We analyze the effect of adding a weak, localized, inhomogeneity to a two dimensional array of oscillators with nonlocal coupling. We propose and also justify a model for the phase dynamics in this system. Our model is a generalization of a viscous eikonal equation that is known to describe the phase modulation of traveling waves in reaction-diffusion systems. We show the existence of a branch of target pattern solutions that bifurcates from the spatially homogeneous state when $\varepsilon$, the strength of the inhomogeneity, is nonzero and we also show that these target patterns have an asymptotic wavenumber that is small beyond all orders in $\varepsilon$.
The strategy of our proof is to pose a good ansatz for an approximate form of the solution and use the implicit function theorem to prove the existence of a solution in its vicinity. The analysis presents two challenges. First, the linearization about the homogeneous state is a convolution operator of diffusive type and hence not invertible on the usual Sobolev spaces. Second, a regular perturbation expansion in $\varepsilon$ does not provide a good ansatz for applying the implicit function theorem since the nonlinearities play a major role in determining the relevant approximation, which also needs to be "correct" to all orders in $\varepsilon$. We overcome these two points by proving Fredholm properties for the linearization in appropriate Kondratiev spaces and using a refined ansatz for the approximate solution, which obtained using matched asymptotics.
Nonlocal elliptic equations in bounded domains: a survey
在本文中，我们综述了关于非局部算子形式为\[Lu(x)=\textrm{PV}\int_{\mathbb R^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)dy.\]的狄利克雷问题\[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }Ω\\ u&=&g&\textrm{in }\mathbb R^n\backslashΩ\end{array}\right.\]的一些结果。我们从最基础的内容开始，证明解的存在性、最大值原理，并构造一些有用的障碍。然后，我们集中在正则性的解决方案，在内部和边界上的区域。为了在正则性理论中包括一些自然算子$L$，我们不假设核上有任何正则性。这导致了一些有趣的功能，是纯粹的非本地，在这个意义上，没有类似的本地方程。我们希望，这项调查将是有用的新的和更有经验的研究人员在该领域。
