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 orbits of an isometric action. In this paper, we investigate variational problems in compact Riemannian manifolds equipped with SRF with special properties, e.g. isoparametric foliations, SRF on fibers bundles with Sasaki metric, and orbit-like foliations. More precisely, we prove two results analogous to Palais' Principle of Symmetric Criticality, one is a general principle for $\mathcal{F}$ symmetric operators on the Hilbert space $W^{1,2}(M)$, the other one is for $\mathcal{F}$ symmetric integral operators on the Banach spaces $W^{1,p}(M)$. These results together with a $\mathcal{F}$ version of Rellich Kondrachov Hebey Vaugon Embedding Theorem allow us to circumvent difficulties with Sobolev's critical exponents when considering applications of Calculus of Variations to find solutions to PDEs. To exemplify this we prove the existence of weak solutions to a class of variational problems which includes $p$-Kirschoff problems.