Solvability of the Neumann problem for elliptic equations in chord-arc domains with very big pieces of good superdomains

Technical Summary: Solvability of the Neumann Problem for Elliptic Equations in Chord-Arc Domains with Very Big Pieces of Good Superdomains

Problem Statement
The paper addresses the solvability of the Neumann problem for elliptic operators in divergence form, $L = -\text{div}(A\nabla)$, within bounded chord-arc domains $\Omega \subset \mathbb{R}^{n+1}$. The coefficients of the matrix $A$ are assumed to have Dini mean oscillation (DMO). The central challenge is to establish $L^p$ solvability for the Neumann problem, denoted $(N_p)_L$, where $1 < p \le 2$.

While the solvability of the Dirichlet problem $(D_p)_L$ and the regularity problem $(R_p)_L$ in chord-arc domains has been extensively studied and settled for the Laplacian and certain classes of operators (e.g., DKP operators), the Neumann problem remains significantly more difficult. Unlike the Dirichlet problem, the Neumann problem lacks a maximum principle that allows for the transfer of solvability from subdomains to the original domain. Furthermore, the Neumann function does not possess a definite sign, preventing the use of positive harmonic measure techniques. The authors specifically target the open question of whether $(N_p)_L$ is solvable in chord-arc domains, a problem highlighted by Kenig and later by Toro, particularly in the context where the boundary supports a weak $p$-Poincaré inequality.

Methodology
The authors employ a "good $\lambda$" inequality argument combined with a bootstrapping procedure and duality. The methodology relies on several key technical components:

1. Dual Formulations and Rough Neumann Problems: The authors introduce variants of the Neumann problem, including the "rough Neumann problem" and $\rho$-smooth versions. They utilize duality arguments to relate the solvability of the Neumann problem $(N_p)_L$ to the solvability of a "rough" Neumann problem for the adjoint operator $L^*$ in Lorentz spaces ($L^{p', \infty}$). Specifically, they establish that $(N_p)_L$ is solvable if and only if the rough Neumann problem $(N^R_{L^{p'}})_{L^*}$ is solvable, provided the regularity problem $(R_q)_L$ is solvable for some $q > p$.

2. Localization and Superdomains: A core innovation is the use of "very big pieces" of superdomains. The authors assume that for every ball $B(\xi, r)$ on the boundary, there exists a "superdomain" $U_{\xi,r}$ (a $C^2$-chord-arc domain) such that $\Omega \cap B(\xi, r) \subset U_{\xi,r}$. Crucially, the Neumann problem is assumed to be uniformly solvable in these superdomains. The condition requires that the "bad" part of the boundary, $\partial \Omega \setminus \partial U_{\xi,r}$, has small measure relative to the ball (controlled by a parameter $\varepsilon$).

3. Good $\lambda$ Inequality and Interpolation: The proof of the main theorem relies on establishing an estimate of the form:
   $$C_L(N^R_{L^{p'}, L^{p', \infty}}) \le K(1 + \varepsilon^a C_L(N^R_{L^{p'}, L^{p', \infty}}))$$
   where $C_L$ represents the solvability constant. By choosing $\varepsilon$ sufficiently small, the term involving the unknown constant on the right-hand side can be absorbed, yielding a uniform bound. This estimate is derived using a localization lemma (Lemma 4.3) which decomposes the solution into local and far-field parts, utilizing the Neumann function and its properties (Moser estimates, Hölder continuity).

4. Approximation and Density: The authors use $\rho$-smooth approximations of the identity ($S_\rho$) to handle the lack of smoothness in the boundary data and the operator coefficients. They prove that solvability for the smooth versions implies solvability for the rough versions via density arguments and interpolation between Lorentz spaces.

Key Contributions and Results
The primary result is Theorem 1.1, which states:
Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded $C^1$-chord-arc domain and $L \in \text{EDMO}(\mathbb{R}^{n+1})$. Let $p \in (1, 2)$. If:

1. The regularity problem $(R_q)_L$ is solvable in $\Omega$ for some $q > p$;
2. $\partial \Omega$ supports a weak $p$-Poincaré inequality;
3. $\Omega$ has "very big pieces" of superdomains $U_{\xi,r}$ (specifically, $C^2$-chord-arc domains) for which the Neumann problem $(N_q)_L$ is uniformly solvable, and the measure of the boundary difference is small ($\varepsilon$-small);

Then, the Neumann problem $(N_p)_L$ is solvable in $\Omega$.

A specific corollary (Corollary 1.2) applies this to the Laplacian ($\Delta$). It asserts that if $(R_q)_\Delta$ is solvable in a $C^1$-chord-arc domain and the domain has very big pieces of Lipschitz superdomains with uniformly solvable Neumann problems, then $(N_p)_\Delta$ is solvable.

Significance and Claims
The paper claims that Theorem 1.1 is a new result even for the Laplace operator. The authors note that while the solvability of $(N_2)_\Delta$ in Lipschitz domains is a classical result (Jerison-Kenig), extending this to $L^p$ for $p \neq 2$ in rougher domains (chord-arc) has been an open problem.

The significance lies in:

• Overcoming the lack of Maximum Principle: The authors provide a method to transfer solvability from "good" superdomains to the "rough" domain $\Omega$ without relying on the maximum principle, which is unavailable for Neumann problems.
• Addressing Kenig's Open Problem: The work makes progress on the question posed by Kenig (1991) and Toro (2010) regarding the existence of $p > 1$ for which the Neumann problem is solvable in chord-arc domains.
• Duality and Rough Neumann Problems: The paper introduces and utilizes a "rough" Neumann problem formulation and its duality with the regularity problem, offering a new perspective on the solvability of boundary value problems in non-smooth settings.
• Optimality of Conditions: The authors acknowledge that their proof requires the "very big pieces" condition (small $\varepsilon$) to absorb the error terms in the good $\lambda$ inequality. They discuss that an iterative application of their theorem might eventually allow for larger $\varepsilon$ (closer to 1), potentially covering all chord-arc domains, though this remains a potential difficulty due to the dependence of constants in the iterative process.

The paper does not claim to solve the Neumann problem for all chord-arc domains unconditionally, but rather establishes solvability under the specific geometric condition of having "very big pieces" of superdomains where the problem is already known to be solvable. The result bridges the gap between the solvability in Lipschitz (or $C^2$) domains and the more general chord-arc setting.