Yamabe equations on Carnot groups

Dimiter Nickolov Vassilev, Purdue University

Abstract

The variational problem of finding the best constant in the Folland-Stein embedding theorem on Carnot groups is considered. The existence of a best constant in the above embedding is proved with the method of concentration compactness. The solutions of the Euler-Lagrange equation are shown to be bounded. A special case of the embedding leads to a Yamabe type equation which is considered in detail on various domains. Boundedness of the horizontal gradient, derivatives along the radial vector field or along vectors from the center of the Lie algebra are proved. This is done under certain assumptions on the boundary of the domain. The introduced “convexity” condition gives an example of domains for which barrier functions are constructed and the above questions are answered. The proven regularity of solutions is used to show non-existence of non-trivial weak non-negative solutions on bounded star-like domains with boundaries satisfying the required assumptions. On groups of lwasawa type the Kelvin transform is used to extend the results to unbounded domains. The symmetries of solutions on such groups is studied using the moving plane method. It is shown that all solutions with partial symmetry are cylindrical. On the other hand all solutions with cylindrical symmetry are determined by reducing the problem to a non-linear equation in the first quadrant of the plane, where an argument similar to the one used by Jerison and Lee and the P-function method of Weinberger is employed. This allows to find the precise value of the best constant in the presence of partial symmetry in this special case of the Folland-Stein embedding.

Degree

Ph.D.

Advisors

Garofalo, Purdue University.

Subject Area

Mathematics

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