Department of Mathematics (MU)
https://hdl.handle.net/10355/251
2024-03-29T06:18:56ZAbsolute continuity of parabolic measure and the initial-dirichlet problem
https://hdl.handle.net/10355/69946
Absolute continuity of parabolic measure and the initial-dirichlet problem
Genschaw, Alyssa
This thesis is devoted to the study of parabolic measure corresponding to a divergence form parabolic operator. We first extend to the parabolic setting a number of basic results that are well known in the elliptic case. Then following a result of Bennewitz-Lewis for non-doubling harmonic measure, we prove a criterion for non-doubling caloric measure to satisfy a weak reverse Holder inequality on an open set [omega] R(n+1), assuming as a background hypothesis only that the essential boundary of [omega] satisfies an appropriate parabolic version of Ahlfors-David regularity (which entails some backwards in time thickness). We then show that the weak reverse Holder estimate is equivalent to solvability of the initial Dirichlet problem with "lateral" data in [Lp], for some p< [infinity]. Finally, we prove that for the heat equation, BMO-solvability implies scale invariant quantitative absolute continuity of caloric measure with respect to surface measure, in an open set [omega] with time-backwards ADR boundary. Moreover, the same results apply to the parabolic measure associated to a uniformly parabolic divergence form operator (L), with estimates depending only on dimension, the ADR constants, and parabolicity, provided that the continuous Dirichlet problem is solvable for (L) in [omega]. By a result of Fabes, Garofalo and Lanconelli [FGL], this includes the case of [C1]-Dini coefficients.
2019-01-01T00:00:00ZThe absolute functional calculus for sectorial operators
https://hdl.handle.net/10355/4155
The absolute functional calculus for sectorial operators
Kucherenko, Tamara
We introduce the absolute functional calculus for sectorial operators. This notion is stronger than the common holomorphic functional calculus. We are able to improve a key theorem related to the maximal regularity problem and hence demonstrate the power and usefulness of our new concept. In trying to characterize spaces where sectorial operators have absolute calculus, we find that certain real interpolation spaces play a central role. We are then extending various known results in this setting. The idea of unifying theorems about sectorial operators on real interpolation spaces permeates our work and opens paths for future research on this subject.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.; Title from title screen of research.pdf file viewed on (July 18, 2006); Includes bibliographical references.; Vita.; Thesis (Ph. D.) University of Missouri-Columbia 2005.; Dissertations, Academic -- University of Missouri--Columbia -- Mathematics.
2005-01-01T00:00:00ZAcceleration-induced nonlocality: kinetic memory versus dynamic memory
https://hdl.handle.net/10355/8577
Acceleration-induced nonlocality: kinetic memory versus dynamic memory
Chicone, Carmen Charles; Mashhoon, Bahram
The characteristics of the memory of accelerated motion in Minkowski spacetime are discussed within the framework of the nonlocal theory of accelerated observers. Two types of memory are distinguished: kinetic and dynamic. We show that only kinetic memory is acceptable, since dynamic memory leads to divergences for nonuniform accelerated motion.
DOI: 10.1002/1521-3889(200204)11:4<309
2001-01-01T00:00:00ZAcceleration-induced nonlocality: uniqueness of the kernel
https://hdl.handle.net/10355/8576
Acceleration-induced nonlocality: uniqueness of the kernel
Chicone, Carmen Charles; Mashhoon, Bahram
We consider the problem of uniqueness of the kernel in the nonlocal theory of accelerated observers. In a recent work, we showed that the convolution kernel is ruled out as it can lead to divergences for nonuniform accelerated motion. Here we determine the general form of bounded continuous kernels and use observational data regarding spin-rotation coupling to argue that the kinetic kernel given by $K(\tau ,\tau')=k(\tau')$ is the only physically acceptable solution.
DOI: 10.1016/S0375-9601(02)00439-5
2002-01-01T00:00:00Z