Papers

This paper concerns the Helmholtz scattering problem for open waveguide networks, extending the analysis from [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674] to plane-wave incident data. We first reduce the problem to a Fredholm second-kind integral equation on an infinite interface. This produces an integral representation of the solution to the original scattering problem, which is then used to obtain detailed asymptotic estimates of the solution at infinity. To leading order and away from the waveguides, the scattered field is itself a plane wave. Its amplitude and phase shift exhibit a jump across a critical polar angle depending on the direction of the incident plane-wave. Our estimates also imply that the scattered field is outgoing in a weak sense, allowing us to establish a uniqueness result. We conclude by calculating the waveguide-mode contributions to the scattered field in a perturbative regime.

We develop a Sobolev theory for the kinetic Fokker-Planck equation subject to a singular Morrey-form $L^\infty+M^q_K$ force field which can produce local blow-ups due to strong attraction to a submanifold. Our approach is based on the fractional representation for the Duhamel series of the equation and on establishing kinetic Adams-type estimates. Accordingly, we construct weak solutions of Brownian-driven kinetic SDEs in a critical Morrey set-up, obtain kinetic Krylov-type estimates and a strongly continuous Feller propagator, and establish uniqueness criteria for a large class of weak solutions.

We study the rough differential equation $$ \frac{du_t}{dt}= F(u_t) \zeta_t, \ \ u(0) = u_0 \in \mathbb{R}^d, $$ where $F : \mathbb{R}^d \to L(\mathbb{R}^d, \mathbb{R}^d)$ is a sufficiently regular vector field, and the noise $\zeta$ is a distribution in $\mathscr{C}^{\alpha-1}(\mathbb{R})$ for $\alpha \in (\frac{1}{4}, \frac{1}{3})$. We construct solutions to this equation at lower regularity than previous work of Gubinelli-Imkeller-Perkowski and Prömel-Trabs using ideas from the high-order paracontrolled viewpoint of Bailleul-Bernicot. The key tool in our work is the development of a harmonic-analytic framework for proving a set of iterated commutator estimates which are necessary for the solution theory. We also show the procedure of enhancement when the noise $\zeta = dZ$, where $Z$ is a realization of fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{3})$.

We consider the Neumann Laplacian on a three-dimensional periodic network of thin rods with edge thickness of order $h$ and vertex diameter of order $h^{2/3}$. We extend the fixed-graph resonant reduction to obtain a finite-$h$ periodic graph approximation with generalised norm-resolvent error $O(h)$, uniformly in the Floquet quasi-momentum. An exact gauge transformation identifies the phase-corrected graph fibres with the canonical lattice Floquet fibres, giving the same rate for the global operators. Collapsing the contact points while retaining the ambient edge tangents gives an approximation with error $O(h^{2/3})$. The proof uses a Ryzhov boundary triple and a Schur–Feshbach reduction. The estimates yield quantitative approximation of Bloch bands and stability of open gaps on bounded energy intervals. For the simple-cubic lattice, we refine the known resonant dispersion calculation to obtain a complete band-gap decomposition and $O(h)$ control of the persistence of each fixed finite collection of gaps in the thin structure.

We prove an improved fractal Weyl upper bound for resonances of infinite-area convex cocompact hyperbolic surfaces, expressed in terms of the Hausdorff dimension of the limit set and the Fourier decay exponent of the Patterson–Sullivan measure. This bound also yields a sharper upper bound for the essential spectral gap for these surfaces.

In this paper, we study the following magnetic nonlinear Schrödinger equation \[ \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(|u|^{2})u\,\ \text{in}\,\ \mathbb{R}^{2}, \] where $\varepsilon>0$ is a parameter, $A\in C(\mathbb{R}^{2}, \mathbb{R}^{2})$ is a magnetic potential, $V\in C(\mathbb{R}^{2}, \mathbb{R})$ is a multi-well potential, and $f\in C^1(\mathbb{R}, \mathbb{R})$ exhibits exponential critical growth. By variational methods and a penalization technique, we prove the existence and multiplicity of nontrivial solutions for sufficiently small $\varepsilon>0$. If, in addition, $A\in C^1(\mathbb{R}^2,\mathbb{R}^2)$, we show that these solutions have multiple peaks and we establish their concentration behavior and exponential decay.

