method for the fully nonlinear parabolic equation u_t + F(D^2 u, Du, u, x, t) = 0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Holder in space, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.

Theorems & Definitions (14)

• Lemma 2.1: First Bound
• Theorem 2.2: First Lipschitz Bound
• Theorem 2.3: Pucci Inequalities
• Theorem 2.4: Basic Sup-Convolution Properties
• Theorem 2.5: Sup-Convolutions are Subsolutions
• Remark 2.6
• Remark 2.7
• Theorem 2.8: Aleksandrov's Theorem
• Lemma 2.9: Jensen's Lemma
• Lemma 2.10: Discrete Gronwall Inequality