Qualitative properties of solutions to elliptic singular problems

We investigate the singular boundary value problem Δu+u−γ=0 in D, u=0 on ∂D, where γ>0. For γ>1, we find the estimate |u(x)−b0δ2/(γ+1)(x)| <βδ(γ−1)/(γ+1)(x), where b0 depends on γ only, δ(x) denotes the distance from x to ∂D and is β suitable constant. For γ>0, we prove that the function u(1+γ)/2 is concave whenever D is convex. A similar result is well known for the equation Δu+up=0, with 0≤p≤1. For p=0, p=1 and γ≥1 we prove convexity sharpness results

Problems for elliptic singular equations with a quadratic gradient term

AbstractWe investigate the homogeneous Dirichlet problem for a class of second-order elliptic partial differential equations with a quadratic gradient term and singular data. In particular, we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation

Second-order boundary estimates for solutions to singular elliptic equations in borderline cases

Let \Omega \subsetR^N be a bounded smooth domain. We investigate the effect of the mean curvature of the boundary \partial \Omega on the behaviour of the solution to the homogeneous Dirichlet boundary value problem for the equation \Delta u + f(u) = 0. Under appropriate growth conditions on f(t) as t approaches zero, we find asymptotic expansions up to the second order of the solution in terms of the distance from x to the boundary \partial \Omega

Problems for P-Monge-Ampere Equations

2010 Mathematics Subject Classification: 35A23, 35B51, 35J96, 35P30, 47J20, 52A40.We consider the homogeneous Dirichlet problem for a class of equations which generalize the p-Laplace equations as well as the Monge- Amp`ere equations in a strictly convex domain ⊂ Rn, n ≥ 2. In the sub-linear case, we study the comparison between quantities involving the solution to this boundary value problem and the corresponding quantities involving the (radial) solution of a problem in a ball having the same η1- mean radius as . Next, we consider the eigenvalue problem for such a p-Monge-Amp`ere equation and study a comparison between its principal eigenvalue (eigenfunction) and the principal eigenvalue (eigenfunction) of the corresponding problem in a ball having the same η1-mean radius as . Symmetrization techniques and comparison principles are the main tools used to get our results

harnack inequality for non divergence structure semi linear elliptic equations

AbstractIn this paper we establish a Harnack inequality for non-negative solutions of {Lu=f(u)} where L is a non-divergence structure uniformly elliptic operator and f is a non-decreasing function that satisfies an appropriate growth conditions at infinity

Boundary estimates for solutions to singular elliptic equations

We deal with the Dirichlet problem in a bounded smooth domain.[omissis

Minimization of the first eigenvalue in problems involving the bi-laplacian

This paper concerns the minimization of the first eigenvalue in problems involvingthe bi-Laplacian under either homogeneous Navier boundary conditions or homogeneousDirichlet boundary conditions. Physically, in case of N = 2, our equation modelsthe vibration of a non homogeneous plate  which is either hinged or clamped alongthe boundary. Given several materials (with different densities) of total extension ||,we investigate the location of these materials inside  so to minimize the first modein the vibration of the corresponding plate.Keywords: bi-Laplacian, first eigenvalue, minimization.Este art´?culo trata de la minimizaci´on del primer autovalor en problemas relativosal bi-Laplaciano bajo condiciones de frontera homog´eneas de tipo Navier o Dirichlet.F´?sicamente, en el problema bi-dimensional, nuestra ecuacin modela la vibraci´on deuna placa inhomog´enea  fija con goznes a lo largo de su borde. Dados varios materiales(de diferentes densidades) y extensi´on total ||, investigamos cu´al debe serla localizaci´on de tales materiales en la placa para minimizar el primer modo de suvibraci´on.Palabras clave: bi-Laplaciano, primer autovalor, minimizaci´on

Overdetermined problems for p-Laplace and generalized Monge–Ampére equations

We investigate overdetermined problems for p-Laplace and generalized Monge-Amp´ere equations. By using the theory of domain derivative we find duality results and a characterization of the overdetermined boundary conditions via minimization of suitable functionals with respect to the domain