Sharp inequalities for derivatives and potentials in the critical cases of the Sobolev embedding theorem

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

Many real world phenomena are modeled using partial differential equations, equations that involve functions and their derivatives. For practical purposes it is extremely important to understand the behavior of the solutions of such equations. Even though it is often impossible to compute these solutions explicitly, a great deal of information can be garnered from quantitative estimates on their size, usually by means of inequalities. The purpose of this research is to obtain many new optimal inequalities related to fundamental equations in mathematics, physics, and geometry: from equations describing the curvature of a surface, to mean-field equations arising in vortex models for turbulent flows, and more. The estimates to be obtained are sharp (i.e., they cannot be improved), and they incorporate deep information about the underlying geometric and physical models. In this project, the principal investigator seeks to find best constants in several Adams, Moser-Trudinger, and Onofri inequalities in various settings, sharp asymptotics for fundamental solutions of differential and pseudodifferential operators, and optimal embeddings of Sobolev spaces in rearrangement invariant spaces. On the Cauchy-Riemann sphere, sharp inequalities are to be obtained for rather general spectrally defined pseudodifferential operators, featuring best constants that depend explicitly on the eigenvalues. The method includes a new asymptotic analysis of the fundamental solutions of such operators. On Euclidean spaces, sharp inequalities will be obtained on domains of infinite measure, filling a twenty-five-year-old gap that was left since Adams's seminal work. The methods are new, and they are powerful enough to allow extensions to other noncompact settings such as the Heisenberg group and spaces endowed with hyperbolic metrics. Sharp Brezis-Merle-type inequalities and optimal embeddings are to be obtained on reduced Sobolev spaces of barely integrable functions. The method is based on a new theory of Adams- and Orlicz-type inequalities on general measure spaces, for integral operators with slowly varying kernels.