[Paper Review] Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces

Hongjie Dong, Doyoon Kim+1 more|arXiv (Cornell University)|Feb 19, 2015

Advanced Mathematical Modeling in Engineering10 references3 citations

TL;DR

This paper establishes $L_p$ estimates and unique solvability in weighted Sobolev spaces for non-divergence form elliptic and parabolic equations with Neumann boundary conditions, where the leading coefficients are measurable in time and have small mean oscillations in space (BMO$_x$) with respect to a weighted measure. The key result is an $L_p$ estimate with a parameter $\lambda$ that ensures solvability for large $\lambda$, extending Krylov's approach to weighted spaces and BMO$_x$ coefficients with applications to SPDEs.

ABSTRACT

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for {\em stochastic} partial differential equations with BMO$_x$ coefficients.

Motivation & Objective

• To establish $L_p$ estimates and unique solvability for non-divergence form elliptic and parabolic equations with Neumann boundary conditions in weighted Sobolev spaces.
• To extend Krylov's $L_p$ theory for BMO$_x$ coefficients to the Neumann problem setting with weighted measures.
• To handle leading coefficients that are measurable in time and have small mean oscillations in spatial variables with respect to a weighted measure $x_1^{\theta-d}$.
• To provide a framework applicable to stochastic partial differential equations with BMO$_x$ coefficients.
• To generalize prior results on Dirichlet problems by allowing Neumann boundary conditions and broader coefficient classes.

Proposed method

• Adopt a perturbation argument based on mean oscillation estimates for equations with coefficients only measurable in time.
• Use the Fefferman–Stein sharp maximal function theorem and Hardy–Littlewood maximal function theorem in weighted $L_p$ spaces to derive $L_p$ estimates.
• Treat the second-order derivatives $D^2u$ in two parts: $D_1u$ via a divergence-type equation after differentiation, and $D_{x'}^2u$ via separate oscillation estimates in $x_1$ and $x'$ directions.
• Apply a scaling and compactness argument to reduce the problem to a localized setting with $R_0 = 0$ for the time-independent coefficient case.
• Introduce a cut-off and oscillation-based function $\hat{u}(t,z) = u(t,x)\zeta(y)\cos(\sqrt{\lambda}y)$ to reduce the problem to a higher-dimensional setting with periodic structure.
• Use a bootstrapping argument to absorb lower-order terms and obtain the final estimate with $\lambda\|u\|_{p,\theta} \leq C_0\|f\|_{p,\theta}$ for large $\lambda$.

Experimental results

Research questions

• RQ1 Can $L_p$ estimates be established for non-divergence form parabolic and elliptic equations with Neumann boundary conditions when the leading coefficients are measurable in time and have small mean oscillations in space (BMO$_x$)?
• RQ2 What is the sharp range of the weight parameter $\theta$ for which the weighted Sobolev space theory remains valid in the Neumann setting?
• RQ3 How can the Krylov-type mean oscillation approach be adapted to the Neumann problem with non-divergence form equations?
• RQ4 Can the solvability theory be extended to include coefficients with spatial BMO$_x$ regularity and time measurability, beyond the VMO or Dini-continuous classes?
• RQ5 What is the role of the weight $x_1^{\theta-d}$ in ensuring the boundedness of the solution operator in weighted $L_p$ norms?

Key findings

• The paper proves $L_p$ estimates of the form $\lambda\|u\|_{p,\theta} \leq C_0\|f\|_{p,\theta}$ for all $\lambda$ sufficiently large, with $C_0$ depending only on $d$, $\delta$, $p$, $\theta$, and $R_0$.
• The unique solvability of the Neumann problem is established in weighted Sobolev spaces $H_{p,\theta}^2$ and $\mathbb{H}_{p,\theta}^2$ for $\theta \in (d-1, d-1+p)$, which is sharp even for the heat equation.
• The method successfully handles coefficients that are measurable in time and have small mean oscillations in space with respect to the weighted measure $x_1^{\theta-d}dx$, extending previous results to the Neumann case.
• The proof relies on a novel reduction using a periodic cut-off function in an auxiliary variable, allowing the use of higher-dimensional $L_p$ estimates.
• The result applies to stochastic PDEs with BMO$_x$ coefficients, providing a theoretical foundation for SPDEs with irregular coefficients.
• The estimate is robust under lower-order perturbations: the presence of $b_i$ and $c$ terms does not break the estimate as long as their norms are bounded and $\lambda$ is chosen large enough.