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

Exponential elliptic boundary value problems on a solid torus in the critical of supercritical case

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 \Delta\upsilon+\gamma=f(x)e^\upsilon, \upsilon>0\quad \mathrm{on} \quad T, \quad\upsilon |_{_{\partial T}}=0. and {ll}\bf{(P_2)} & \displaystyle \Delta\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

A Neumann problem with the qq-Laplacian on a solid torus in the critical of supercritical case

Following the work of Ding [21] we study the existence of a nontrivial positive solution to the nonlinear Neumann problem $$displaylines{ Delta_qu+a(x)u^{q-1}=lambda f(x)u^{p-1}, quad u>0quad hbox{on } T,cr abla u|^{q-2}frac{partial u}{partial u}+b(x) u^{q-1} =lambda g(x)u^{ilde{p}-1} quadhbox{on }{partial T},cr p =frac{2q}{2-q}>6,quad ilde{p}=frac{q}{2-q}>4,quad frac{3}{2}<q<2, }$$ on a solid torus of $mathbb{R}^3$. When data are invariant under the group $G=O(2)imes I subset O(3)$, we find solutions that exhibit no radial symmetries. First we find the best constants in the Sobolev inequalities for the supercritical case (the critical of supercritical)

CLASSICAL AND RELAXED DISCRETIZATION METHODS FOR SEMILINEAR PARABOLIC OPTIMAL CONTROL PROB-

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