Singular Riemannian Foliations, variational problems and Principles of Symmetric Criticalities

Abstract

A singular foliation $\mathcal{F}$ on a complete Riemannian manifold $M$ is called Singular Riemannian foliation (SRF for short) if its leaves are locally equidistant, e.g., the partition of $M$ into the orbits of a Lie group action by isometries. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRFs with special properties, which we name as AVP. Examples of such SRFs being considered include isoparametric foliations, SRFs on Euclidean fiber bundles, and the partition of $M$ into the orbits of a Lie group acting by isometries. More precisely, we prove an analog to Palais' Principle of Symmetric Criticality for $\mathcal{F}$-symmetric integral operators on the Banach spaces $W^{1,p}(M)$. This result together with a version of the Rellich--Kondrachov--Hebey--Vaugon Embedding Theorem for $\mathcal{F}$-basic Sobolev functions allows us to circumvent difficulties with Sobolev's critical exponents when considering applications of techniques from Calculus of Variations to find solutions to PDEs. To exemplify this, we prove the existence of countably infinite many weak solutions to a class of variational problems, which includes $p$-Kirchhoff problems for manifolds equipped with AVP.