Sharp Nash inequalities on manifolds with boundary in the presence of symmetries

In this paper we establish the best constant $\widetilde A_{opt}(\bar{M})$ for the Trace Nash inequality on a $n-$dimensional compact Riemannian manifold in the presence of symmetries, which is an improvement over the classical case due to the symmetries which arise and reflect the geometry of manifold. This is particularly true when the data of the problem is invariant under the action of an arbitrary compact subgroup $G$ of the isometry group $Is(M,g)$, where all the orbits have infinite cardinal

An alternative approach to critical PDEs

In this article, we use an alternative method to prove the existence of an infinite sequence of distinct, non-radial nodal G-invariant solutions for critical nonlinear elliptic problems defined in the whole the Euclidean space. Our proof is via approximation of the problem on symmetric bounded domains. The base model problem of interest originating from Physics is stated below: -∆u = |u|4/n-2u, u ∈ C2 (ℝn), n ≥ 3.Mathematic

Best constants in Sobolev inequalities on manifolds with boundary in the presence of symmetries and applications

AbstractIn this paper we establish the best constants for a Sobolev inequality and a Sobolev trace inequality on compact Riemannian manifolds with boundary, the functions being invariant under the action of a compact subgroup G of the isometry group I(M,g) and we give applications to some nonlinear PDEs with upper critical Sobolev exponent