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We construct three families of singular critical points for a variational free boundary problem. These critical points are homogeneous solutions of degree one to some overdetermined boundary value problem. The intersections of the level sets of these solutions with the unit sphere are isoparametric hypersurfaces and their focal submanifolds. [ABSTRACT FROM AUTHOR]

Using the Cartan isoparametric cubic, we construct a singular critical point of a variational free boundary problem in 5 dimensions which is homogeneous of degree 1. The intersection of the free boundary with S 4 consists of certain level sets of the Cartan cubic, and thus admits a three dimensional automorphism group. [ABSTRACT FROM AUTHOR]

Abstract: We consider the minimum problem for the functional in three dimensional space, where is a constant. We will establish a Liouville type theorem for this variational problem: if is a nonnegative and nonzero global minimizer, then for some point and some unit vector . [Copyright &y& Elsevier]