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Cornell University

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

Degree: 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: Healey, Timothy James (committeeMember), Cao, Xiaodong (committeeMember).*

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 . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/67362

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

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

Liu, Jingbo. “HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS .” 2019. Thesis, Cornell University. Accessed August 10, 2020. http://hdl.handle.net/1813/67362.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Liu, Jingbo. “HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS .” 2019. Web. 10 Aug 2020.

Vancouver:

Liu J. HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS . [Internet] [Thesis]. Cornell University; 2019. [cited 2020 Aug 10]. Available from: http://hdl.handle.net/1813/67362.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

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

Not specified: Masters Thesis or Doctoral Dissertation

Cornell University

2. Tasena, Santi. Heat Kernal Analysis On Weighted Dirichlet Spaces .

Degree: 2011, Cornell University

URL: http://hdl.handle.net/1813/33627

This thesis is concerned with heat kernel estimates on weighted Dirichlet spaces. The Dirichlet forms considered here are strongly local and regular. They are defined on a complete locally compact separable metric space. The associated heat equation is similar to that of local divergence form differential operators. The weight functions studied have the form of a function of the distance from a closed set [SIGMA], that is, x [RIGHTWARDS ARROW] a(d( x, [SIGMA])). We place conditions on the geometry of the set [SIGMA] and the growth rate of function a itself. The function a can either blow up at 0 or [INFINITY] or both. Some results include the case where [SIGMA] separates the whole spaces. It can also apply to the case where [SIGMA] do not separate the space, for example, a domain Ω and its boundary [SIGMA] = ∂Ω. The condition on [SIGMA] is rather mild and do not assume differentiability condition.
*Advisors/Committee Members: Cao, Xiaodong (committeeMember), Gross, Leonard (committeeMember).*

Subjects/Keywords: poincare inequality; doubling; remotely constant

Record Details Similar Records

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

APA (6^{th} Edition):

Tasena, S. (2011). Heat Kernal Analysis On Weighted Dirichlet Spaces . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/33627

Not specified: Masters Thesis or Doctoral Dissertation

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

Tasena, Santi. “Heat Kernal Analysis On Weighted Dirichlet Spaces .” 2011. Thesis, Cornell University. Accessed August 10, 2020. http://hdl.handle.net/1813/33627.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Tasena, Santi. “Heat Kernal Analysis On Weighted Dirichlet Spaces .” 2011. Web. 10 Aug 2020.

Vancouver:

Tasena S. Heat Kernal Analysis On Weighted Dirichlet Spaces . [Internet] [Thesis]. Cornell University; 2011. [cited 2020 Aug 10]. Available from: http://hdl.handle.net/1813/33627.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Tasena S. Heat Kernal Analysis On Weighted Dirichlet Spaces . [Thesis]. Cornell University; 2011. Available from: http://hdl.handle.net/1813/33627

Not specified: Masters Thesis or Doctoral Dissertation

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

Degree: 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: Healey, Timothy James (committeeMember), Cao, Xiaodong (committeeMember).*

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 . (Thesis). Cornell University. Retrieved from http://hdl.handle.net/1813/67578

Not specified: Masters Thesis or Doctoral Dissertation

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

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces .” 2019. Thesis, Cornell University. Accessed August 10, 2020. http://hdl.handle.net/1813/67578.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Hou, Qi. “Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces .” 2019. Web. 10 Aug 2020.

Vancouver:

Hou Q. Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces . [Internet] [Thesis]. Cornell University; 2019. [cited 2020 Aug 10]. Available from: http://hdl.handle.net/1813/67578.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

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

Not specified: Masters Thesis or Doctoral Dissertation