[Paper Review] A Quantitative Regularity Estimate for Nonnegative Supersolutions of Fully Nonlinear Uniformly Parabolic Equations

Jessica Lin|arXiv (Cornell University)|May 31, 2013

Advanced Mathematical Modeling in Engineering13 references3 citations

TL;DR

This paper establishes a quantitative lower bound for nonnegative viscosity supersolutions of fully nonlinear uniformly parabolic equations, using the Fabes-Stroock estimate to derive a parabolic analogue of the elliptic regularity result by Caffarelli, Souganidis, and Wang. The key contribution is a precise lower bound involving the $L^{N+1}$-norm of the right-hand side, with exponential decay in the log of the norm, valid in a positive fraction of the domain.

ABSTRACT

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a nonlinear, quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of the Fabes-Stroock estimate. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.

Motivation & Objective

• To establish a quantitative interior lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations.
• To provide a parabolic analogue of the elliptic regularity estimate by Caffarelli, Souganidis, and Wang for stochastic homogenization.
• To extend the lower bound estimate to linear nondivergence form equations with bounded measurable coefficients, which is new in this context.
• To recover the elliptic result in the limit as $t \to \infty$, demonstrating consistency across settings.
• To demonstrate that the lower bound exhibits a power-logarithmic decay structure in the $L^{N+1}$-norm of the source term.

Proposed method

• Apply the Fabes-Stroock estimate for parabolic cylinders to control the heat kernel and derive intrinsic lower bounds.
• Use viscosity solution theory and Pucci's extremal operators $\mathcal{M}^{-}$ and $\mathcal{M}^{+}$ to linearize the fully nonlinear equation.
• Construct a barrier function $w(x,t)$ via iterative scaling in dyadic parabolic cylinders to propagate lower bounds.
• Employ a tower of cylinders with controlled height and radius to ensure the lower bound propagates into the interior domain.
• Derive estimates for the infimum of $u$ in terms of $\|f + F(0,\cdot,\cdot)\|_{L^{N+1}(Q_1)}$, incorporating logarithmic corrections.
• Use comparison principles and the structure of the parabolic boundary to control the decay rate of the lower bound.

Experimental results

Research questions

• RQ1 Can a quantitative lower bound be established for nonnegative supersolutions of fully nonlinear uniformly parabolic equations, analogous to the elliptic result of Caffarelli, Souganidis, and Wang?
• RQ2 How does the lower bound depend on the $L^{N+1}$-norm of the source term $f$ in the parabolic setting?
• RQ3 What is the precise dependence of the lower bound on the ellipticity constants $\lambda, \Lambda$, dimension $N$, and the distance to the boundary?
• RQ4 Can the parabolic result be used to recover the known elliptic regularity estimate in the limit as $t \to \infty$?
• RQ5 Does the method yield a nontrivial lower bound for linear uniformly parabolic equations with bounded measurable coefficients?

Key findings

• A quantitative lower bound is established: for $|x| \leq \kappa$ and $t \in [-\kappa/2 \|f + F(0,\cdot,\cdot)\|_{L^{N+1}}^{N+1}, 0]$, the solution satisfies $u(x,t) \geq c \|f + F(0,\cdot,\cdot)\|_{L^{N+1}}^{\rho} \exp(-\beta \|f + F(0,\cdot,\cdot)\|_{L^{N+1}}^{-2(N+1)})$.
• The lower bound is valid in a positive fraction of the original domain, specifically for $t$ in a time interval proportional to $\|f\|_{L^{N+1}}^{N+1}$.
• The exponent $\rho$ and the constant $\beta$ depend only on $\lambda, \Lambda, N, \kappa$, and are independent of $f$.
• The result is new for linear nondivergence form equations with bounded measurable coefficients, extending known results to the parabolic case.
• The elliptic result of Caffarelli, Souganidis, and Wang is recovered in the limit as $t \to \infty$, confirming consistency with the elliptic theory.
• Corollary 4.3 provides a power-type decay bound when the set $\{f > \alpha\}$ is strictly interior, yielding $u(x,t) \geq c m^{\beta}$ with $m = |\Gamma|/|Q_1|$.