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Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations. (English) Zbl 1435.35148

The Dirichlet boundary value problem for degenerate elliptic equations in the upper half-space \(\mathbb{R}^{n+1}_{+}\) when \(n\geq 2\) is considered. In the uniformly elliptic case, the solvability of the Dirichlet problem has been established for a variety of complex coefficient structures. In contrast to the proof of solvability in the uniformly elliptic case, in the present paper the approach of C. Kenig et al. [J. Geom. Anal. 26, No. 3, 2383–2410 (2016; Zbl 1386.35070)] is adapted to the degenerate elliptic case.
First, technical preliminaries concerning weights and degenerate elliptic operators are considered. Then, a Carleson measure estimate is obtained and a degenerate elliptic measure is constructed. The solvability of the Dirichlet problem is deduced and an uniqueness result is also obtained. The paper is technical and interesting and the proofs of the main results are well presented.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J70 Degenerate elliptic equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory

Citations:

Zbl 1386.35070
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References:

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