Papers

We study steady solutions of the two dimensional incompressible Euler equations in the exterior of a compact domain. We assume that fluid enters the exterior domain through every point of the obstacle boundary. Under suitable assumptions on the direction and decay of the velocity at infinity, we prove that the flow is irrotational if and only if the velocity has no stagnation points.

We consider a general smooth bounded strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq2$ and describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. Our results hold more generally on compact smooth Riemannian manifolds with smooth strictly convex boundary. More specifically, we construct a sharp local in (semiclassical) time parametrix and then proceed to obtain dispersion estimates: our fixed-time decay rate for the Green function exhibits a loss of $1/4$ in the exponent of $h/t$ with respect to the boundaryless case. The loss is sharp for dispersion, as shown earlier by the first author in the case of a model convex domain. On compact three-dimensional manifolds with smooth strictly convex boundary, the resulting spectrally localized Strichartz estimates yield global well-posedness in the energy space for the defocusing cubic nonlinear Schrödinger equation, thus matching the corresponding result for boundaryless manifolds due to Burq-Gérard-Tzvetkov. Moreover, we extend bounds on the time growth of higher Sobolev norms from Planchon-Visciglia-Tzvetkov to our setting.

Physics-informed machine learning as recently emerged as a new paradigm for the resolution of partial differential equations, including functionalities for parametric modeling and inverse design, using neural networks to represent the solution fields. However, several difficulties have been reported concerning the training of such networks, resulting in slow convergence, low accurate solutions and sensitivity to the choice of hyper-parameters. In order to better understand the mechanisms underlying these difficulties, the present work proposes a rigourous comparison of neural networks and B-Splines representations within the same physics-informed learning framework. Four problems with closed-form solutions are numerically studied to establish a rigorous analysis of the accuracy and efficiency of the two representations, in terms of asymptotic convergence rates, sampling requirements, loss function minimization and arithmetic precision needed.

In this paper we prove the stability and the continuity of the free boundaries of two-phase and one-phase weak solutions to the Stefan problem in any dimension. Our proof has two main ingredients. The first one is an energy conservation principle for weak solutions that we derive for general domains in both the one-phase and the two-phase settings; this energy conservation principle provides the modulus of continuity for the displacement of the free boundary and is a flexible tool for analyzing long-time behavior of solutions. The second key ingredient for the control of the displacement of the free boundary is a new lower density bound for the melting region at free boundary points that move with maximal velocity. We also discuss some applications, showing a series of qualitative and quantitative results on the asymptotic behavior of weak solutions.

We study the fundamental solution operators associated to Kolmogorov equations with symmetric non-local diffusion with rough kernels. We prove that these operators are induced by measurable kernels and prove sharp upper bounds in terms of a kinetic profile governed by the interplay of transport and non-local diffusion. Our proof combines the small-jump with loss decomposition of P.-A. Meyer with the ideas of Nash. The proof is purely analytic.

We study the $\Gamma$-convergence of a general class of anisotropic phase-field functionals modelling cohesive fracture in the multidimensional, vector-valued, and geometrically nonlinear setting. The limiting energy comprises a quasiconvexified bulk term, a Cantor part governed by its recession function, and a cohesive surface energy density defined via an asymptotic cell problem. After establishing the structural properties and equivalent formulations of the limiting surface density, we show that, whenever the recession function of the elastic energy is given by the square of an operator norm compatible with the phase-field anisotropy, the vectorial cell problem explicitly reduces to a one-dimensional variational problem. As a consequence, we extend the cohesive law reconstruction procedure of \cite{alessi2025phasefieldpart2} to the vectorial framework, explicitly designing phase-field potentials that recover prescribed classes of anisotropic, jump-dependent traction-separation laws of the form $g(\zeta,\nu)=\gamma(\nu)g_{\scal}(|\zeta|)$.

Spectral Barron spaces are Fourier-Lebesgue-type spaces that arise in the approximation of functions by neural networks. Conventional spectral Barron spaces are mostly considered on the whole space or on a bounded domain with certain smoothness assumptions on the boundary. In this work, we investigate spectral Barron spaces for periodic functions, thereby extending the study of spectral Barron spaces to high-dimensional tori and, more generally, to locally compact Abelian groups. These extensions are carried out by considering Fourier series expansions that are isometrically isomorphic to weighted sequence spaces built upon the $\ell^1$ sequence space. We discuss the well-posedness of PDEs in these newly introduced spaces. Two specific inverse problems for evolution equations are investigated, yielding conditional stability within the framework of spectral Barron spaces on the torus.

This paper proves propagation of energy moments for the three-dimensional plasma-charge model, namely, the repulsive Vlasov-Poisson system with a moving point charge. For every \(K>2\), an initially finite \(K\)th energy moment remains finite on every bounded time interval, without any higher-order moment assumption, smallness condition on the plasma mass, or initial separation from the point charge. The constructed global weak solution conserves total energy for almost every time. The proof relies on a critical space-time density estimate and a direct control of the electric field along the moving charge.

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_\gamma $, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $\gamma\in [0,n)$, between the classical Lorentz sequence spaces $\lambda_p(v)$ and $\lambda_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_\gamma x$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_\gamma$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the authors' recent paper, which enables us to obtain the final characterization.

Boscaggin, Feltrin and Zanolin conjectured that the Neumann and periodic problems associated with the equation \begin{equation*} u”+a(t)u^\gamma=0 \end{equation*} with a two-step indefinite weight $a(t)$ have at most one positive solution for every $\gamma\in\mathbb{R}\setminus\left\{-1,0,1\right\}$. We prove this conjecture and complete the uniqueness picture by also including the logarithmic potential case $\gamma=-1$. More generally, for every $p>1$, we consider the equation \begin{equation*} \left(\left\lvert u'\right\rvert^{p-2}u'\right)'+a(t)u^\gamma=0 \end{equation*} under Neumann or periodic boundary conditions and establish a complete existence and uniqueness classification for all $\gamma\in\mathbb{R}\setminus\left\{0,p-1\right\}$. The necessary mean-value condition $\gamma\int_{0}^{T}a(t)\,\mathrm{d} t<0$ is also sufficient for existence when $\gamma\leq-1$ or $\gamma>p-1$, while an additional sharp condition on the mean of the weight is required in the intermediate range $\gamma\in(-1,p-1)\setminus\{0\}$. Finally, a numerical example provides evidence of multiple positive solutions for the Neumann problem whose weight has the same single change of sign but a non-constant internal profile, highlighting the role of the two-step assumption.

We establish transference results for multilinear Fourier multiplier operators from Euclidean spaces to product spaces of the form $\mathbb R^{d_1}\times\mathbb T^{d_2}$. We prove strong and weak-type transference principles that preserve the corresponding Euclidean operator bounds. As applications, we obtain bounds for certain multilinear operators, as well as for some multiplier operators arising naturally in PDEs.

We study the Cauchy problem for the one-dimensional isothermal compressible Navier–Stokes system with the van der Waals equation of state, and analyze the well-posedness and asymptotic stability of steady-state phase-transition solutions at subcritical temperatures. Working in Lagrangian coordinates, we use variational techniques to construct piecewise smooth steady-state solutions corresponding to two-phase equilibrium states under finite-mass perturbations. Introducing artificial-viscosity regularization, we first establish existence of the global minimizer of the regularized variational problem, unique within each fixed symmetry class; passing to the vanishing artificial-viscosity limit then yields a physically admissible steady-state solution with two phase interfaces. For initial perturbations that are small with respect to energy, piecewise-gradient, total-variation, and integral norms but may exhibit large pointwise jumps across phase boundaries, we derive uniform a priori bounds for weak solutions near the two-phase equilibrium profile. We further prove that these global weak solutions converge to the steady-state two-phase equilibrium in $L^\infty(\mathbb{R})$ as $t\to\infty$, thereby characterizing their large-time asymptotic behavior. Physically, finite mass perturbations break the uniqueness of the uniform steady state and generate infinitely many piecewise constant two-phase equilibria satisfying the Maxwell equal-area rule. Artificial viscosity acts as a selection principle that singles out the physically relevant two-interface steady-state configuration, and this two-phase equilibrium is asymptotically stable under small integral perturbations that allow large-amplitude discontinuities at phase interfaces.