We prove local well-posedness for the modified Benjamin-Ono equation in $H^s(\mathbb{R})$ for $s>3/2$. This equation arises as a modified counterpart of the Benjamin-Ono equation through a Miura-type transformation, in close analogy with the classical relation between mKdV and KdV equations. One of the main obstacles is the apparent derivative loss in the quadratic part of the equation. To overcome this difficulty, we introduce a Cole-Hopf transform adapted to the structure of the equation. The map removes the quadratic derivative term and converts the equation into a Schrödinger equation with a transport and a commutator terms that can be controlled without derivative loss. Solutions are then constructed by parabolic regularization, while frequency envelopes are used to obtain the continuity of the data-to-solution map. In the degenerate case $\gamma=0$, the transform completely linearizes the equation and yields local well-posedness for every $s>1/2$, down to the scaling-critical threshold.

We study periodic homogenization of fully-nonlinear second-order Hamilton–Jacobi–Bellman and Isaacs equations subject to a homogeneous Dirichlet boundary condition and the numerical approximation of the effective Hamiltonian, where the diffusion matrix is assumed to satisfy the Cordes condition. In the first part, assuming the Cordes parameter is sufficiently large, we show existence and uniqueness of $W^{2,p}$ solutions to the true cell problem, prove homogenization under minimal regularity assumptions, and establish a quantitative homogenization result. In the second part, we propose and rigorously analyze $W^{2,p}$-conforming and mixed finite element methods for the approximation of the corrector, which are then used in the approximation of the effective Hamiltonian. The key tool used throughout this work is Campanato's concept of near operators.

In this paper we deal with the theory of Regular Lagrangian Flows, as initiated in [29] and axiomatized in [2]. In the first part of the paper we show in Theorem 4.3 (see also Corollary 4.5) that the decomposition of $\mathcal{L}^{1+d}$-almost all the space time in a family of (essentially) disjoint characteristics, the technique at the basis of the well posedness in [1] (for planar, divergence-free and autonomous vector fields), and in [13] (for vector fields having a suitable regularity in space, including BV vector fields) is not only sufficient for well posedness, but also necessary. This shows that in many cases the typical uniqueness statement of the DiPerna-Lions theory, namely that the path $\mathbb{X}(\cdot,x)$ is (essentially) unique given the initial position $x$, can, as in the classical theory, be strengthened by saying that the paths $\mathbb{X}(\cdot,x)$ (essentially) do not branch. As the simple Example 1.1 shows, when we consider vector fields with singular divergence well posedness results may hold only for solutions that do not change sign. In the second part of the paper we revisit [8,9] obtaining a well posedness result for special vector fields of bounded deformation and nonnegative solutions, under a mild regularity assumption (5.3) with respect to the time variable, weaker than the space-time $SBD$ regularity assumption made in [9] for nearly incompressible vector fields. Finally, we revisit carefully the constructions of Section 9 of [13], to see how they provide existence and uniqueness for (1.2) assuming $BV$ regularity in space, in the class of nonnegative bounded solutions.

