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1. Liu, Jingbo. HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS.

Degree: PhD, Mathematics, 2019, Cornell University

URL: http://hdl.handle.net/1813/67362

In this thesis we study the properties of the Schrodinger operator L=−∆+q on a Harnack-type Dirichlet space for q belonging to Kato class K or Kato-infinity class K∞. To be specific, it consists of three parts as follows: The first part is a generalization of [27]. For any Harnack-type Dirichlet space we give conditions under which there exists a positive Dirichlet solution (the profile) in an unbounded uniform domain for the operator L. In this setting, we further give the two-sided heat kernel estimate using the famous h-transform technique. The idea of second part comes from [64]. In the exterior of a compact set in a non- parabolic Harnack-type space, we can prove some equivalent statements connect- ing subcrilicality, positiveness of the Green function, gaugeability and the bound- edness of the Dirichlet-type solution provided the potential q ∈ K∞. Particularly, we can apply the boundedness result of the profile to the first part and conclude a more precise heat kernel estimate. In the third part we provide some typical examples and explore some properties when the potential decays faster than the quadratic one. Some other examples are given in the domain outside an unbounded domain and we propose some hypothesis as an supplement to the second part.
*Advisors/Committee Members: Saloff-Coste, Laurent Pascal (chair), Healey, Timothy James (committee member), Cao, Xiaodong (committee member).*

Subjects/Keywords: Mathematics; Dirichlet; Heat Kernel Estimate; Uniform Domain

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Liu, J. (2019). HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS. (Doctoral Dissertation). Cornell University. Retrieved from http://hdl.handle.net/1813/67362

Chicago Manual of Style (16^{th} Edition):

Liu, Jingbo. “HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS.” 2019. Doctoral Dissertation, Cornell University. Accessed September 29, 2020. http://hdl.handle.net/1813/67362.

MLA Handbook (7^{th} Edition):

Liu, Jingbo. “HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS.” 2019. Web. 29 Sep 2020.

Vancouver:

Liu J. HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS. [Internet] [Doctoral dissertation]. Cornell University; 2019. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/1813/67362.

Council of Science Editors:

Liu J. HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS. [Doctoral Dissertation]. Cornell University; 2019. Available from: http://hdl.handle.net/1813/67362

2. Hou, Qi. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.

Degree: PhD, Mathematics, 2019, Cornell University

URL: http://hdl.handle.net/1813/67578

This thesis studies some qualitative properties of local weak solutions of the heat equation in Dirichlet spaces. Let ≤ ft(X,𝓔,𝓕)) be a Dirichlet space where X is a metric measure space, and ≤ ft(𝓔,𝓕)) is a symmetric, local, regular Dirichlet form on L^{2} ≤ ft(X)). Let -P and ≤ ft(H_{t}))_{t>0} denote the corresponding generator and semigroup. Consider the heat equation ≤ ft(\partial_{t+P}))u=f in ℝ × 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 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 ≤ ft(\partial_{t+P}))u=f on some time-space cylinder I × Ω.\ (i) If the time derivative of f is locally in L^{2} ≤ ft(I × Ω)), then the time derivative of u is a local weak solution to ≤ ft(\partial_{t+P}))\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 ≤ ft(t,x,y)), 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 ≤ ft(t,x,y)). As a special case, on the infinite torus \mathbb{T}^∞, local boundedness of h ≤ ft(t,x,y)) implies automatically the continuity of h ≤ ft(t,x,y)), 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.
*Advisors/Committee Members: Saloff-Coste, Laurent Pascal (chair), Healey, Timothy James (committee member), Cao, Xiaodong (committee member).*

Subjects/Keywords: Dirichlet space; heat equation; heat kernel; heat semigroup; local weak solution; Mathematics

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Hou, Q. (2019). Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. (Doctoral Dissertation). Cornell University. Retrieved from http://hdl.handle.net/1813/67578

Chicago Manual of Style (16^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.” 2019. Doctoral Dissertation, Cornell University. Accessed September 29, 2020. http://hdl.handle.net/1813/67578.

MLA Handbook (7^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces.” 2019. Web. 29 Sep 2020.

Vancouver:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. [Internet] [Doctoral dissertation]. Cornell University; 2019. [cited 2020 Sep 29]. Available from: http://hdl.handle.net/1813/67578.

Council of Science Editors:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces. [Doctoral Dissertation]. Cornell University; 2019. Available from: http://hdl.handle.net/1813/67578