A New Approach to Absolute Continuity of Elliptic Measure, with Applications to Non-symmetric Equations☆

By Carlos E. Kenig, Herbert Koch, Jill Pipher, T. Toro. 2000

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In Applied Mathematics

Citations in the corpus

Homogenization of elliptic boundary value problems in Lipschitz domains (2009, background): “Related work on L p boundary value problems for operators with tindependent, but non-symmetric coefficients may be found in [ 19 ,22].”
Regularity theory for solutions to second order elliptic operators with complex coefficients and the L Dirichlet problem (2016, background): “u(z)dz. Proof. We will be brief as the hard part was the good-λ inequality (Lemma 6.5). For q > p (6.65) follows immediately by a standard argument from the good-λ inequality. For more details see [20]. Furthermore as in [20, Theorem 3.18] the global result for q > p implies the local estimate”
The Dirichlet problem for elliptic equations with drift terms (2001, methodology): “The functions hj(x)were defined in [ KKPT ] and are inspired by a stopping time argument of [Fe-St].”
Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems (2007, background): “The corresponding elliptic equation (1.1) in R2+ was studied by Kenig, Koch, Pipher and Toro in [20], where they showed that (Dir-A) fails to be well posed for certain values of k.”
The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients (2013, methodology): “For real, nonsymmetric coe fficients, the case n = 1 has been treated by Kenig, Koch, Pipher and Toro [36] (Dirichlet problem), and by Kenig and Rule [ 38] (Neumann and Regularity).”
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I (2009, background): “The recent thesis of Barton [14] deals with complex, t-independent L∞ perturbations of the situation in [31, 36], and she obtains well-posedness of the Neumann and regularity problems in Lp for p − 1 > 0 small and even at p = 1 with data in the classical Hardy space.”
Analyticity of layer potentials and $L^{2}$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty}$ coefficients (2007, background): “t-independent coeffi cients, the counter-example of [ KKPT ] shows that our perturbation results are in the nature of best possible.”
The solution of the Kato problem in two dimensions (2002, methodology): “We perform a stopping time decomposition in the spirit of Carleson’s “Corona” construction, and reminiscent also of a stopping time argument in recent work of Kenig, Koch, Pipher and Toro [ 12 ], beginning with the two estimates that we have just noted.”
Uniform Rectifiability, Carleson measure estimates, and approximation of harmonic functions (2014, background): “nsion Garnett’s ε-approximability result, observing that (iv) implies (ii)1. The explicit connection of ε-approximability with theA∞ property of harmonic measure, i.e., that (ii) =⇒ (iii), appears in [KKPT] (where this implication is established not only for the Laplacian, but for general divergence”
Solvability of elliptic systems with square integrable boundary data (2008, background): “pticity condition, (Dir-A) and (Reg-A) where solved by Gao [22]. As for non-symmetic systems, Verchota and Vogel [35] obtained Lp solvability results for (Dir-A), (Reg-A) and (Neu-A) in the spirit of [26] and [28] for certain non-symmetric constant coeﬃcient Legendre–Hadamard systems of two equation”
Local Profiles for Elliptic Problems at Different Scales: Defects in, and Interfaces between Periodic Structures (2015, background): “Note that [31, Theorem 5] is stated only for symmetric matrices, but its proof is based only on the Harnack inequality, which is true for non-symmetric matrices (see [18, 24]).”