Mathematical Sciences: Gain of Regularity

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One central theme of this project focuses on nonlinear partial differential equations representing the evolution of certain physical phenomena. An evolution equation whose solutions are smoother than the initial data is said to have a gain of regularity. An example of this occurs in the Schrodinger equation if the initial data decay faster than any polynomial. Some success has also been obtained in the nonlinear area, particularly in the case of the Korteweg-de Vries equation. In this work, a general class of equations will be investigated for gain in regularity. It will concentrate on equations in three space dimensions and seek to prove that gain in regularity is governed by the sign of the third derivative of the solution. Method of proof uses the technique of nonlinear multipliers and a fundamental energy (integral) estimate which shows the gain. Efforts will also be made in applying similar ideas to dispersive equations such as nonlinear Schrodinger equations. A second line of investigation concerns spectral and inverse spectral problems of the Laplace operator or the Laplace operator plus a potential defined on compact Riemannian manifolds. Three objectives of this work are to: (1) describe the distribution of the eigenvalues, (2) identify the properties of a Riemannian metric or a potential which are spectral invariants, and (3) determine the metrics or potentials which can be isospectrally deformed in a mathematically significant manner. Particular emphasis will be placed on measuring the pattern of gaps between eigenvalues and the density properties of potentials with a finite number of double eigenvalues.