On Kato’s conjecture and mixed boundary conditions
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Beschreibung
T. Kato‘s square root problem, first formulated in 1961, was amongst
the most notorious problems of 20th century analysis and remained
unsolved until 2001. In his doctoral thesis, Moritz Egert addresses
Kato‘s conjecture for elliptic second-order divergence-form operators
subject to mixed boundary conditions. He is first to resolve this
problem on rough domains, even beyond the Lipschitz-class. To
this end, he combines common techniques from harmonic analysis
and functional calculus with tools from potential theory and geometric
measure theory in an innovative way. At the heart of the matter
lies a comprehensive theory for the Sobolev spaces capturing mixed
boundary conditions, including trace and extension theorems,
interpolation theory, Hardy‘s inequality, and the Calderón-Zygmund
decomposition.
As an application to the theory of partial differential equations, the
author considers elliptic boundary value problems on cylindrical
domains and obtains new a priori representations for solutions and
well-posedness for specific classes of coefficients.
The necessary mathematical background on potential theory and
functional calculus for bisectorial operators is provided in two
comprehensive introductory sections.
Autor*in
Moritz Egert
Themen in »On Kato’s conjecture and mixed boundary conditions«
(bi)sectorial theory DB formalism Hardy’s inequality Kato’s conjecture Lions problem Sobolev extension operators divergence-form operators elliptic boundadary value problems fractional Sobolev spaces functional calculus mixed boundary conditions