Open Access Dissertation
Date of Award
In section 2 of part I, We study the maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We
obtain the radial symmetry and monotonicity properties for
nonnegative viscosity solutions of fully nonlinear equations under some asymptotic decay rate at infinity. Our symmetry and monotonicity results also
apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum
principle for viscosity solutions to fully nonlinear elliptic equations is established. In section 3, We establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma, we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature.
Part II is to address two open questions raised by W. Reichel on characterizations of balls in terms of the Riesz potential and fractional Laplacian. These results answer two open questions W. Reichel to some extent.
Zhu, Jiuyi, "Qualitative Properties Of Solutions Of Fully Nonlinear Equations And Overdetermined Problems" (2013). Wayne State University Dissertations. 814.