The Kato Square Root Problem for Mixed Boundary Value Problems

By A. Axelsson, S. Keith, A. Mcintosh. 2004

88 citations recorded by the source corpus.

We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by Lions in 1962. To do this we develop a general theory of quadratic estimates and functional calculi for complex perturbations of Dirac‐type operators on Lipschitz domains.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/369132ab070c69eb1f9540e27984e81c3800f565

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Numerical Approximation of Fractional Powers of Regularly Accretive Operators (2015, unknown): “This is a deep theorem which has been intensively studied, see [1, 2, 39, 37] and the references in [39].”
Maximal parabolic regularity for divergence operators including mixed boundary conditions (2008, background): “The isomorphism property is known for the Hilbert space case L(2) (see [13]) in case of mixed boundary conditions and even complex coefficients, but the proof fundamentally rests on the Hilbert space structure, so that we do not see a possibility of directly generalizing this to the L case.”
Solvability of elliptic systems with square integrable boundary data (2008, background): “Further elaborations of the ideas presented in this paper, along the lines of thought in [3] and [9], working with general inhomogeneous differential forms taking values in the full exterior algebra and allowing lower order terms, one should be able to extend the theory to cover both Dirac’s and Max”
On a quadratic estimate related to the Kato conjecture and boundary value problems (2008, methodology): “Further results on inhomogeneous problems are in [9].”
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I (2009, methodology): “his estimate was already available from [14] as a consequence of the strategy to prove the Kato conjecture on Rn. We shall need to prove it on the sphere, essentially by localization and reduction to [13], where such estimates were proved for ﬁrst order operators with boundary conditions. An implica”