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Damien Galant

Publications and source records attributed to Damien Galant.

10 recordsLinked to original sources
arXiv2026

The energy of fractional Allen--Cahn layers in dimension one

We study the energy $\mathcal{E}(s) := E_{s}[\Phi_s]$ of the one-dimensional fractional Allen--Cahn layer solution $\Phi_s$, defined as the unique odd, increasing solution of $(-\Delta)^{s} \Phi_s = \Phi_s - \Phi_s^{3}$ with $\Phi_s(\pm\infty)=\pm 1$ and $\Phi_s(0)=0$, for $s\in(1/2,1]$. Our main results are a sharp qualitative and quantitative description of the energy $\mathcal{E}$ on this interval. We show that the energy is continuous and strictly decreasing, and we obtain explicit asymptotic expansions at both endpoints. At the upper endpoint we prove $\mathcal{E}(s) = \frac{2\sqrt{2}}{3} + \kappa_1 (1-s) + o(1-s)$ with an explicit formula for $\kappa_1$. At the lower endpoint we prove that the energy goes to infinity as $\mathcal{E}(s)=\frac{1}{\pi(s-1/2)}+O(1)$. The strict decrease is proved with computer assistance. On an interior subinterval it is reduced to finitely many inequalities verified by interval-arithmetic computations. The proofs combine the Cabr\'e--Sire construction of the layer, the minimality theorem of Palatucci--Savin--Valdinoci, an identity for the $s$ derivative of the energy, and a computer-assisted coercivity estimate at explicit approximate layers.

math.AP↗
arXiv2026

Blow-up analysis and a priori bounds for NLS equations on metric graphs

We consider, on a connected metric graph $\mathcal{G}$, a family of nonlinear Schrödinger equations $$ -u'' + W_n(x) u + λ_n u = ρ_n(x)|u|^{p-2}u, \quad n \in \mathbb{N}. \qquad (*) $$ We assume that $p > 2$, $(W_n)$, $(ρ_n) \subseteq L^{\infty}(\mathcal{G})$ with $ρ_n \geq 0$, $|W_n|_{L^\infty(\mathcal{G})}$ and $|ρ_n|_{L^\infty(\mathcal{G})}$ are bounded and $λ_n \to +\infty$. Given $n \in \mathbb{N}$, we call "solution" a function $u_n \in H^1(\mathcal{G})$ which satisfies (*) for that $n\in \mathbb{N}$ together with the Kirchhoff conditions at the vertices. Focusing on the limiting behavior of sequences $(u_n) \subseteq H^1(\mathcal{G})$ of solutions as $λ_n \to + \infty$ and assuming that the Morse index $m(u_n)$ of $u_n$ is uniformly bounded, we establish, the existence of a finite subset of blow-up points away from which, up to a subsequence, $|u_n|$ has a global exponential decay. These points are generally a strict subset of the blow-up points, and their number is estimated by the bound on the Morse index of $(u_n)$. It is the first time that this global exponential decay property is established on graphs even if one consider only signed solutions. In the last part of the paper we derive various results of a priori bounds on the solutions in $L^\infty$ and $L^2$. Our blow-up analysis, combined with ODE arguments allows, for frequently considered classes of graphs, to obtain a fairly complete picture of the relationships between the number of nodal regions, Morse index, $L^\infty$ and $L^2$ norms of solutions.

math.AP↗
arXiv2026

Normalized solutions of Nehari-Pankov type to mass-supercritical indefinite variational problems

We consider abstract nonlinear equations of the form $A u = λu + I'(u)$, where $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$, $λ\in \mathbb{R}$ is a parameter, and $u \mapsto I'(u)$ is a superlinear term of variational nature. In this abstract setting, we develop a new approach to detect prescribed norm solutions in $H$ which does not rely on any mass-subcriticality assumptions. We then consider various applications of this approach. First, we obtain, under general assumptions including the full mass-supercritical parameter regime, the existence of (infinitely many) solutions to a class of nonlinear Schrödinger equations on a compact graph $\mathcal{G}$ with prescribed arbitrarily large mass, thereby improving previous results which only cover small masses. Moreover, we derive a similar result for a biharmonic Schrödinger equation in the $2$-torus. For a larger class of second order and higher order equations in a bounded domain with Dirichlet boundary conditions, we also show the existence of multiple solutions with prescribed small mass. The solutions we obtain are detected as ground states of Nehari-Pankov type for the associated $λ$-dependent action functional, where $λ$ varies in a spectral gap between sufficiently large eigenvalues of $A$. The key new observation in this abstract framework is the fact that the $H$-norms of these $λ$-dependent solution families form connected sets even though the solution families themselves may be disconnected. To estimate the size of these connected sets in specific settings, we use Weyl type estimates for the length of spectral gaps, variational characterizations of eigenvalues, bounds for associated eigenfunctions and a bound from analytic number theory.

