[Paper Review] Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients

Н. В. Крылов|arXiv (Cornell University)|Jan 22, 2012

Advanced Harmonic Analysis Research1 references3 citations

TL;DR

This paper establishes $L_p$-estimates for second-order elliptic and parabolic equations with measurable coefficients, proving Hölder-type bounds on second-order derivatives and resolvent operator estimates in $L_p$ spaces. It provides rigorous proofs for key results used in fully nonlinear equations, relying on Krylov-Safonov theory and probabilistic methods, with applications to VMO coefficients and $L_q$-drift conditions.

ABSTRACT

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order equations in \cite{DKL}. The first type is an estimate of the $γ$th norm of the second-order derivatives, where $γ\in(0,1)$, and the second type deals with estimates of the resolvent operators in $L_{p}$ when the first-order coefficients are summable to an appropriate power.

Motivation & Objective

• To establish $L_p$-estimates for second-order elliptic and parabolic operators with measurable coefficients, particularly in the context of fully nonlinear equations.
• To provide a complete proof of $L_p$-estimates for $|D^2u|^\gamma$ with $\gamma \in (0,1)$, filling a gap in prior literature where the result was cited without proof.
• To derive uniform resolvent estimates in $L_p$ for operators with $L_q$-drift coefficients, extending results used in the theory of fully nonlinear equations.
• To generalize and rigorously prove estimates involving the resolvent operator $R_\mu = (\mu - L)^{-1}$, with bounds independent of $\mu$ for large $\mu$.
• To present new nontrivial estimates, such as Lemma 1.6 and its probabilistic counterpart Theorem 3.1, which support the main results.

Proposed method

• Uses the Krylov-Safonov Harnack inequality and its version from [8], derived from [6], to control the oscillation and integrability of solutions.
• Applies the maximum principle and Alexandrov estimate to control the $L_d$-norm of the resolvent operator $R_\mu$.
• Employs mollification techniques and $\lambda$-convex functions $\psi_\lambda$ to construct subsolutions satisfying $L\psi^\varepsilon_\lambda - \mu\psi^\varepsilon_\lambda \leq -f^\varepsilon + (|b|\sqrt{\lambda} + \lambda\,{\rm tr}\,a - \mu)\psi^\varepsilon_\lambda$, enabling norm estimates.
• Uses the Marcinkiewicz interpolation theorem to extend $L_p$-estimates from $p = d$ to general $p \geq d$, ensuring uniform bounds.
• Applies Fatou’s lemma and approximation arguments to pass from smooth to general $L_p$ data, preserving norm inequalities.
• Relies on probabilistic counterparts such as occupation measures and fundamental solutions to derive $L_1$-norm estimates for the resolvent, showing the necessity of dichotomous behavior in $\mu$-dependence.

Experimental results

Research questions

• RQ1 Can $L_p$-estimates for $|D^2u|^\gamma$ with $\gamma \in (0,1)$ be rigorously established for elliptic and parabolic operators with measurable coefficients, especially when $b$ and $c$ are bounded?
• RQ2 What conditions on the drift coefficient $b$ allow for uniform resolvent estimates $\|R_\mu g\|_{L_p} \leq N\|g\|_{L_p}$ independent of $\mu$ for large $\mu$?
• RQ3 How can the $L_p$-norm of the resolvent operator $R_\mu = (\mu - L)^{-1}$ be bounded uniformly in $\mu$ when $b \in L_q$ for $q \leq p$?
• RQ4 What is the precise dependence of the constant $N$ in the resolvent estimate on the parameters $\delta$, $K$, $d$, and $\mu$?
• RQ5 Can the dichotomy in the $\mu$-dependence of the resolvent norm be shown to be unavoidable, and what does it imply for the solvability of parabolic equations?

Key findings

• For any $L \in \mathfrak{L}_{\delta,K}$, there exist constants $\gamma \in (0,1]$ and $N$ depending only on $\delta$, $K$, and $d$ such that $\int_{C_{1,1}(1,0)} |D^2u|^\gamma \,dxdt \leq N\left(\int_{C_{2,1}} |Lu|^{d+1} \,dxdt\right)^{\gamma/(d+1)} + N\sup_{\partial'C_{2,1}} |u|^\gamma$, proving Hölder regularity of second derivatives.
• For $p = d$, the resolvent norm satisfies $\|R_\mu\|_{L_d \to L_d} \leq N\lambda^{-1}$ for $\mu \geq K\lambda + \mu_\theta\lambda^{1/2}$, with $\theta$ chosen so that $N'\theta \leq 1/2$, ensuring uniform boundedness.
• The estimate $\|R_\mu f\|_{L_d} \leq N\lambda^{-1}\|f\|_{L_d}$ holds for all nonnegative $f \in L_d$, derived via subsolution construction and maximum principle.
• For general $p \geq d$, the Marcinkiewicz interpolation theorem extends the $L_p$-bound to $\|R_\mu g\|_{L_p} \leq N\|g\|_{L_p}$ with $N$ independent of $\mu$ for large $\mu$, under $b \in L_q$ with $q \leq p$.
• The $L_1$-norm of the fundamental solution with pole at the origin is $1/\nu^2 = [\sqrt{M^2 + 4\mu} + M]^2 / (4\mu^2)$, showing that the resolvent norm cannot be bounded uniformly in $\mu$ without a dichotomous dependence.
• The dichotomy in the $\mu$-dependence of the resolvent norm is unavoidable and matches the form in Corollary 5.5, confirming sharpness of the estimates.