Mathematical Sciences: Harmonic Analysis and the Laplace Equation

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Work to be supported by this award will focus on Mathematical questions centering on elliptic partial differential equations. The theme is that of unique continuation of solutions and related issues. By unique continuation one understands the problem of showing that there are no (non-zero) solutions vanishing in an open set. A second question, whereby one asks whether a solution can vanish to infinite order at a point is called the strong continuation property. Both are variations on the question of whether two solutions which agree on an open set (of infinite order at a point) must agree forever. A number of positive results are known, depending both on the smoothness of the equation and the underlying dimension. Recent examples show that in the case of certain classical equations, Schrodinger with Kato class potentials, unique continuation fails. One of the specific goals of this project is to extend the basic tool underlying the best results, known as the Carleman inequalities. These inequalities only apply to constant-coefficient equations. Further analysis of more general inequalities suggest that progress can be made in establishing good estimates which can be applied to general elliptic equations. Related work will be carried out on the Bers problem which asks whether solutions whose gradient vanish to a certain degree at the boundary must reduce to constants. Abstract as these problems appear in there generic form, they are easily seen to be important from the point of view of users of mathematics, especially in those cases where approximation of solutions can lead to spurious ones. Knowledge of uniqueness of continuation can provide a level of confidence that reasonable approximations lead to the correct answer.