A fundamental issue in the theory of fully nonlinear elliptic equations is to establish quantitative, possibly sharp, Hessian integrability of their solutions. The problem is particularly relevant, both from pure and applied viewpoints, for models involving a very large number of independent variables arising from fields such as financial engineering, machine learning, and deep learning; among others. In this dissertation work, we establish new, improved universal bounds for the Hessian integrability exponent of viscosity super-solutions to fully nonlinear elliptic PDEs. Such estimates yield a substantial quantitative improvement on the decay of this exponent with respect to the dimension of the model. In particular, we solve, in the negative, the Armstrong-Silvestre-Smart Conjecture on the optimal Hessian integrability exponent for n grater or equal than 3. The flatland case, n = 2, is a bit more involved, as the maximum exponent is known to be sharp. In this case, we show that the best universal Hessian integrability exponent remains at least 81.45% of the theoretical maximum, uniformly as the ellipticity ratio goes to zero.
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Doctor of Philosophy (Ph.D.)
College of Sciences
Length of Campus-only Access
Doctoral Dissertation (Campus-only Access)
Moura Do Nascimento, Thialita, "On Quantitative Hessian Estimates for Fully Nonlinear Elliptic Equations" (2023). Electronic Theses and Dissertations, 2020-. 1795.
Restricted to the UCF community until August 2026; it will then be open access.