On Kato’s conjecture and mixed boundary conditions

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.