[Paper Review] Parabolic equations with variably partially VMO coefficients

Hongjie Dong|ArXiv.org|Nov 25, 2008

Advanced Mathematical Modeling in Engineering8 references3 citations

TL;DR

This paper establishes the $W^{1,2}_{p}$-solvability of second-order parabolic equations in nondivergence form for all $p \in (1,\infty)$, under the assumption that the leading coefficients are measurable in one spatial direction and have vanishing mean oscillation (VMO) in the remaining variables and time, with the direction of measurability varying per cylinder. The key advance is removing the prior restriction $p > 2$ by introducing a change of variables to transform the equation into divergence form, enabling the use of recent results on divergence-form equations with partially VMO coefficients.

ABSTRACT

We prove the $W^{1,2}_{p}$-solvability of second order parabolic equations in nondivergence form in the whole space for $p\in (1,\infty)$. The leading coefficients are assumed to be measurable in one spatial direction and have vanishing mean oscillation (VMO) in the orthogonal directions and the time variable in each small parabolic cylinder with the direction depending on the cylinder. This extends a recent result by Krylov [17] for elliptic equations and removes the restriction that $p>2$.

Motivation & Objective

• To extend the $W^{1,2}_p$-solvability theory for nondivergence form parabolic equations to all $p \in (1,\infty)$, including $p \leq 2$, under partially VMO coefficient assumptions.
• To resolve the longstanding restriction $p > 2$ in prior results on equations with partially VMO coefficients, which arose from limitations in sharp function estimates based on $L^2$-theory.
• To generalize the recent result by Krylov on elliptic equations with variably partially VMO coefficients to the parabolic setting, including mixed-norm Sobolev spaces.
• To establish solvability in mixed-norm spaces $W^{1,2}_{q,p}$ for $q \geq p$ under the same coefficient assumptions.

Proposed method

• Introduce a change of variables to transform the nondivergence form parabolic equation with coefficients depending only on one spatial variable into a divergence form equation.
• Apply a recent result on divergence-form parabolic equations with partially VMO coefficients to obtain $W^{1,2}_p$-solvability for coefficients depending solely on one spatial variable.
• Use pointwise sharp function estimates and generalized Fefferman-Stein theorems to control the oscillation of second-order derivatives in small parabolic cylinders.
• Employ maximal function estimates and dyadic decomposition techniques to handle the mixed-norm $L_{q,p}$-spaces, particularly for $q \geq p$, by controlling the $L^p$-norms of derivatives via maximal functions of lower-order terms.
• Construct a partition of unity and use localized maximal functions to control the $L^p$-norms of $D^2u$ and $u_t$ in terms of $Pu$, $Du$, and $f$, leveraging the VMO condition on coefficients.

Experimental results

Research questions

• RQ1 Can the $W^{1,2}_p$-solvability of nondivergence form parabolic equations be extended to all $p \in (1,\infty)$ when the leading coefficients are measurable in one spatial direction and VMO in the remaining variables and time?
• RQ2 Is the restriction $p > 2$ in prior results on partially VMO coefficients removable through a different analytical approach?
• RQ3 Can the solvability theory be extended to mixed-norm Sobolev spaces $W^{1,2}_{q,p}$ for $q \geq p$ under the same coefficient assumptions?
• RQ4 Does the use of a change of variables to convert nondivergence to divergence form enable $L^p$-estimates for $p < 2$ in the context of partially VMO coefficients?

Key findings

• The paper establishes $W^{1,2}_p$-solvability for all $p \in (1,\infty)$ for parabolic equations in nondivergence form with leading coefficients measurable in one spatial variable and VMO in the remaining variables and time, with the direction of measurability varying per parabolic cylinder.
• The restriction $p > 2$ is removed by transforming the equation into divergence form via a change of variables when coefficients depend only on one spatial variable, allowing the use of $L^p$-theory for divergence-form equations.
• The result extends Krylov’s recent work on elliptic equations with variably partially VMO coefficients to the parabolic setting, confirming the same solvability holds for all $p \in (1,\infty)$.
• The paper proves $W^{1,2}_{q,p}$-solvability for $q \geq p$ under the same coefficient assumptions, with the estimate $\|u_t\|_{L_{q,p}} + \|D^2u\|_{L_{q,p}} \leq N\|Pu\|_{L_{q,p}} + N\|Du\|_{L_{q,p}}$ holding for $1 < p < q < \infty$, valid for sufficiently small $R$ and large $\kappa$.
• The proof relies on sharp function estimates and maximal function bounds, with constants depending only on $d$, $p$, $q$, and the ellipticity constant $\delta$, and the final estimate is achieved by choosing $\kappa$ large and $\gamma$, $R$ small enough to absorb the error terms.