We study a stationary scattering problem related to the nonlinear Helmholtz equation $-\Delta u - k^2 = f(x,u) \ \text{in $\mathbb{R}^N$,}$ where $N \ge 3$ and $k>0$. For given incident free wave $\varphi \in L^\infty(\mathbb{R}^N)$, we prove existence of complex-valued solutions form $u=\varphi+u_{\text{sc}}$, $u_{\text{sc}}$ satisfies Sommerfeld outgoing radiation condition. Since neither var...

Processes that can be modeled with numerical calculations of acoustic pressure fields include medical and industrial ultrasound, echo sounding, and environmental noise. We present two methods for making these calculations based on Helmoltz equation. The first method is based directly on the complex-valued Helmholtz equation and an algebraic multigrid approximation of the discretized shifted-Lap...

The “potato kugel” theorem of Aharonov, Schiffer, and Zalcman, which concerns an inverse property harmonic functions, is extended to the settings modified Helmholtz equation that describe nuclear forces acoustic waves respectively.

The mean flux theorems are proved for solutions of the Helmholtz equation and its modified version. Also, their converses considered along with some other properties which generalize those that guarantee harmonicity.

finite and boundary element methods have been used by many authors to solve mathematicalphysics problems. however, the coupling of these two methods happens to be more efficient as it combinestheir merits. in this paper, the mathematical analysis of the coupling of finite and boundary element methodsfor the helmholtz equation is presented.