[Paper Review] Weighted mixed-norm $L_p$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem

Hongjie Dong, Tuoc Phan|arXiv (Cornell University)|Mar 14, 2021

Nonlinear Partial Differential Equations23 references4 citations

TL;DR

This paper establishes weighted mixed-norm $L_p$ estimates for non-divergence form elliptic and parabolic equations with singular coefficients in the upper half-space, under Dirichlet boundary conditions. It proves existence and uniqueness of strong solutions in intrinsic weighted Sobolev spaces, even when the weight $x_d^ u$ is outside the $A_p$-Muckenhoupt class, by controlling the combined term $D_d^2u + \alpha x_d^{-1}D_d u$ rather than $D_d^2u$ alone.

ABSTRACT

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the $A_p$-Muckenhoupt class.

Motivation & Objective

• To establish weighted mixed-norm $L_p$ estimates for non-divergence form elliptic and parabolic equations with singular first-order coefficients in the upper half-space.
• To prove existence and uniqueness of strong solutions under the Dirichlet boundary condition, even when the weight $x_d^\gamma$ is not in the $A_p$-Muckenhoupt class.
• To develop a theory in intrinsic weighted Sobolev spaces where the key estimate controls $D_d^2u + \alpha x_d^{-1}D_d u$ rather than $D_d^2u$ alone.
• To extend classical Calderón-Zygmund estimates to equations with singular coefficients, such as those arising in fractional Laplace and heat equations.
• To handle coefficients with small weighted mean oscillations in parabolic cylinders, ensuring solvability under minimal regularity assumptions.

Proposed method

• Use of intrinsic weighted Sobolev spaces with weights $x_d^\gamma$, where $\gamma \in (\alpha p - 1, 2p - 1)$, to control the singular behavior near the boundary.
• Adaptation of Calderón-Zygmund type arguments in the context of non-divergence form equations with singular coefficients.
• Introduction of a weighted maximal function and dyadic decomposition via cutoff functions $\eta_k$ to localize and estimate the solution in parabolic cylinders.
• Employment of a functional analytic approach via weighted $L_{q,p}$-norms and the use of a modified operator $\mathfrak{M} = x_d^{-1}$ to handle the singular terms.
• Application of a change of variables $u = x_d^\beta v$ to transform the original equation into a form amenable to known estimates, particularly when zeroth-order terms $b x_d^{-2} v$ are present.
• Use of a bootstrapping argument via summation over dyadic levels $k$, absorbing higher-order terms through careful choice of parameters $\lambda_k$ to achieve summable decay.

Experimental results

Research questions

• RQ1 Can weighted mixed-norm $L_p$ estimates be established for non-divergence form equations with singular coefficients in the half-space when the weight is outside the $A_p$-Muckenhoupt class?
• RQ2 How can one control the second-order derivative $D_d^2u$ in the presence of a singular $x_d^{-1}$ coefficient, especially when $D_d^2u$ alone is not integrable?
• RQ3 What conditions on the coefficients (e.g., small weighted mean oscillation) ensure existence and uniqueness of strong solutions in weighted Sobolev spaces?
• RQ4 To what extent can the classical Calderón-Zygmund theory be extended to equations with singular lower-order terms like $\alpha x_d^{-1} D_d u$?
• RQ5 Can the method be extended to include zeroth-order terms of the form $b x_d^{-2} u$, and if so, under what conditions on $b$ and $\alpha$?

Key findings

• The paper proves that for $\alpha \in (-\infty, 1)$, $p \in (1,\infty)$, and $\gamma \in (\alpha p - 1, 2p - 1)$, the Dirichlet problem for $\Delta u + \alpha x_d^{-1} D_d u - \lambda u = f$ admits a unique strong solution satisfying the weighted $L_p$ estimate with weight $x_d^\gamma$.
• The key estimate controls $|D_d^2 u + \alpha x_d^{-1} D_d u|^p$ rather than $|D_d^2 u|^p$, which is essential due to the singularity of the coefficient.
• The results are new even when coefficients are constant, and the weight $x_d^\gamma$ is not required to be in the $A_p$-Muckenhoupt class.
• The existence and uniqueness of solutions are established under smallness conditions on the weighted mean oscillations of the coefficients in parabolic cylinders.
• The method extends to equations with additional zeroth-order terms $b x_d^{-2} u$ via a transformation $u = x_d^\beta v$, allowing the use of Theorem 2.3 after choosing $\beta$ to eliminate the singular term.
• The proof relies on a dyadic decomposition and summation argument over $k$, where the parameter $\lambda_k = \lambda_0 \rho_0^{-2} + (5N 2^k)^2$ ensures decay of the higher-order terms, leading to a finite sum and absorption of the main term.