Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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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



Inserted: 19 mar 2024
Last Updated: 19 mar 2024

Year: 2023

ArXiv: 2311.07058 PDF


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.

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