math.AP↗
arXiv2026

On the defocusing stationary nonlinear Schrödinger equation on metric graphs

We study the defocusing nonlinear Schrödinger equation on noncompact metric graphs under general self-adjoint vertex conditions ensuring the existence of a negative eigenvalue of the Hamiltonian operator. First, we focus on the existence of energy ground states with prescribed mass. We show that existence and stability always hold for small masses and fail for large masses in the $L^2$-subcritical regime. For $δ$-type vertex conditions, we provide more precise results: ground states exist for all masses in the $L^2$-critical and supercritical cases, while in the subcritical case, for one vertex graphs, there exists a sharp mass threshold such that ground states exist below it and do not exist above it. Moreover, we show that the ground state bifurcates from the vanishing solution at the bottom of the Hamiltonian spectrum. Finally, we present multiplicity results for stationary solutions, both in the fixed-frequency and fixed-mass settings.

math.AP↗
arXiv2024

An action approach to nodal and least energy normalized solutions for nonlinear Schrödinger equations

We develop a new approach to the investigation of normalized solutions for nonlinear Schrödinger equations based on the analysis of the masses of ground states of the corresponding action functional. Our first result is a complete characterization of the masses of action ground states, obtained via a Darboux-type property for the derivative of the action ground state level. We then exploit this result to tackle normalized solutions with a twofold perspective. First, we prove existence of normalized nodal solutions for every mass in the $L^2$-subcritical regime, and for a whole interval of masses in the $L^2$-critical and supercritical cases. Then, we show when least energy normalized solutions/least energy normalized nodal solutions are action ground states/nodal action ground states.

math.AP↗
arXiv2023

Constant sign and sign changing NLS ground states on noncompact metric graphs

We investigate existence and nonexistence of action ground states and nodal action ground states for the nonlinear Schrödinger equation on noncompact metric graphs with rather general boundary conditions. We first obtain abstract sufficient conditions for existence, typical of problems with lack of compactness, in terms of ``levels at infinity'' for the action functional associated with the problems. Then we analyze in detail two relevant classes of graphs. For noncompact graphs with finitely many edges, we detect purely topological sharp conditions preventing the existence of ground states or of nodal ground states. We also investigate analogous conditions of metrical nature. The negative results are complemented by several sufficient conditions to ensure existence, either of topological or metrical nature, or a combination of the two. For graphs with infinitely many edges, all bounded, we focus on periodic graphs and infinite trees. In these cases, our results completely describe the phenomenology. Furthermore, we study nodal domains and nodal sets of nodal ground states and we show that the situation on graphs can be totally different from that on domains of $\mathbb{R}^N$.

math.AP↗
arXiv2023

On the notion of ground state for nonlinear Schrödinger equations on metric graphs

We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.

math.AP↗
arXiv2022

Geography of bilinearized Legendrian contact homology

We study the geography of bilinearized Legendrian contact homology for closed, connected Legendrian submanifolds with vanishing Maslov class in 1-jet spaces. We show that this invariant detects whether the two augmentations used to define it are DGA homotopic or not. We describe a collection of graded vector spaces containing all possible values for bilinearized Legendrian contact homology and then show that all these vector spaces can be realized.

math.SG↗
arXiv2021

A note on optimal degree-three spanners of the square lattice

In this short note, we prove that the degree-three dilation of the square lattice $\mathbb{Z}^2$ is $1+\sqrt{2}$. This disproves a conjecture of Dumitrescu and Ghosh. We give a computer-assisted proof of a local-global property for the uncountable set of geometric graphs achieving the optimal dilation.

cs.CG↗