Archive for Rational Mechanics and Analysis· 2026Q1
Solvability of the Neumann Problem for Elliptic Equations in Chord-Arc Domains with Very Big Pieces of Good Superdomains
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- Q1SCImago
- 2026year
Short summary
The Neumann problem for elliptic operators with Dini coefficients is solvable in L^p within chord-arc domains if the problem is solvable in L^q (q>p) and the domain has large superdomains where the problem is uniformly solvable.
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Key points
- Solvability of the Neumann problem for elliptic operators with Dini coefficients is shown for L^p spaces in chord-arc domains.
- Conditions include L^q solvability (q>p) for the regularity problem and a weak p-Poincaré inequality on the boundary.
- The presence of large superdomains with uniform L^q Neumann problem solvability is a key requirement.
- The elliptic operator is of the form -div(A∇) with A having Dini mean oscillation coefficients.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract Let $$\Omega \subset {\mathbb {R}}^{n+1}$$ Ω ⊂ R n + 1 be a bounded chord-arc domain, let $${\mathcal {L}}=-\textrm{div} A\nabla $$ L = - div A ∇ be an elliptic operator in $$\Omega $$ Ω associated with a matrix A having Dini mean oscillation coefficients, and let $$1<p\le 2$$ 1 < p ≤ 2 . In this paper we show that if the regularity problem for $${\mathcal {L}}$$ L is solvable in $$L^q$$ L q for some $$q>p$$ q > p in $$\Omega $$ Ω , $${\partial \Omega }$$ ∂ Ω supports a weak p -Poincaré inequality, and $$\Omega $$ Ω has very big pieces of superdomains for which the Neumann problem for $${\mathcal {L}}$$ L is solvable uniformly in $$L^q$$ L q , then the Neumann problem for $${\mathcal {L}}$$ L is solvable in $$L^p$$ L p in $$\Omega $$ Ω .
The authors' abstract, as published at the source. Archive for Rational Mechanics and Analysis, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics