Shiferaw Berhanu

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)−b 0 δ 2/(γ+1) (x)| (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.

Global properties of a class of planar vector fields of infinite type

This paper studies some global and semi global properties of infinite type, planar, C-valued real analytic vector fields that are invariant under the rotation group. Results are proved on the integrability, kernel, range and classification of such operators.

Orbits and global unique continuation for systems of vector fields

SupposeM is a smooth manifold andV is a locally integrable vector subbundle of the complexified tangent bundle ofM. This paper explores the global unique continuation property of distribution solutions ofV, i.e., the distributionsu onM such thatLu = 0 wheneverL is a section ofV, and the closely related problem of the structure of the Sussmann orbits ofHV.

An F. and M. Riesz theorem for planar vector fields

Abstract. We prove that solutions of the homogeneous equation Lu=0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if

The trace problem for vector fields satisfying Hörmander's condition

\Omega

On rotationally invariant vector fields in the plane

is an open subset of the plane with smooth boundary,

Liouville's theorem and the maximum modulus principle for a system of complex vector fields

u\in C^1(\Omega)

Uniqueness for locally integrable solutions of overdetermined systems

satisfiesLu=0 on

A Class of FBI Transforms

\Omega

A similarity principle for complex vector fields and applications

, has tempered growth at the boundary, and its weak boundary value is a measure