We consider a family of non-local and non-convex functionals depending on three real parameters $(\gamma,p,\lambda)$, which was introduced by H. Brezis, A. Seeger, J. Van Schaftingen and P. L. Yung. In the case $\gamma<-p$ we prove a lower bound for the Gamma-liminf of these functionals as $\lambda\to 0^+$ and a Poincaré-type inequality. We also show that such inequality cannot hold if $\gamma>0$, and we establish a modified version in this case. These results answer some open questions posed by these authors and by H. M. Nguyen.

We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles $\theta_\pm$, the angle $\theta$ between the skew lines, and the initial data $\gamma_0$, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.

We study rigidity for the weighted $p$-Laplace equation on smooth bounded domains in dimensions three and higher. We show that a smooth positive weight sufficiently close to $1$ is identically $1$ if the derivative of its Dirichlet-to-Neumann map at one affine boundary value agrees with that for the unit weight. We also prove local rigidity at every smooth positive reference weight which is constant in one direction. The reference weight need not be close to $1$, and the unknown weight is only required to be sufficiently close to it, without a directional restriction. No convexity, analyticity, or agreement near the boundary is assumed. The proof combines linearization at a solution without critical points with recent rigidity results for the anisotropic Calderón problem. An identity for the linearized conductivity tensors removes the remaining diffeomorphism. This last step does not require smallness of either weight.

In this paper, we investigate nonexistence results for stable solutions of the semilinear elliptic problem $$(-\Delta)^s u+\lambda|x|^\alpha u=|x|^\beta |u|^{p-1}u \; \mbox{in}\; \mathcal{H}, $$ where $\mathcal{H}=\mathbb{R}^n$ or $\mathcal{H}=\mathbb{R}^n_+=\{x=(x',x_n),\, x'\in\mathbb{R}^{n-1},x_n>0\}$ , with Dirichlet conditions in the half-space case. We assume that $n\geq2s$, $p>1$, $\lambda>0,$ $\alpha, \beta>-2s$ and $0<s\leq1$. Our aim is to analyze the influence of the Hardy-type power potential $\lambda|x|^\alpha u$ in the presence of the Hénon weight $|x|^\beta$, with particular emphasis on the existence and classification of weak stable solutions, including solutions that may be unbounded or sign-changing. We establish nonexistence results for nontrivial stable solutions and, more generally, for solutions that are stable outside a compact set, covering critical and supercritical regimes without additional assumptions, and the subcritical regime under a suitable $L ^\infty$-smallness condition on a weighted version of the solution. Our results apply to both the local case $s = 1$ and the nonlocal case $0 < s < 1 $, in the whole space and in the half-space. The proofs are based on energy estimates combined with a Pohozaev identity. In the supercritical regime, we further establish a monotonicity formula, following the approach of Dávila et al. (Trans. Amer. Math. Soc. 369, 6087-6104, 2017).

We establish large and moderate deviation principles for a broad class of nonlinear stochastic partial differential equations driven by multiplicative Poisson random measures in the small-noise regime. Our abstract framework is formulated for locally monotone evolution equations over a Hilbert triple and encompasses several important models arising in fluid dynamics and turbulence theory, including the two-dimensional Navier–Stokes equations, magnetohydrodynamics (MHD), the GOY shell model of turbulence, the two-dimensional tidal equations, and Navier–Stokes equations with nonlinear viscosities. A key analytical ingredient is the well-posedness of the associated controlled equations, established using the method of local monotonicity. The large and moderate deviation principles are then proved via the weak convergence approach.

