## Marcos M. Alexandrino - Leonardo F. Cavenaghi - D. Corro - Marcelo K. Inagaki

# Singular Riemannian Foliations, variational problems and Principles of
Symmetric Criticalities

created by corro on 19 Mar 2024

[

BibTeX]

*preprint*

**Inserted:** 19 mar 2024

**Last Updated:** 19 mar 2024

**Year:** 2023

**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.