Analysis of PDEs (math.AP)

Abstract: We study the optimal partition problem for the prescribed constant $Q$-curvature equation induced by the higher order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed $Q$-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant $Q$-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of $GJMS$-operators on Ricci solitons, compute the $Q$-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with prescribed number of nodal domains on the Koiso-Cao Ricci soliton.

Abstract: In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about: the existence of a solution to the first eigentriplet problem; the geometry of the principal eigenvalue problem; the asymptotic stability of the first eigenvector with possible constructive rate of convergence.This abstract theory is motivated and illustrated by several examples of differential, intro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for: some parabolic equations in a bounded domain and in the whole space; some transport equations in a bounded or unbounded domain, including some growth-fragmentationmodels and some kinetic models; the kinetic Fokker-Planck equation in the torus and in the whole space; some mutation-selection models.

Abstract: Anyons with a statistical phase parameter $\alpha\in(0,2)$ are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the $y$-direction than in the $x$-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the $x$-direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on $\alpha$, and no trace left of the long-range interactions of the 2D model.

Abstract: We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calder\'on's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann-to-Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.

Abstract: In this work we study a priori bounds for weak solution to elliptic problems with nonstandard growth that involves the so-called $g-$Laplace operator. The $g-$Laplacian is a generalization of the $p-$Laplace operator that takes into account different behaviors than pure powers. The method to obtain this a priori estimates is the so called ``blow-up'' argument developed by Gidas and Spruck. Then we applied this a priori bounds to show some existence results for these problems.

Abstract: We prove quantitative unique continuation estimates for relatively dense sets and spectral subspaces associated to small energies of Schr\"odinger operators on Riemannian manifolds with Ricci curvature bounded below. The upper bound for the energy range and the constant appearing in the estimate are given in terms of the lower bound of the Ricci curvature and the parameters of the relatively dense set.

Abstract: We provide a full series expansion of a generalization of the so-called $u$-capacity related to the Dirichlet-Laplacian in dimension three and higher, extending previous results of the authors, and of the authors together with Virginie Bonnaillie-No\"el, dealing with the planar case. We apply the result in order to study the asymptotic behavior of perturbed eigenvalues when Dirichlet conditions are imposed on a small regular subset of the domain of the eigenvalue problem.

Abstract: In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems \begin{align*} \left\{ \begin{aligned} &-\epsilon^2\Delta u+V(x)u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N, &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}\epsilon^N, \end{aligned} \right. \end{align*} where $a,\epsilon>0$, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V:\mathbb{R}^N \to [0,\infty)$ is a continuous function, and $f$ is a continuous function with $L^2$-subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential $V$ attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer.

Abstract: In this paper we study the focusing inhomogeneous 3D nonlinear Schr\"odinger equation with inverse-square potential in the mass-supercritical and energy-subcritical regime. We first establish local well-posedness in $\dot{H}_a^{s_c}\cap \dot{H}_a^1$, with $s_c=3/2-(2-b)/2\sigma$. Next, we prove the blow-up of the scaling invariant Lebesgue norm for radial solutions and also, with an additional restriction, in the non-radial case.