We prove a weighted Wasserstein spectral gap for the three-dimensional primitive equations on $\mathscr E=\{u\in H^1\cap L^8: \partial_zu\in L^4\}$ under finite-rank additive Brownian forcing that is phase-complete finite Fourier and satisfies a quantitative quadratic saturation condition. In particular, four real forcing directions suffice and this number is optimal within the phase-complete class. Consequently, the system is uniquely ergodic and exponentially mixing in the $H^1$ topology. The main difficulties stem from the lack of sufficiently strong moment bounds and the highly degenerate nature of the noise, placing the problem outside the direct reach of standard methods. We introduce a new quantitative stable–compact asymptotic coupling criterion that captures the delicate balance between the available control of the growth of initial perturbations and the accuracy–cost of compensating them by perturbing the noise. For the primitive equations, this leads to a quantitative infinite-dimensional Hörmander analysis requiring tracking the degree and coefficient cost of Lie-bracket polynomials with the Fourier frequency.

We consider the focusing nonlinear fourth-order Schrödinger equation \begin{equation*} i \partial_t u + \mu \Delta u - \Delta^2 u = - |u|^{p-1} u, \ \ \ (t,x) \in \mathbb{R} \times \mathbb{R}^d, \end{equation*} where $\mu \ge 0$. The scattering of radial solutions below the ground state is shown by Guo (2016) and Dinh (2021) in the case where $d \ge 2$ and $1 + 8/d < p < 1 + 8/(d-4)_{+}$. In the present paper, we prove the scattering of non-radial solutions below the ground state in the case where $d \ge 2$ and $1 + 8/(d-1) \le p < 1 + 8/(d-4)_{+}$. Our proof is based on the concentration compactness argument by Kenig–Merle (2006). To overcome difficulties in the non-radial case, we use the virial identity in the direction orthogonal to the momentum vector. The condition $p \ge 1 + 8/(d-1)$ is required to show the positivity of the functional in that virial identity.

We study a dynamical optimal transport problem on path-space with kinetic cost and nonlocal interaction. For a Gaussian interaction kernel, cyclical monotonicity and the regularity of the effective endpoint cost yield a conservative vector field associated with the initial momentum of the optimal trajectories. Separately, the Gaussian interaction potential $W_{\pi_0}$ satisfies a uniform gradient estimate and, under a quantitative condition on the interaction strength, is a classical and hence viscosity subsolution of an eikonal equation. We also derive the Euler–Lagrange dynamics directly from path-space optimality by means of endpoint-preserving perturbations of the optimal path measure. The associated phase-space marginals satisfy a measure-valued Liouville-type, or nonlocal kinetic, equation driven by the smooth Gaussian self-consistent force. Conversely, a superposition principle lifts suitable measure-valued solutions of the phase-space continuity equation to measures on phase-space trajectories. Thus path-space optimal transport provides a variational connection between endpoint geometry, eikonal behavior of the Gaussian interaction potential, and kinetic mean-field dynamics.

Let $n\geq2$, $s\in(0,1)$, and let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we establish optimal global Sobolev estimates for the fractional Dirichlet problem \begin{equation*} \left\{\begin{aligned} (-\Delta)^su&=f & & \text{in}\ \ \Omega, u&=0 & & \text{in}\ \ \mathbb{R}^n\setminus\Omega. \end{aligned}\right. \end{equation*} More precisely, for any given $t\in[s,\min\{2s,\,s+1/2\})$, we prove that there exists a small positive constant $\eta=\eta(n,s,t,\Omega)$ such that the following holds. If $n=2$ and $s\in(1/2,1)$, then, for any $q\in(1,\frac{4}{1+2s}+\eta)$ and $p\in(1,\frac{nq}{n-(2s-t)q}]$, \begin{equation*} \|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}, \end{equation*} and, if either $n=2$ and $s\in(0,1/2]$ or $n\ge 3$ and $s\in(0,1)$, then, for any $q\in(1,\frac{n(2s+1)}{nt+(2s-t)(2s+1)}+\eta)$ and $p\in(1,\frac{nq}{n-(2s-t)q}]$, \begin{equation*} \|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}. \end{equation*} Here the positive constant $C$ depends only on $n$, $s$, $t$, $p$, $q$, and $\Omega$. The stated universal baselines are optimal in Dahlberg's sense: for any larger target exponent $p$ there is a bounded Lipschitz domain whose solution with right-hand side identically $1$ is not in $W^{t,p}(\mathbb{R}^n)$. The counterexamples are constructed using regular Cantor sets on the boundary and, in the planar exceptional case, homogeneous solutions in sectors.

