Analysis of PDEs (math.AP)

Abstract: We study finite total curvature solutions of the Liouville equation $\Delta u+e^{2u}=0$ on a complete surface $(M,g)$ with nonnegative Gauss curvature. It turns out that the asymptotic behavior of the solution separates two extremal cases: on the one end, if the solution decays not too fast, then $(M,g)$ must be isometric to the standard Euclidean plane; on the other end, if $(M,g)$ is isometric to the flat cylinder $\mathbb{S}^1\times \mathbb{R}$, then solutions must decay linearly and are completely classified.

Abstract: In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

Abstract: This paper deals with the existence of guided waves and edge states in particular two-dimensional media obtained by perturbing a reference periodic medium with honeycomb symmetry. This reference medium is a thin periodic domain (the thickness is denoted $\delta$ > 0) with an hexagonal structure, which is close to an honeycomb quantum graph. In a first step, we show the existence of Dirac points (conical crossings) at arbitrarily large frequencies if $\delta$ is chosen small enough. We then perturbe the domain by cutting the perfectly periodic medium along the so-called zigzag direction, and we consider either Dirichlet or Neumann boundary conditions on the cut edge. In the two cases, we prove the existence of edges modes as well as their robustness with respect to some perturbations, namely the location of the cut and the thickness of the perturbed edge. In particular, we show that different locations of the cut lead to almost-non dispersive edge states, the number of locations increasing with the frequency. All the results are obtained via asymptotic analysis and semi-explicit computations done on the limit quantum graph. Numerical simulations illustrate the theoretical results.

Abstract: We consider a Serrin-type problem in convex cones in the Euclidean space and motivated by recent rigidity results we study the quantitative stability issue for this problem. In particular, we prove both sharp Lipschitz estimates for an $L^2-$pseudodistance and estimates in terms of the Hausdorff distance.

Abstract: In this paper, we consider a fourth order inner-controlled parabolic equation on an open bounded subset of $R^d$, or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskii-Sapiro condition. We derive a spectral inequality for the solution of the parabolic system that yields a null-controllability result. The spectral inequality is a consequence of an interpolation inequality obtained via a Carleman inequality for the bi-Laplace operator under the considered boundary conditions.

Abstract: The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional $p-$Laplace equation. In doing so, with the help of tools from potential analysis, such as fractional relative Sobolev capacities, Wiener type integrals, Wolff potentials, $(\alpha,p)-$barriers, and $(\alpha,p)-$balayages, we first prove the characterizations of the fractional thinness and the Perron boundary regularity. Then, we establish a Wiener test and a generalized fractional Wiener criterion. Furthermore, we also prove the continuity of the fractional superharmonic function, the fractional resolutivity, a connection between $(\alpha,p)-$potentials and $(\alpha,p)-$Perron solutions, and the existence of a capacitary function for an arbitrary condenser.