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Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations
Annali di Matematica Pura ed Applicata ( IF 1 ) Pub Date : 2021-03-31 , DOI: 10.1007/s10231-021-01086-6
Tobias Weth , Tolga Yeşil

We establish weighted \(L^p\)-Fourier extension estimates for \(O(N-k) \times O(k)\)-invariant functions defined on the unit sphere \({\mathbb {S}}^{N-1}\), allowing for exponents p below the Stein–Tomas critical exponent \(\frac{2(N+1)}{N-1}\). Moreover, in the more general setting of an arbitrary closed subgroup \(G \subset O(N)\) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation

$$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$

where Q is a nonnegative bounded and G-invariant weight function.



中文翻译:

对称函数的傅里叶扩展估计及其在非线性亥姆霍兹方程中的应用

我们建立加权\(L ^ p \)-傅立叶扩展估计\(O(Nk)\ times O(k)\)-在单位球面上定义的不变函数\({\ mathbb {S}} ^ {N- 1} \),允许指数p低于Stein–Tomas临界指数\(\ frac {2(N + 1)} {N-1} \)。此外,在任意封闭子组\(G \ subset O(N)\)G-不变函数的更一般的设置中,我们研究加权傅里叶扩展估计对相应加权亥姆霍兹的有界性和不消失性的影响解析运算符。最后,我们使用这些属性导出G的新存在结果Helmholtz方程的定不变解

$$ \ begin {aligned}-\ Delta u-u = Q(x)| u | ^ {p-2} u,\ quad u \ in W ^ {2,p}({\ mathbb {R}} ^ {N}),\ end {aligned} $$

其中Q是非负有界G不变权重函数。

更新日期:2021-04-01
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