Topological Sensitivity Analysis for the Anisotropic Laplace Problem

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.

Electrostatic description of 3d $$ \mathcal{N} $$ = 4 linear quivers

We present the holographic dual for the strongly coupled, low energy dynamics of balanced$$ \mathcal{N} $$ N = 4 field theories in (2 + 1) dimensions. The infinite family of Type IIB backgrounds with AdS4× S2× S2 factors is described in terms of a Laplace problem with suitable boundary conditions. The system describes an array of D3, NS5 and D5 branes. We study various aspects of these Hanany-Witten set-ups (number of branes, linking numbers, dimension of the Higgs and Coulomb branches) and encode them in holographic calculations. A generic expression for the Free Energy/Holographic Central Charge is derived. These quantities are then calculated explicitly in various general examples. We also discuss how Mirror Symmetry is encoded in our Type IIB backgrounds. The connection with previous results in the bibliography is made.

A multiparameter fractional Laplace problem with semipositone nonlinearity

<p style='text-indent:20px;'>In this paper we prove the existence of at least one positive solution for nonlocal semipositone problem of the type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (P_\lambda^\mu)\left\{ \begin{array}{rcl} (-\Delta)^s u&amp; = &amp; \lambda(u^{q}-1)+\mu u^r \text{ in } \Omega\\ u&amp;&gt;&amp;0 \text{ in } \Omega\\ u&amp;\equiv &amp;0 \text{ on }{\mathbb R^N\setminus\Omega}. \end{array}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>when the positive parameters <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> belong to certain range. Here <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is assumed to be a bounded open set with smooth boundary, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0, 1), N&gt; 2s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;q&lt;1&lt;r\leq \frac{N+2s}{N- 2s}. $\end{document}</tex-math></inline-formula> First we consider <inline-formula><tex-math id="M6">\begin{document}$ (P_ \lambda^\mu) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \mu = 0 $\end{document}</tex-math></inline-formula> and prove that there exists <inline-formula><tex-math id="M8">\begin{document}$ \lambda_0\in(0, \infty) $\end{document}</tex-math></inline-formula> such that for all <inline-formula><tex-math id="M9">\begin{document}$ \lambda&gt; \lambda_0 $\end{document}</tex-math></inline-formula> the problem <inline-formula><tex-math id="M10">\begin{document}$ (P_ \lambda^0) $\end{document}</tex-math></inline-formula> admits at least one positive solution. In fact we will show the existence of a continuous branch of maximal solutions of <inline-formula><tex-math id="M11">\begin{document}$ (P_\lambda^0) $\end{document}</tex-math></inline-formula> emanating from infinity. Next for each <inline-formula><tex-math id="M12">\begin{document}$ \lambda&gt;\lambda_0 $\end{document}</tex-math></inline-formula> and for all <inline-formula><tex-math id="M13">\begin{document}$ 0&lt;\mu&lt;\mu_{\lambda} $\end{document}</tex-math></inline-formula> we establish the existence of at least one positive solution of <inline-formula><tex-math id="M14">\begin{document}$ (P_\lambda^\mu) $\end{document}</tex-math></inline-formula> using variational method. Also in the sub critical case, i.e., for <inline-formula><tex-math id="M15">\begin{document}$ 1&lt;r&lt;\frac{N+2s}{N-2s} $\end{document}</tex-math></inline-formula>, we show the existence of second positive solution via mountain pass argument.</p>

An indirect BIE free of degenerate scales

<p style='text-indent:20px;'>Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.</p>

Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in {\mathbb{R}^{3}}, i.e. nonlocal minimal surfaces with a graphical structure.We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum.This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error.We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.