Mathematical Sciences: Nonlinear Partial Differential Equations and Nonconvex Variational Problems

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Much of the work in this project represents continuing investigations into problems of mathematical analysis, generally related to some particular application or physical model. There are four categories of study, the first concerned with singular perturbations of nonconvex variational problems. By this one means locating functions which minimize an integral of some nonconvex transformation of a class of functions. Solutions are generally not unique. To single out more reasonable solutions, singular perturbations are introduced to penalize the formation of interfaces which occur among piecewise constant solutions. This adaptation of the problem leads to a form of "preferred" solutions which have been identified in certain cases. There are many other important cases which have not been treated; two in particular are the cases where the unknown functions are gradients and the case where the unknowns are matrix- valued functions. Additional work will be done on constrained reaction- diffusion processes, analyzing the front propagation of solutions to nonlinear parabolic equations. The fronts typically represent phase boundaries and their velocity often depends on their local mean curvature such as occur in the motion of grain boundaries in metals. The problem to be analyzed here is a gradient flow subject to a mass constraint. The fronts will develop rapidly and then propagate with a curvature-dependent normal velocity. The mass constraint will be inhibiting so that the actual form of the propagation rule remains to be established. In a nonconvex problem arising in optimal design, one seeks to minimize the cost of material while maintaining the necessary strength. Mathematically, the form of the minimization problems is that of a constrained least gradient problem. A prescription for the construction of a solution has been given by Kohn and Strang for the two-dimensional case. This work aims at developing a rigorous analysis of this method as well as looking for a means to analyze the question in higher dimensions. Work will also be done on branching phenomena of solitary waves. In 1954, Friedrichs and Hyers established the existence of branch solutions of the water wave equation for sufficiently small speed and amplitude. It has not been established whether or not secondary bifurcations of the first branches can occur. The object of this study will be to show that this phenomenon is not possible and to establish conditions which guarantee the uniqueness of solutions having given amplitude or speed.