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Sobolev-Type Inequalities on Manifolds in the Presence of Symmetries and Applications

In: Contributions in Mathematics and Engineering

Author

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  • Athanase Cotsiolis

    (University of Patras, Department of Mathematics)

  • Nikos Labropoulos

    (University of Patras, Department of Mathematics)

Abstract

In this article, we present some Sobolev-type inequalities on compact Riemannian manifolds with boundary, the data and the functions being invariant under the action of a compact subgroup of the isometry group. We investigate the best constants for the Sobolev, trace Sobolev, Nash, and trace Nash inequalities. By developing particular geometric properties of the manifold as well as of the solid torus, we can calculate the precise values of the best constants in the presented Sobolev-type inequalities. We apply these results to solve nonlinear elliptic, type Dirichlet and Neumann, PDEs of upper critical Sobolev exponent.

Suggested Citation

  • Athanase Cotsiolis & Nikos Labropoulos, 2016. "Sobolev-Type Inequalities on Manifolds in the Presence of Symmetries and Applications," Springer Books, in: Panos M. Pardalos & Themistocles M. Rassias (ed.), Contributions in Mathematics and Engineering, pages 45-68, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-31317-7_3
    DOI: 10.1007/978-3-319-31317-7_3
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