Harnack type inequality on Riemannian manifolds of dimension 5

We give some estimates of type sup * inf on Riemannian manifold of dimension 5.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1103.019

Analogue of Gidas-Ni-Nirenberg result in hyperbolic space and sphere

Let $u\epsilon C^{2}\left(\overline{\Omega}\right)$be a positive solution of the differential equation $\Delta u+f\left(u\right)=0$ in $\Omega$ with boundary condition u=0 on $\partial\Omega$ where f is a C$^{1}$ function and $\Omega$ is a geodesic ball in the hyperbolic space $\mathbf{H}^{\mathbf{n}}$ $\left(\textrm{respectively}\:\textrm{sphere}\:\mathbf{S^{\mathbf{n}}}\right)$. Further in case of sphere we assume that $\overline{\Omega}$ is contained in a hemisphere. Then we prove that u is radially symmetric

Symmetry in rearrangement optimization problems

This article concerns two rearrangement optimization problems. The first problem is motivated by a physical experiment in which an elastic membrane is sought, built out of several materials, fixed at the boundary, such that its frequency is minimal. We capture some features of the optimal solutions, and prove a symmetry property. The second optimization problem is motivated by the physical situation in which an ideal fluid flows over a seamount, and this causes vortex formation above the seamount. In this problem we address existence and symmetry

Positive solutions for elliptic problems with critical indefinite nonlinearity in bounded domains

In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely $$- Delta u =lambda u + h (x) u^{(n+2)/(n-2)}$$ in a smooth open bounded domain $Omegasubseteq mathbb{R}^n$, $n > 4$ with Dirichlet boundary conditions and for $lambda geq 0$. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue $lambda_1(Omega)$. For $n=2$, we get similar results for $-Delta u =lambda u + h (x)phi(u)e^u$ where $phi$ is bounded and superlinear near zero