Analysis of singularities in elliptic equations : the Ginzburg-Landau model of superconductivity, the Lin-Ni-Takagi problem, the Keller-Segel model of chemotaxis, and conformal geometry

Abstract : This thesis is devoted to the analysis of singularities in nonlinear elliptic partial differential equations arising in mathematical physics, mathematical biology, and conformal geometry. The topics treated are the Ginzburg-Landau model of superconductivity, the Lin-Ni-Takagi problem, the Keller-Segel model of chemotaxis, and the prescribed scalar curvature problem. The Ginzburg-Landau model is a phenomenological description of superconductivity. An essential feature of type-II superconductors is the presence of vortices, which appear above a certain value of the strength of the applied magnetic field called the first critical field. We are interested in the regime of small epsilon, where epsilon is the inverse of the Ginzburg-Landau parameter (a material constant). In this regime, the vortices are at main order co-dimension 2 topological singularities. We provide a quantitative three-dimensional vortex approximation construction for the Ginzburg-Landau energy, which gives an approximation of vortex lines coupled to a lower bound for the energy, which is optimal to leading order and valid at the epsilon-level. By using these tools we then analyze the behavior of global minimizers below and near the first critical field. We show that below this critical value, minimizers of the Ginzburg-Landau energy are vortex-free configurations and that near this value, minimizers have bounded vorticity. The Lin-Ni-Takagi problem arises as the shadow of the Gierer-Meinhardt system of reaction-diffusion equations that models biological pattern formation. This problem is that of finding positive solutions of a critical equation in a bounded smooth three-dimensional domain, under zero Neumann boundary conditions. In this thesis, we construct solutions to this problem exhibiting single bubbling behavior at one point of the domain, as a certain parameter converges to a critical value. Chemotaxis is the influence of chemical substances in an environment on the movement of organisms. The Keller-Segel model for chemotaxis is an advection-diffusion system consisting of two coupled parabolic equations. Here, we are interested in radial steady states of this system. We are then led to study a critical equation in the two-dimensional unit ball, under zero Neumann boundary conditions. In this thesis, we construct several families of radial solutions which blow up at the origin of the ball and concentrate on the boundary and/or an interior sphere, as a certain parameter converges to zero. Finally, we study the prescribed scalar curvature problem. Given an n-dimensional compact Riemannian manifold, we are interested in finding bubbling metrics whose scalar curvature is a prescribed function, depending on a small parameter. We assume that this function has a critical point which satisfies a suitable flatness assumption. We construct several metrics, which blow-up as the parameter goes to zero, with prescribed scalar curvature.
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Carlos Román. Analysis of singularities in elliptic equations : the Ginzburg-Landau model of superconductivity, the Lin-Ni-Takagi problem, the Keller-Segel model of chemotaxis, and conformal geometry. Mathematical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066343⟩. ⟨tel-01956978v1⟩

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