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The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions

In this paper, we study the superconvergence of the error for the discontinuous Galerkin (DG) finite element method for linear conservation laws when upwind fluxes are used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the DG solution and the exact solution is ($k+2$)-th order superconvergent at the downwind-biased Radau points with suitable initial discretization. Moreover, we also prove the DG solution is ($k+2$)-th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution. The superconvergence result in this paper leads to a new {\em a posteriori} error estimate. Our analysis is valid for arbitrary regular meshes and for $\mathcal{P}^k$ polynomials with arbitrary $k\geq1$, and for both periodic boundary conditions and for initial-boundary value problems. We perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal.

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This article studies the problem of image restoration of observed images corrupted by impulse noise and mixed Gaussian impulse noise. Since the pixels damaged by impulse noise contain no information about the true image, how to find this set correctly is a very important problem. We propose two methods based on blind inpainting and $\ell_0$ minimization that can simultaneously find the damaged pixels and restore the image. By iteratively restoring the image and updating the set of damaged pixels, these methods have better performance than other methods, as shown in the experiments. In addition, we provide convergence analysis for these methods; these algorithms will converge to coordinatewise minimum points. In addition, they will converge to local minimum points (or with probability one) with some modifications in the algorithms.