[Paper Review] Boundary value problems in Lipschitz domains for equations with drifts

Georgios Sakellaris|arXiv (Cornell University)|Jan 28, 2017

Advanced Mathematical Modeling in Engineering13 references3 citations

TL;DR

This paper establishes the unique solvability of the $D_2$ Dirichlet and $R_2$ Regularity problems for second-order elliptic operators with drift terms $L = -\operatorname{div}(A\nabla\cdot) + b\nabla\cdot$ in bounded Lipschitz domains, using harmonic measure and layer potential methods. Key results show that $D_2$ for $L^t$ and $R_2$ for $L$ are uniquely solvable under minimal assumptions ($A$ Lipschitz, $b$ bounded), while $R_2$ for $L^t$ requires stronger regularity on $A$ and $\operatorname{div}b$ in $L^d$.

ABSTRACT

In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{ m div}(A abla\cdot)+b abla\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{ m div}(A^t abla\cdot)-{ m div}(b\,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${ m div}b$ in order to obtain solvability. On the other hand, we show that $R_2$ for $L$ and $D_2$ for $L^t$ are uniquely solvable, assuming only that $A$ is Lipschitz continuous (and not necessarily symmetric) and $b$ is bounded.

Motivation & Objective

• To establish the unique solvability of the $D_2$ Dirichlet and $R_2$ Regularity problems for second-order elliptic operators with drift terms in bounded Lipschitz domains.
• To analyze the role of the drift term $b$ and its divergence $\operatorname{div}b$ in determining solvability conditions for boundary value problems.
• To extend solvability results to both the operator $L$ and its adjoint $L^t$, distinguishing between symmetric and non-symmetric coefficients.
• To derive quantitative estimates for solutions in terms of the $L^p$-norms of boundary data and the regularity of coefficients.
• To unify the treatment of boundary value problems through harmonic measure and layer potential techniques, particularly for non-symmetric operators.

Proposed method

• The paper employs harmonic measure techniques to analyze the $D_2$ problem for $L$, leveraging estimates on the density of harmonic measure and maximal function control.
• Layer potential methods are used to study the $R_2$ problem, including the invertibility of single and double layer potentials and jump relations on the boundary.
• A decomposition of the drift $b$ into a divergence-free part $\tilde{b}$ and a remainder is used to reduce the non-symmetric case to symmetric settings via integration by parts.
• Rellich-type estimates are applied to control the $L^2$-norm of the gradient on the boundary, crucial for $R_2$ solvability.
• The authors use weighted $L^p$ estimates and maximal function techniques to control nontangential maximal functions of solutions.
• The analysis relies on the Lipschitz regularity of the domain $\Omega$, the uniform ellipticity of $A$, and integrability conditions on $b$ and $\operatorname{div}b$.

Experimental results

Research questions

• RQ1 Under what conditions on the drift $b$ and its divergence $\operatorname{div}b$ is the $D_2$ Dirichlet problem uniquely solvable for $L = -\operatorname{div}(A\nabla\cdot) + b\nabla\cdot$ in a bounded Lipschitz domain?
• RQ2 Is the $R_2$ Regularity problem uniquely solvable for $L$ when $A$ is only Lipschitz continuous and $b$ is bounded, without requiring $\operatorname{div}b$ to be in $L^p$?
• RQ3 What additional regularity is required for the $R_2$ problem to be uniquely solvable for the adjoint operator $L^t$ when $A$ is non-symmetric?
• RQ4 How does the decomposition of $b$ into a divergence-free part and a remainder affect the solvability of boundary value problems?
• RQ5 To what extent do the solvability results depend on the symmetry of the coefficient matrix $A$?

Key findings

• The $D_2$ Dirichlet problem for $L = -\operatorname{div}(A\nabla\cdot) + b\nabla\cdot$ is uniquely solvable in bounded Lipschitz domains when $A$ is uniformly elliptic and Lipschitz continuous, $b \in L^\infty$, and $\operatorname{div}b \in L^{p_d}(\Omega)$ with $p_d = 2$ for $d=3$ and $p_d = d/2$ for $d \geq 4$.
• The $R_2$ Regularity problem for $L$ is uniquely solvable under the same assumptions on $A$ and $b$, with constants depending only on $d$, the ellipticity of $A$, the Lipschitz norm of $A$, $\|b\|_\infty$, the diameter of $\Omega$, and the Lipschitz character of $\Omega$.
• The $D_2$ Dirichlet problem for the adjoint operator $L^t = -\operatorname{div}(A^t\nabla\cdot) - \operatorname{div}(b\cdot)$ is uniquely solvable assuming only that $A$ is Lipschitz and $b$ is bounded.
• The $R_2$ Regularity problem for $L^t$ is uniquely solvable when $A$ is $C^{1,\alpha}$ and $b$ is in $\operatorname{Dr}_{d,\alpha}(\mathbb{R}^d)$, with $\operatorname{div}b \in L^d(\mathbb{R}^d)$, and constants depending on $\|A\|_{C^{1,\alpha}}$, $\|b\|_{\operatorname{Dr}_{d,\alpha}}$, and the domain's diameter and character.
• For $p \in (2 - \varepsilon, \infty)$, the $D_p$ Dirichlet problem for $L$ is uniquely solvable when $\operatorname{div}b \in L^{p_d}(\Omega)$, with $\varepsilon > 0$ depending on the domain and coefficients.
• The solvability results are stable under small perturbations of $b$, as shown by the construction of a divergence-free correction $\tilde{b}$ that preserves the structure of the equation and allows reduction to symmetric cases.