We consider the existence and uniqueness of positive, bounded stationary solutions of the Fisher-KPP equation with Dirichlet boundary conditions. We establish that stationary solutions are unique, and exist if and only if the generalised principal eigenvalue is strictly positive (supercriticality), in dimension at most $6$. This is sharp: we exhibit examples both of non-uniqueness and of existence at criticality in dimension $7$ or higher, including for the classical reaction $f(u)=u-u^2$. This proves a conjecture of Berestycki and Graham in the affirmative for dimension at most $6$, and in the negative for dimension at least $7$. We thereby show that branching Brownian motion cannot survive globally without surviving locally in dimension $6$ or lower, but can do so in dimension $7$ or higher, mirroring the classical transition of Brownian motion from recurrence to transience from dimension $2$ to $3$. We further show that relaxing the strong-KPP condition results in a more general sharp dimension threshold for the uniqueness/non-existence result. These results are valid without any regularity assumptions on the domain boundary. As a corollary, we show that solutions of the evolution reaction-diffusion equation are always convergent in the long run in dimension at most $6$, whilst we exhibit non-convergent solutions in dimension larger than $6$.

This paper establishes a local turnpike property for a two-population, second-order mean-field game system with a quadratic Hamiltonian, in which the running cost for each population may depend on both population densities. In the case of two populations, imposing a scalar stability condition separately on each species is not sufficient because it neglects cross-interaction effects. The appropriate requirement is instead a matrix-valued coercivity condition on vector perturbations $(\mu_1,\mu_2)$, which involves the symmetric part of the linearized interaction matrix together with weighted Poincaré inequalities associated with the two stationary measures. Consequently, in the multi-population framework, diffusion still acts independently on each population, but the interactions are controlled by the full coupling matrix. A naive scalar energy functional fails to capture mixed second-variation terms arising from costs that depend on both densities. We prove that matrix coercivity is precisely equivalent to the strict positivity of this second variation, and we provide a spectral characterization in terms of the principal spectral value of the coupled stationary linearized operator. For linear models with cross-interactions, this yields explicit algebraic conditions that separate stable parameter ranges from unstable ones. In the non-potential framework, the skew-symmetric part of the interaction matrix leaves the coercivity threshold unchanged, but it can induce transient oscillatory dynamics. Moreover, the scalar turnpike theory extends in a natural, though nontrivial, manner to multi-population systems, where stability is governed by the positivity of a coupled matrix-valued second variation that encodes both self-interactions and cross-interactions.

We study the boundary corrector $\theta_\epsilon$ in the two-scale expansion for periodic elliptic problems with Dirichlet conditions. By results of Gérard-Varet–Masmoudi and Prange, on smooth uniformly convex domains $\theta_\epsilon$ converges to the solution $\theta^*$ of a homogenized problem with boundary data given by the boundary layer tail $d(n)$, which is determined by a half-space problem for each normal $n$. In the lifted form of Gérard-Varet and Masmoudi, $d(n)$ is given explicitly by the cell correctors and the Dirichlet-to-Neumann operator $N_n$ of this half-space problem. Our main result is that $N_n$ satisfies a nonlinear equation on the period cell alone, so that $d(n)$ and the boundary density that represents it can be computed on the cell. This equation is a Riccati equation in which the derivative in depth is replaced by a commutator with the derivative along $n$. This cell problem can therefore be solved by a Riccati doubling algorithm, and we can compute $d(n)$ at irrational normals without rational approximation. Dropping the commutator term in the cell equation gives an operator square root approximation for $N_n$ that captures the leading term of $d(n)$ for small contrast, for the truncated problem. Finally, we approximate $\theta_\epsilon$ at finite $\epsilon$ by the sum of two solutions of constant coefficient problems, one with oscillatory boundary data and the other $\theta^*$, and test it on the disk.