We construct a continuous function $u$ on the closed unit ball in $\mathbb{R}^4$ and a smooth autonomous Hamiltonian $H:\mathbb{R}^4\to[1,2]$ for which $u$ solves $H(Du)=c$ in the unit ball in the viscosity sense for every $c\in[1,2]$. This reveals a striking and distinctive pathology of viscosity solutions: a fixed function and a fixed Hamiltonian need not determine the right-hand side of the equation.

Mean-field games provide a continuum framework for modeling the dynamics of large, interacting populations of non-cooperative agents. This paper studies symmetry-breaking in a finite-horizon, two-choice min-LQG mean-field game in which identical agents with linear stochastic dynamics choose one of two equally desirable terminal destinations, while trading off control effort against social pressure to conform. The model has an odd symmetry between the two destinations and therefore always admits a symmetric, dynamic Nash equilibrium in which the population splits evenly between them, producing a deadlock collective state at final time. Numerical studies have suggested that, as the penalty for social nonconformity increases, this symmetric equilibrium loses stability as a fixed point of an associated scalar, self-consistency map, and asymmetric consensus Nash equilibria emerge, where most agents select the same destination. By analyzing the linearized forward-backward PDE system through the scalar map representation, we provide a proof of this loss of stability of the symmetric equilibrium. Together with the odd symmetry of the map, this implies the existence of symmetry-broken mean-field game equilibria corresponding to consensus on either destination.

We study the field concentration phenomenon between two closely spaced inclusions with imperfect bonding interfaces of low conductivity type. The inclusions are assumed to have finite conductivities. The problem is governed by a system of elliptic equations coupled with Robin-type boundary conditions. While it is known that finite-conductivity inclusions with ideal interfaces yield bounded gradients, in this paper we show that with imperfect bonding interfaces, the gradient of the solution may blow up as $\varepsilon$ (the distance between two inclusions) tends to zero when the bonding parameter $\gamma$ is large, while it is uniformly bounded independently of $\varepsilon$ when the bonding parameter $\gamma$ is small. Moreover, we identify the threshold of $\gamma$ and the optimal blow-up rates under certain symmetry assumptions. Compared to the case when the inclusions are perfect conductors, we find a novel logarithmic blow-up phenomenon at the critical value of $\gamma$.

This paper study observability for the Dirichlet heat equation on a bounded interval with a real bounded potential depending on space and time. We prove an observability inequality from every measurable subset of spacetime with positive measure. The constant is uniform over potentials with a prescribed $L^\infty$ bound. By duality, positive spacetime measure is equivalent to null controllability by square-integrable distributed controls. Controls can also be chosen bounded in time with values in $L^2$. If almost every spatial slice of the observation set has a fixed positive lower bound on its measure, the observability constant is bounded above by $C_0e^{C_1/T}$, uniformly in the locations of the slices. The proof combines observation estimates on finite-dimensional spaces transported by the evolution with decay estimates for the distance to those spaces.

We study the large time asymptotics of the Feynman-Kac semigroups of the symmetric Lévy process on unbounded open sets. Our main result proves the exact exponential asymptotic decay rate for the survival probability given in terms of the bottom of the spectrum of the associated nonlocal Schrödinger operator. We also prove quantitative upper bounds with an explicit polynomial correction. The proof is probabilistic and is done by decomposing the Lévy process into a finite-range jump process collecting the small jumps and an independent compound Poisson process describing the large jumps. Our approach gives a direct link between the spectral properties of nonlocal Schrödinger operators and pointwise decay of survival probabilities, which was previously unknown for jump processes.

