Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

Variational approach to scattering by unbounded rough surfaces with Neumann and generalized impedance boundary conditions

Pages: 511 – 537



Guanghui Hu (Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany)

Xiaodong Liu (Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China)

Fenglong Qu (School of Mathematics and Information Science, Yantai University, Yantai, Shandong, China)

Bo Zhang (LSEC and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, China)


This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an infinite penetrable medium with a finite height modeled by the Helmholtz equation. On the lower boundary of the rough layer, the Neumann or generalized impedance boundary condition is imposed. The scattered field in the unbounded homogeneous medium is required to satisfy the upward angular-spectrum representation. Using the variational approach, we prove uniqueness and existence of solutions in the standard space of finite energy for inhomogeneous source terms, and in appropriate weighted Sobolev spaces for incident point source waves in $\mathbb{R}^m (m=2,3)$ and incident plane waves in $\mathbb{R}^2$. To avoid guided waves, we assume that the penetrable medium satisfies certain non-trapping and geometric conditions.


rough surface scattering, Helmholtz equation, generalized impedance boundary condition, Neumann boundary condition, angular-spectrum representation, uniqueness and existence, weighted Sobolev space

2010 Mathematics Subject Classification

35J05, 35J20, 35J25, 42B10, 78A45

Published 3 December 2014