0 Se p 20 06 Boundary singularities of N - harmonic functions ∗

Solution of Harmonic Problems with Weak Singularities Using Equilibrated Basis Functions in Finite Element Method

In this paper, Equilibrated Singular Basis Functions (EqSBFs) are implemented in the framework of the Finite Element Method (FEM), which can approximately satisfy the harmonic PDE in homogeneous and heterogeneous media. EqSBFs are able to automatically reproduce the terms consistent with the singularity order in the vicinity of the singular point. The newly made bases are used as the compliment...

ar X iv : 0 80 5 . 36 61 v 1 [ m at h . A P ] 2 3 M ay 2 00 8 Boundary singularities of solutions of N - harmonic equations with absorption ∗

Abstract We study the boundary behaviour of solutions u of −∆Nu + |u| u = 0 in a bounded smooth domain Ω ⊂ R subject to the boundary condition u = 0 except at one point, in the range q > N − 1. We prove that if q ≥ 2N − 1 such a u is identically zero, while, if N − 1 < q < 2N − 1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such si...

Boundary singularities of N -harmonic functions

ar X iv : m at h / 06 05 53 3 v 2 [ m at h . PR ] 1 9 Ju n 20 06 Potential Theory of Truncated Stable Processes

R d with a Lévy density given by c|x| 1{|x|<1} for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic nonnegative functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded co...

Se p 20 06 Asymptotic stability of harmonic maps under the Schrödinger flow ∗

For Schrödinger maps from R ×R+ to the 2-sphere S, it is not known if finite energy solutions can form singularities (“blowup”) in finite time. We consider equivariant solutions with energy near the energy of the two-parameter family of equivariant harmonic maps. We prove that if the topological degree of the map is at least four, blowup does not occur, and global solutions converge (in a dispe...