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Thesis.ShengGuo.NeumannProblem.pdf (1.03 MB)

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On Neumann Problems for Fully Nonlinear Elliptic and Parabolic Equations on Manifolds

Guo, Sheng
http://orcid.org/0000-0001-8710-4133
http://rave.ohiolink.edu/etdc/view?acc_num=osu1571696906482925

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis, we study second order fully nonlinear elliptic and parabolic equations with Neumann boundary conditions on compact Riemannian manifolds with smooth boundary. We mainly focus on the elliptic equations with Neumann boundary condition. We derive oscillation bounds for solutions under the assumption of existence of certain C-subsolutions. We use a parabolic approach to derive the solutions of above elliptic equations. We obtain a priori C^2 estimates for parabolic equations with Neumann boundary conditions. We require some geometric assumptions of background manifolds to derive the second order estimates for non-uniformly-elliptic equations. We obtain long-time existence and uniform convergence results, through which we derive the solutions of above elliptic equations. In the appendix we derive a parabolic Harnack inequality for linear uniformly parabolic operators with vanishing Neumann boundary condition (it does not require any curvatures assumption), which is essential for the uniform convergence results. In application, we apply the Neumann problem for the elliptic equations above to study the Calabi-Yau type problem for compact Kahler manifolds with smooth boundary.
Bo Guan (Advisor)
Barbara Keyfitz (Committee Member)
King Yeung Lam (Committee Member)
106 p.
Mathematics
Neumann problem; fully nonlinear; PDE; Geometric Analysis

Recommended Citations

Citations

  • Guo, S. (2019). On Neumann Problems for Fully Nonlinear Elliptic and Parabolic Equations on Manifolds [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1571696906482925

    APA Style (7th edition)

  • Guo, Sheng. On Neumann Problems for Fully Nonlinear Elliptic and Parabolic Equations on Manifolds. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1571696906482925.

    MLA Style (8th edition)

  • Guo, Sheng. "On Neumann Problems for Fully Nonlinear Elliptic and Parabolic Equations on Manifolds." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1571696906482925

    Chicago Manual of Style (17th edition)

osu1571696906482925
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This open access ETD is published by The Ohio State University and OhioLINK.