Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions
Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions
复制标题
非线性亥姆霍兹本征函数的存在性和渐近性
DOI：
10.1137/19m1307238
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Junyong Zhang
中科院分区：
文献类型：
--
作者：
Jesse Gell;Andrew Hassell;Jacob Shapiro;Junyong Zhang
We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta - \lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an operator, $\lambda$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p \geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutierrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic.
We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, \omega) = r^{-(n-1)/2} \Big( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} g(\omega) + O(r^{-\epsilon}) \Big), \text{ as } r \to \infty, \quad \epsilon > 0, \end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.
We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta - \lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an operator, $\lambda$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p \geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutierrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic.
We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, \omega) = r^{-(n-1)/2} \Big( e^{-i\lambda r} f(\omega) + e^{+i\lambda r} g(\omega) + O(r^{-\epsilon}) \Big), \text{ as } r \to \infty, \quad \epsilon > 0, \end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.