dc.contributor.author Hou, Qi dc.date.accessioned 2019-10-15T16:48:25Z dc.date.available 2019-10-15T16:48:25Z dc.date.issued 2019-08-30 dc.identifier.other Hou_cornellgrad_0058F_11677 dc.identifier.other http://dissertations.umi.com/cornellgrad:11677 dc.identifier.other bibid: 11050561 dc.identifier.uri https://hdl.handle.net/1813/67578 dc.description.abstract This thesis studies some qualitative properties of local weak solutions of the heat equation in Dirichlet spaces. Let $\left(X,\mathcal{E},\mathcal{F}\right)$\ be a Dirichlet space where $X$\ is a metric measure space, and $\left(\mathcal{E},\mathcal{F}\right)$\ is a symmetric, local, regular Dirichlet form on $L^2\left(X\right)$. Let $-P$\ and $\left(H_t\right)_{t>0}$\ denote the corresponding generator and semigroup. Consider the heat equation $\left(\partial_t+P\right)u=f$\ in $\mathbb{R}\times X$. Examples of such heat equations include the ones associated with (i) Dirichlet forms associated with uniformly elliptic, second order differential operators with measurable coefficients on $\mathbb{R}^n$, and Dirichlet forms on fractal spaces;\\ (ii) Dirichlet forms associated with product diffusions and product anomalous diffusions on infinite products of compact metric measure spaces, including the infinite dimensional torus, and the infinite product of fractal spaces like the Sierpinski gaskets.\\ We ask the following qualitative questions about local weak solutions to the above heat equations, which in spirit are generalizations of the notion of hypoellipticity: Are they locally bounded? Are they continuous? Is the time derivative of a local weak solution still a local weak solution? Under some hypotheses on existence of cutoff functions with either bounded gradient or bounded energy, and sometimes additional hypotheses on the semigroup, we give (partially) affirmative answers to the above questions. Some of our key results are as follows. Let $u$\ be a local weak solution to $\left(\partial_t+P\right)u=f$\ on some time-space cylinder $I\times \Omega$.\\ (i) If the time derivative of $f$\ is locally in $L^2\left(I\times \Omega\right)$, then the time derivative of $u$\ is a local weak solution to $\left(\partial_t+P\right)\partial_t u=\partial_t f$.\\ (ii) If the semigroup $H_t$\ is locally ultracontractive, and satisfies some Gaussian type upper bound, and if $f$\ is locally bounded, then $u$\ is locally bounded.\\ (iii) Besides satisfying local contractivity and some Gaussian type upper bound, if the semigroup $H_t$\ further admits a locally continuous kernel $h\left(t,x,y\right)$, then $u$\ is locally continuous.\\ (iv) If the semigroup is locally ultracontractive and satisfies some Gaussian type upper bound, then it admits a locally bounded function kernel $h\left(t,x,y\right)$. As a special case, on the infinite torus $\mathbb{T}^\infty$, local boundedness of $h\left(t,x,y\right)$\ implies automatically the continuity of $h\left(t,x,y\right)$, and hence of all local weak solutions.\\ (v) The needed Gaussian type upper bounds can often be derived from the ultracontractivity conditions. We also discuss such implications under existence of cutoff functions with bounded gradient or bounded energy.\\ The results presented in this thesis are joint work with Laurent Saloff-Coste. dc.language.iso en_US dc.subject Dirichlet space dc.subject heat equation dc.subject heat kernel dc.subject heat semigroup dc.subject local weak solution dc.subject Mathematics dc.title Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces dc.type dissertation or thesis thesis.degree.discipline Mathematics thesis.degree.grantor Cornell University thesis.degree.level Doctor of Philosophy thesis.degree.name Ph.D., Mathematics dc.contributor.chair Saloff-Coste, Laurent Pascal dc.contributor.committeeMember Healey, Timothy James dc.contributor.committeeMember Cao, Xiaodong dcterms.license https://hdl.handle.net/1813/59810 dc.identifier.doi https://doi.org/10.7298/x9mb-8p29
﻿