Square function/non-tangential maximal function estimates and the\n dirichlet problem for non-symmetric elliptic operators

2012/02/10 by Steve Hofmann, Carlos E. Kenig, Hofmann, Steve +5 · 5 citations
Computer Science · Mathematics · #35J25 #42B20 #42B25 #42B99 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1202.2405

openalex publication_date 2012/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider divergence form elliptic operators L = - div A(x)\∇, defined\nin the half space Rⁿ⁺¹₊, n \≥ 2, where the coefficient matrix A(x) is\nbounded, measurable, uniformly elliptic, t-independent, and not necessarily\nsymmetric. We establish square function/non-tangential maximal function\nestimates for solutions of the homogeneous equation Lu = 0, and we then combine\nthese estimates with the method of "\ε-approximability" to show that\nL-harmonic measure is absolutely continuous with respect to surface measure\n(i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant\nsense: more precisely, that it belongs to the class A_\∞ with respect to\nsurface measure (equivalently, that the Dirichlet problem is solvable with data\nin L^p, for some p < \∞). Previously, these results had been known only in\nthe case n = 1.\n

Citations

• Some new function spaces and their applications to Harmonic analysis
• A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations
• Hp spaces of several variables

Cited by

• A sign-changing Poisson kernel for a non-symmetric elliptic operator in a bounded domain
• Tangential approach in the Dirichlet problem for elliptic equations
• Equivalence between solvability of the Dirichlet and Regularity problem under an L¹ Carleson condition on ∂_t A
• The L^p regularity problem for parabolic operators with transversally independent coefficients
• Layer potentials beyond singular integral operators

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• Square function and maximal function estimates for operators beyond divergence form equations