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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 investigate an overdetermined problem for the torsion equation with two kinds of free boundary conditions in an unbounded domain and present a symmetry result in the case of 2D. The proof is based on the maximum principle. In the proof, we use the sliding method and some convex arguments. [Copyright &y& Elsevier]

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]