We study the geometry of positive Dirichlet solutions on smooth, uniformly geodesically convex domains $\Omega\subsetneq\mathbb S^n$, $n\ge2$. Set $p_n^*=2n/(2n-3)$. For the Lane–Emden equation $-\Delta_g u=u^p$, $1<p\le p_n^*$, we prove that the set of positive solution pairs $(p,u)$ has a connected component containing the near-linear solution curve and projecting onto $(1,p_n^*]$. Every pair on this component satisfies $\nabla_g^2(u^{-(p-1)/3})>0$ in $\Omega$. For the torsion equation $-\Delta_g u=1$, we establish $\nabla_g^2\sqrt{u}<0$ for its unique Dirichlet solution in all dimensions $n\ge2$, without an additional diameter or circumradius assumption. In both settings, the solutions specified above have a unique nondegenerate maximum and strictly geodesically convex positive superlevel sets. For the non-power reaction $f(s)=s^p/(1+\lambda s^\delta)$, with $1<p\le p_n^*$, $\lambda>0$, and $0<\delta\le(p-1)/3$, we also prove the existence of at least one positive solution $u$ for which the transform $\left(\int_u^\infty ds/f(s)\right)^{1/3}$ has positive definite Hessian. The proofs combine a spherical Schur-complement estimate with a priori bounds and continuation.

We study vector-valued local minimizers of the borderline double phase functional whose integrand is $|D\mathbf{w}|^p+a(x)|D\mathbf{w}|^p\log^\gamma(e+|D\mathbf{w}|)$, where $p>1$, $\gamma>0$ and the coefficient $a(\cdot)$ is nonnegative and bounded. If the modulus of continuity $\omega(\cdot)$ of $a(\cdot)$ satisfies $\omega(r)\log^\gamma(1/r)\to0$ as $r\to0$, then minimizers are locally Hölder continuous for every exponent less than one. If $a(\cdot)$ is Hölder continuous, then the gradients of minimizers are locally Hölder continuous. In the vector-valued setting these results were previously available only for $\gamma=1$. The proof combines a comparison with frozen functionals, whose regularity theory applies uniformly in the frozen coefficient, with the higher integrability of minimizers.

We establish the existence and uniqueness, up to translation, of monotone viscous-dispersive shock profiles for the one-dimensional Navier-Stokes-Fourier-Korteweg (NSFK) system, provided the two constant end states satisfy the Rankine-Hugoniot and Lax entropy conditions and the shock strength is small. Treating the small shock strength as a singular perturbation parameter, we construct the profiles via Fenichel's geometric singular perturbation theory. We further provide pointwise estimates for the profiles, which are used in the nonlinear stability analysis of generic Riemann solutions to the NSFK system in a companion work.

We study the amplification and concentration of wave energy in smooth, periodic space-time media through the dynamics of the characteristic curves of the associated wave equation. In an impedance-matched setting, we show that stable limit cycles attract neighbouring characteristics, focusing the wave energy density. The exponential attraction rate is estimated analytically from the system's Lyapunov exponent. We further show that the attraction of characteristics and energy accumulation persist under a range of nonperiodic perturbations. Finally, we present a Floquet-Fourier numerical method for computing stable and unstable limit cycles and their attraction rates. Throughout, we exemplify the results using the example of modulation given by a superposition of travelling waves, revealing saddle–node bifurcations where limit cycles may appear depending on the relative wave speeds. Full-wave simulations confirm that the characteristic mechanism continues to describe energy accumulation beyond the impedance-matched case.

This paper studies fundamental solutions of uniformly parabolic systems with stationary random coefficients depending on space and time. Under ergodicity and local regularity, we establish quenched Gaussian bounds for the kernel and its spatial derivatives, with stationary random prefactors. A spectral gap assumption gives moments of every finite order for these prefactors and hence annealed Gaussian bounds. We also prove quantitative two-scale expansions for the kernel, its spatial gradient, and its mixed spatial derivative. The error estimates hold in every finite probability moment and retain Gaussian decay. Their dimension-dependent growth is described by a corrector weight, with a single additional logarithmic factor in the derivative estimates. The proofs combine large-scale regularity, extended correctors, weighted estimates, and a parabolic weak-norm estimate. The latter supplies the passage from deterministic pairings to local annealed estimates without interchanging expectation and a supremum over test functions. The mixed-derivative estimate builds on the gradient expansion and does not introduce a further logarithmic loss.

