arXiv subjects
Publications and source records attributed to Athanase Cotsiolis.
In this paper both we establish the best constants for the Nash inequalities on the standard unit sphere $\mathbb{S}^n$ of $\mathbb{R}^{n+1}$ and we give answers on the existence of extremal functions on the corresponding problems. Also we study the problem of the best constants in the case, where the data are invariant under the action of the group $G=O(k)\times O(m)$, and we find the best constants.
In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus $\bar{T}$ of $\mathbb{R}^3$, when data are invariant under the group $G=O(2)\times I \subset O(3)$. The model problems of interest are stated below: ${ll} {\bf(P_1)} & \displaystyle Δ\upsilon+γ=f(x)e^\upsilon, \upsilon>0\quad \mathrm{on} \quad T, \quad\upsilon |_{_{\partial T}}=0.$ and ${ll}\bf{(P_2)} & \displaystyle Δ\upsilon+a+fe^\upsilon=0, \upsilon>0\quad \mathrm{on}\quad T, [1.3ex] &\displaystyle \frac{\partial \upsilon}{\partial n}+b+ge^\upsilon=0\quad \mathrm{on} \quad{\partial T}.$ We prove that exist solutions which are $G-$invariant and these exhibit no radial symmetries. In order to solve the above problems we need to find the best constants in the Sobolev inequalities in the exceptional case.
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.