The Kato Square Root Problem for Mixed Boundary Value Problems

Axelsson, Andreas; Keith, Stephen; McIntosh, Alan

doi:10.1112/s0024610706022873

Cited by 68 publications

(79 citation statements)

References 18 publications

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“…On the upper half-space, this estimate was already available from [Axelsson et al 2006b] as a consequence of the strategy to prove the Kato conjecture on ‫ޒ‬ n . We shall need to prove it on the sphere, essentially by localization and reduction to [Axelsson et al 2006a], where such estimates were proved for first-order operators with boundary conditions. An implication of independent interest is the solution to the Kato square root on Lipschitz manifolds.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

Auscher

Rosén

2012

Anal. PDE

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We continue the development, by reduction to a first-order system for the conormal gradient, of L 2 a priori estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence-form second-order complex elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning a priori almost everywhere nontangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying a posteriori a separate work on bounded domains.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

Auscher

Rosén

2012

Anal. PDE

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“…Later mathematicians tried to find new classes of operators satisfying the Kato conjecture. For strongly elliptic differential operators with measurable bounded coefficients, corresponding results were obtained in . The main difficulties in this case were related to an absence of smoothness for generalized solutions.…”

Section: Introductionmentioning

confidence: 93%

Elliptic differential‐difference operators with degeneration and the Kato square root problem

Скубачевский

2018

Mathematische Nachrichten

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“…122;17,Sec. 11]; c can be assumed to be sufficiently large), then Condition 1 holds by virtue of the results in [18].…”

Section: Theorem 2 Let Conditionmentioning

confidence: 99%

L p -solvability of parabolic problems with an operator satisfying the Kato conjecture

Selitskii

2015

Diff Equat

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We study the solvability in the spaces L p (0, T ; X) of an abstract parabolic equation with an operator defined by a sesquilinear form satisfying the Kato conjecture, where X is a Hilbert space obtained by interpolation between the domain of the form and the dual space. We describe the initial data spaces for various values of p and show that in the most common cases they coincide with the Besov spaces.

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The Kato Square Root Problem for Mixed Boundary Value Problems

Cited by 68 publications

References 18 publications

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II

Elliptic differential‐difference operators with degeneration and the Kato square root problem

L p -solvability of parabolic problems with an operator satisfying the Kato conjecture

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