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Estimates for k -Hessian operator and some applications

Dongrui Wan (2013)

Czechoslovak Mathematical Journal

The k -convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations F k [ u ] = 0 , where F k [ u ] is the elementary symmetric function of order k , 1 k n , of the eigenvalues of the Hessian matrix D 2 u . For example, F 1 [ u ] is the Laplacian Δ u and F n [ u ] is the real Monge-Ampère operator det D 2 u , while 1 -convex functions and n -convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we establish an approximation theorem for negative k -convex functions, and give several...

Estimation of Green's function on piecewise Dini-smooth bounded Jordan domains

Mohamed Amine Ben Boubaker, Mohamed Selmi (2013)

Colloquium Mathematicae

We establish inequalities for Green functions on general bounded piecewise Dini-smooth Jordan domains in ℝ². This enables us to prove a new version of the 3G Theorem which generalizes its previous version given in [M. Selmi, Potential Anal. 13 (2000)]. Using these results, we give a comparison theorem for the Green kernel of Δ and the Green kernel of Δ - μ, where μ is a nonnegative and exact Radon measure.

Existence of solutions to the Poisson equation in L p -weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of L p , p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.

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