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On Some Classes of Fully Nonlinear Partial Differential Equations
Sui, Zhenan

2015, Doctor of Philosophy, Ohio State University, Mathematics.
The first topic is the initial boundary value problem for fully nonlinear parabolic equations on compact manifolds. To obtain the long time existence of solutions, a fundamental step is to establish C2+a,1+ß estimates, which, as a result of Evans-Krylov, can be deduced from C2, 1 estimates. In this thesis, we present a new way to derive second order estimates on general Riemannian manifolds without any curvature assumption or geometric restrictions on the boundary. For this, we assume the existence of an admissible subsolution, which also enables us to use the least amount of structure conditions on f . It is worth mentioning that the key lemma, from recent work of Guan for the elliptic case, works equally well for the parabolic case. Our results are new even for equations in Euclidean space.

On a closed manifold, i.e. compact manifold without boundary, an admissible subsolution would be a solution. Hence we introduce a modified version of a subsolution and apply the modified version of the key lemma which originated in recent work of Guan.

Besides, we are interested in second order interior estimates on general Riemannian manifolds. For this, we work on a class of fully nonlinear parabolic equations which is more general but has to satisfy strict concavity on A with respect to the gradient. By constructing a test function with a special choice of cut off function, we are able to obtain such estimates.

The second topic is on existence of entire solutions to conformal Ricci curvature equations, which arises from the geometric problem – the existence of complete conformal metric of negative Ricci curvature on Euclidean space. By constructing a pair of radically symmetric supersolutions and subsolutions, an admissible solution can be obtained by a diagonal process, and hence follows the existence of such metric. In addition, some upper bounds are established for admissible solutions, which, combined with a strong half space technique, result in nonexistence of admissible solutions.
Bo Guan (Advisor)
Barbara Keyfitz (Committee Member)
Yuan Lou (Committee Member)
75 p.

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