Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators
By S. Hofmann, Carlos E. Kenig, Svitlana Mayboroda, Jill Pipher. 2012
118 citations recorded by the source corpus.
A. We consider divergence form elliptic operators L = − div A(x)∇, defined in the half space Rn+1 + , n ≥ 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu = 0, and we then combine these estimates with the method of “!-approximability” to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A∞ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in Lp, for some p < ∞). Previously, these results had been known only in the case n = 1.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/a79e3a029a01f034e6f867725002a56ddcc8f10a
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Square Functions and the A∞ Property of Elliptic Measures (2016, background): “33) by well-known arguments (see [7,8,10], the discussion around (4.”
- Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I (2009, methodology): “Domination of the non-tangential maximal function ‖Ñ∗(u)‖2 by the square function ‖∇t,xu‖L2(tdt;L2) ≈ ‖S(u)‖2, S(u)(x) = ( ∫ |y−x|<t |∇t,yu|2dtdy/tn−1)1/2, is reminiscent of the result of Dahlberg, Jerison and Kenig [20], and also of Dahlberg, Kenig, Pipher and Verchota [21].”