In this paper we consider the 4th order (or biharmonic) nonlinear Schrödinger (NLS) equation in dimensions 1, 2, and 3, where the potential term is expressed as a power non-linearity (for any positive power) and the dispersion operator has the fourth order combined with the lower second order. The fourth order NLS equation has recently attracted attention of researchers since quartic solitons have been experimentally obtained in optics, and thus, mathematical theory of solutions to the 4th order NLS equation in physical dimensions $N=1,2,3$ is timely to develop. In this work we show the local well-posedness of the 4th order or bi-harmonic NLS equation on a weighted subset of a Sobolev space as well as in $H^s$ spaces.

We prove an asymptotically sharp lower bound for the eigenvalues $\lambda_k$ of the Dirichlet Laplacian on bounded open sets in the plane with area $A>0$. The proof employs a trace identity for the commutator between a certain multiplication operator and its adjoint. As a corollary, we obtain $(2/3)\cdot 4\pi k/A\leq \lambda_k$ for every $k$, which is two thirds of the bound conjectured by Pólya.

Let $N\ge1$, $p\in[1,\infty)$, and $\Omega\subset\mathbb R^N$ be a bounded open interval when $N=1$ or a bounded Lipschitz domain when $N\ge2$. For any given $\gamma\in\mathbb R\setminus\{0\}$ and $\lambda\in(0,\infty)$, consider the weak-type nonlocal functional \begin{align*} F_{\lambda,p,\gamma}(u;\Omega):=\lambda^p \underset{|u(x)-u(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}} {\int_\Omega\int_\Omega}|x-y|^{\gamma-N}\,dy\,dx. \end{align*} In this article, we prove that, for $\gamma\in(0,\infty)$ as $\lambda\to\infty$ or for $\gamma\in(-\infty,-1]$ as $\lambda\to0^+$, the family $F_{\lambda,p,\gamma}(\cdot;\Omega)$ converges, in the sense of $\Gamma$-convergence in $L^p(\Omega)$, to the functional \begin{align*} \Psi_{p,\gamma}^{\Omega}(u) :=\begin{cases} C_{N,p,\gamma}^cell\displaystyle\int_\Omega|\nabla u|^p\,dx, &p\in(1,\infty)\ \hbox{and}\ u\in W^{1,p}(\Omega),\\[2mm] C_{N,1,\gamma}^cell|Du|(\Omega), &p=1\ \hbox{and}\ u\in BV(\Omega),\\[1mm] \infty,&\hbox{otherwise}. \end{cases} \end{align*} where the positive constants $C_{N,p,\gamma}^{\mathrm{cell}}$ are independent of $\Omega$. As applications, we further show that, for $\gamma\in(0,\infty)$ as $\lambda\to\infty$ or for $\gamma\in(-\infty,-1]$ as $\lambda\to0^+$, the weak-type nonlocal functional $F_{\lambda,p,\gamma}(\cdot;\mathbb R^N)$ converges, in the sense of $\Gamma$-convergence in $L^1_{\rm loc}(\mathbb R^N)$, to a similar functional; this gives an complete answer to the problem posed by Brezis et al. in [Section~7C, Anal. PDE 2024].

In this paper, we study the multiplicity of smooth limit interfaces for the anisotropic Allen-Cahn equation on closed Riemannian manifolds. We prove that an isolated smooth limit interface has multiplicity one whenever it is a closed, embedded, two-sided, strictly \(F\)-stable \(F\)-minimal hypersurface. The proof uses anisotropic barriers and a sliding argument to confine the nodal sets to a neighborhood of width \(O(\varepsilon)\). We then classify the nonconstant rescaled limits as one-dimensional solutions and obtain an energy upper bound near the limiting hypersurface. Exponential decay controls the remaining energy, yielding the multiplicity-one conclusion.