论文标题
非线性Helmholtz本征函数的散射矩阵的规律性
Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions
论文作者
论文摘要
我们研究非线性Helmholtz方程$（δ-λ^2） Laplace-Beltrami操作员在渐近的欧几里得或圆锥形歧管上，$ V $是短范围的潜力，而$ n [u] $是更一般的多项式非线性。在条件下$（p-1）（n-1）> 4 $和$ k>（n-1）/2 $，每$ f \ in h^k（s^{n-1}_Ω）的每一个足够小规范的$，我们表明有一个非线性helmholtz eigenfunction form \ begin \ begin \ begin \ begin \ begin {等式*equient {等式*} e^{-iλr} f(ω) + e^{+iλr} b(ω) + O(r^{-ε}) \Big), \qquad \text{as } r \to \infty, \end{equation*} for some $b \in H^k(S_ω^{n-1})$ and $ε> 0$.也就是说，散射矩阵$ f \ mapsto b $保留Sobolev的规律性，这是对作者与张的先前工作的改进，这证明了相似的结果，损失了四个衍生物。
We study the nonlinear Helmholtz equation $(Δ- λ^2)u = \pm |u|^{p-1}u$ on $\mathbb{R}^n$, $λ> 0$, $p \in \mathbb{N}$ odd, and more generally $(Δ_g + V - λ^2)u = N[u]$, where $Δ_g$ is the (positive) Laplace-Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1) > 4$ and $k > (n-1)/2$, for every $f \in H^k(S^{n-1}_ω)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} b(ω) + O(r^{-ε}) \Big), \qquad \text{as } r \to \infty, \end{equation*} for some $b \in H^k(S_ω^{n-1})$ and $ε> 0$. That is, the scattering matrix $f \mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work with Zhang, that proved a similar result with a loss of four derivatives.
