Axial symmetry and regularity of solutions to an integral equation in a half-space

Guozhen Lu and Jiuyi Zhu

Vol. 253 (2011), No. 2, 455–473

DOI: 10.2140/pjm.2011.253.455

Abstract

We consider the integral equation

where G(x,y) is the Green’s function of the corresponding polyharmonic Dirichlet problem in a half-space. We prove by the method of moving planes in integral form that, under some integrability conditions, the solutions are axially symmetric with respect to some line parallel to the x_n-axis and nondecreasing in the x_n direction, which further implies the nonexistence of solutions. We also show similar results for a class of systems of integral equations. This appears to be the first paper in which the moving plane method in integral form is employed in a half-space to derive axial symmetry.

We also obtain the regularity of the integral equation in a half-space

by the regularity lifting method. As a corollary, we prove the nonexistence of nonnegative solutions to this equation. Moreover, we show that the nonnegative solutions in this equation only depend on x_n if u ∈ L_loc^{2n∕(n−2m)}(ℝ₊ⁿ) and 1 < p < (n + 2m)∕(n − 2m).

Keywords

axial symmetry, regularity of solutions, half-space, Green’s functions for polyharmonic operators, integral equation, nonexistence of solutions

Mathematical Subject Classification 2010

Primary: 35J60

Secondary: 45G15

Milestones

Received: 9 September 2010

Revised: 30 May 2011

Accepted: 11 July 2011

Published: 21 January 2012