[Paper Review] Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

Soojung Kim, Ki-Ahm Lee|arXiv (Cornell University)|Apr 27, 2013

Nonlinear Partial Differential Equations13 references3 citations

TL;DR

This paper establishes the parabolic Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on Riemannian manifolds with sectional curvature bounded from below. By adapting the Krylov-Safonov approach using intrinsic geometric tools like the parabolic normal map and Pucci extremal operators, the authors prove a Harnack inequality under curvature and integrability conditions on the right-hand side, extending classical results to the viscosity setting on manifolds.

ABSTRACT

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.

Motivation & Objective

• To extend the Krylov-Safonov Harnack inequality to viscosity solutions of nonlinear uniformly parabolic equations in nondivergence form on Riemannian manifolds.
• To establish a Harnack inequality for nonnegative viscosity solutions under a lower bound on sectional curvature and integrability conditions on the source term.
• To generalize prior results on classical solutions to the viscosity solution framework on manifolds with curvature constraints.
• To develop a parabolic ABP-Krylov-Tso estimate using intrinsic geometric constructions such as the parabolic normal map on Riemannian manifolds.
• To prove the Harnack inequality for the elliptic counterpart of the parabolic equation under similar geometric and integrability assumptions.

Proposed method

• Adapt the Krylov-Safonov approach to Riemannian manifolds by replacing affine functions with squared distance functions to handle the lack of global affine structure.
• Define viscosity solutions on Riemannian manifolds using the framework of [AFS, Z], extending the theory of viscosity solutions to non-Euclidean settings.
• Employ Pucci’s extremal operators to characterize uniform ellipticity, replacing the trace operator and enabling treatment of nonlinearities.
• Use sup- and inf-convolution techniques to approximate viscosity solutions by smooth functions, preserving the PDE inequality in the limit.
• Construct the parabolic normal map $\Phi(x,t) := \left(\exp_x(\nabla_x u), -\frac{1}{2}d^2(x,\exp_x(\nabla u)) - u(x,t)\right)$ and compute its Jacobian determinant to derive the ABP-type estimate.
• Apply a covering argument based on Bishop and Gromov's volume comparison theorem to control measure growth and derive the Harnack inequality.

Experimental results

Research questions

• RQ1 Can the Krylov-Safonov Harnack inequality be extended to viscosity solutions of nonlinear uniformly parabolic equations on Riemannian manifolds with bounded sectional curvature?
• RQ2 How can the classical ABP estimate and its parabolic analogue be generalized to Riemannian manifolds without relying on affine functions?
• RQ3 What role does the intrinsic parabolic normal map play in deriving Harnack-type estimates on curved manifolds?
• RQ4 Under what conditions on the source term $f$ and curvature does the Harnack inequality for viscosity solutions remain valid?
• RQ5 To what extent can the elliptic Harnack inequality for viscosity solutions be established under the same geometric and integrability assumptions?

Key findings

• The paper establishes a parabolic Harnack inequality for nonnegative viscosity solutions to uniformly parabolic equations in nondivergence form on Riemannian manifolds with sectional curvature bounded from below by $-\kappa$.
• A key result is that if $\left(\fint_{B_{2R}(x_0)} |R^2 f^+|^{n\theta}\right)^{1/(n\theta)} \leq \epsilon_0$ with $\theta = 1 + \log_2 \cosh(8\sqrt{\kappa}R_0)$, then the Harnack inequality holds with a positive density lower bound $\mu_0 > 0$.
• The Harnack inequality is proven via a parabolic ABP-Krylov-Tso estimate using the intrinsic parabolic normal map and its Jacobian determinant.
• The authors show that the viscosity solution class $\overline{\mathcal{S}}_P(\lambda,\Lambda,f)$ satisfies the Harnack inequality under the stated curvature and integrability conditions.
• The elliptic Harnack inequality for viscosity solutions is also established under the same geometric assumptions, using a similar argument based on volume comparison and covering techniques.
• The proof relies on approximating viscosity solutions by smooth functions via inf-convolution and applying a uniform Harnack estimate to the approximations, then passing to the limit.