For $T>0$, we consider the Benjamin–Ono-type Dirac equation $$ i\partial_t\psi(x,t) + D(-\Delta)^{\frac{1}{2}}\psi(x,t) = D\,G(\psi(x,t)), \qquad (x,t)\in\mathbb{R}^n\times[0,T], $$ where $(-\Delta)^{\frac{1}{2}}$ is the fractional Riesz operator, $D$ is the Dirac operator associated with the Clifford algebra of signature $(0,n)$, and $G(\cdot)$ a scalar-valued quadratic nonlinearity. Under a pointwise anti-commutation condition on $\psi$ and a uniform dyadic regularity assumption on its spatial Fourier profile, we construct a unit-modulus gauge map $u$ by $ u=\exp\bigl(-iw\bigr),$ where $w$ is a solution of $Dw=\psi$. Assuming, in addition, the stated regularity and commutation conditions on $w$, we show that the Plemelj components $u_{\pm}=P_{\pm}u$ solve a coupled Schrödinger system with gauge nonlinearities $P_{\pm}N(\psi,u)$, where $$ N(\psi(x,t),u(x,t)) = -2P_{-}\!\left[ G(\psi(x,t)) \right]u(x,t). $$ We also obtain a priori estimates for the homogeneous Sobolev seminorms $\dot H^\sigma$ of $\psi(\cdot,t)$, with at most exponential growth on $[0,T]$, together with estimates for $P_{\pm}N(\psi(\cdot,t),u(\cdot,t))$. Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for $n\geq 3$ and every sharp Schrödinger admissible pair $(q,p)$, an a priori Strichartz-type estimate for $\psi$ in $L_T^q\dot W_x^{\sigma,p}$, expressed in terms of the initial datum $\psi(\cdot,0)$ and the a priori controls of $\psi$ in $L_T^\infty H_x^{\sigma+1}$ and $L_T^\infty L_x^\infty$.

The purpose of this note is twofold. First, we revisit the classical regularity theory of harmonic maps in the setting of maps defined on a lattice $\mathbb{Z}^d$ instead of $\mathbb{R}^d$ or some other manifold. The target space of our harmonic maps will be spheres $\mathbb{S}^{N-1}, N\geq 2$. A harmonic map $\sigma : \mathbb{Z}^d \to \mathbb{S}^{N-1}$ is a map which in any finite subgraph $\Lambda$ minimizes the internal Dirichlet energy $\frac 1 2 \sum_{i \sim j \in \Lambda} \|\sigma_i -\sigma_j\|^2$ given the boundary condition $\sigma_{|\partial \Lambda}$. One key difference between our setting and the continuum is that our main Hölder regularity statement in a large domain $\Lambda\subset \mathbb{Z}^{d=2}$ does not require any regularity assumption on the map $\sigma$ restricted to the boundary of the domain. For example, when $d=2$ and $N=3$, the Hölder regularity induced in the bulk of the domain holds uniformly in all boundary configurations $\sigma \in (\mathbb{S}^{2})^{\partial \Lambda}$ (this is in contrast with the continuum which requires extensions in suitable Sobolev spaces). The fact we obtain results uniformly in the boundary conditions allows us to extend our results for positive temperatures. Namely, as $T \searrow 0$, we obtain a local regularity theory for all infinite volume Spin $O(N)$ Gibbs measures, at least when $d=2$ and $N\geq 3$. We also discuss implications of our analysis in dimensions $d\geq 3$.

We study two sharp competition thresholds in the Lotka-Volterra competition-diffusion system: the zero-speed threshold for bistable traveling fronts and the linear-determinacy threshold for monostable traveling fronts. Building on our previous results on propagation direction and the classification of traveling waves, we derive explicit Gaussian integral representations of both critical parameters, without involving unknown traveling-wave profiles or speeds. These formulas provide exact characterizations of the optimal competition parameters separating the corresponding propagation regimes. The derivation of these representations was assisted by ChatGPT (OpenAI) Pro, while the underlying theoretical foundations were established in our earlier works.

