In this paper we apply KAM theory and the Aubry-Mather theory for twist maps to the study of bound geodesic dynamics of a perturbed blackhole background. The general theories apply mainly to two observable phenomena: the photon shell (unstable bound spherical orbits) and the quasi-periodic oscillations (QPO). We prove that there is a gap structure in the photon shell that can be used to reveal information of the perturbation.

The Camassa-Holm-Kadomtsev-Petviashvili-I equation (CH-KP-I) is a two dimensional generalization of the Camassa-Holm equation (CH). In this paper, we prove transverse instability of the line solitary waves under periodic transverse perturbations. The proof is based on the framework of [18]. Due to the high nonlinearity, our proof requires necessary modification. Specifically, we first establish the linear instability of the line solitary waves. Then through an approximation procedure, we prove that the linear effect actually dominates the nonlinear behavior.

We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number) of the set$$\left\{\iint_{\{u>0\}\times\{u>0\}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2\sigma}}dxdy:u\in H^{\sigma}(\mathbb{R}^n), \int_{\mathbb{R}^n}u^2=1, |\{u>0\}|\le 1\right\}$$Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $\mathbb{R}^n \times \mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.

We consider the Nemytskii operators $u\to |u|$ and $u\to u^{\pm}$ in a bounded domain $\Omega$ with $C^2$ boundary. We give elementary proofs of the boundedness in $H^s(\Omega)$ with $0\le s<3/2$.