This paper is concerned with space inhomogeneous Fermi-Dirac-Fokker-Planck equation posed on a classical kinetic phase space. The nonlinear factor $f(1-f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the exclusion principle of fermions. Solutions are subject to the physical constraint $0\leq f\leq 1$. Previous existing results have mainly been obtained under suitable smallness or close-to-equilibrium assumptions. A main difficulty stems from the nonlinear and degenerate parabolic structure of the equation. We prove the existence and uniqueness of global solutions without any smallness or close-to-equilibrium assumptions. To construct solutions, we use a vanishing viscosity approximation by adding a Laplacian in the spatial variable. We derive uniform estimates for the regularized problem and obtain compactness in the space of functions of bounded variation $(BV),$ allowing us to pass to the limit as the spatial viscosity vanishes. Since the limiting equation is degenerate with a nonlinear transport, we formulate an appropriate notion of entropy solutions incorporating the kinetic structure of the equation. We prove uniqueness within this class and show that every vanishing-viscosity limit satisfies the entropy formulation. Our result provides a global well-posedness theory beyond the perturbative regime.

We study normalized principal Steklov eigencouples on periodically perforated domains when both the bulk energy and the boundary normalization have general, possibly nonhomogeneous, Orlicz growth. The surface term on the hole boundaries carries the critical surface-to-volume scaling. For each fixed mass level, the constrained minima converge along the whole family, while selected minimizers, fluxes and Lagrange multipliers converge along subsequences to the corresponding objects of a homogenized problem. In the limit, the bulk operator is given by a convex cell problem, and the boundary term becomes a volume term weighted by the total surface density of the reference hole. The key tool is a strong approximation of the microscopic traces by cell averages, measured in the scaled surface modular.

Let $p>1$ and $\Omega\subset\mathbb{R}^n$ be a bounded smooth domain. We study the following overdetermined problem for the p-Laplacian: \begin{equation*} \left\{ \begin{array}{ll} -\Delta_p u=f(x)&x\in\Omega,\\ u=0&x\in\partial\Omega,\\ \frac{\partial u}{\partial\nu}=g(x)&x\in\partial\Omega, \end{array} \right. \end{equation*} where $\nu$ is the inward unit normal to $\partial\Omega$ and both data may depend on position. A boundary-local version of the moving-plane argument is developed for the possibly degenerate p-Laplace operator. The advantage of this argument is that it avoids the analysis of the degenerate set of the p-Laplacian. Under some reasonable structural conditions, we prove that the solvability of the above mentioned problem forces the domain to be a ball, the source to be radial, and the Neumann datum to be a suitable constant. We also give an explicit formula for the solution. Further consequences include one directional domain symmetry, the conclusion that full rigidity follows from reflection information in any set of directions whose linear reflections generate a dense subgroup of the orthogonal group $O(n)$, and a constant mean curvature characterization.

We prove that the singular locus of the geodesic flow on the unit tangent bundle of a nonpositively curved surface enjoys an additional regularity property: its characteristic function belongs to the geometric anisotropic Sobolev space $H^{5/8-}_xH^{-13/16}_\theta$. In particular, this implies that if the geodesic flow is not ergodic with respect to the Liouville measure, then the zero set of the curvature must be thick in the sense that it supports a nonzero, nonnegative function in $H^{5/8-}$. The proof relies on semiclassical analysis and a new radial estimate for parabolic flows.

We determine the minimal observation time for the heat equation on a closed almost-Riemannian surface, for a class of observation sets leaving sufficiently thin collars around simple Baouendi-Grushin circles. The structure and density are analytic near these circles, and the coefficients may depend on both variables. The formula is expressed through a complex phase describing concentration near the singular set and extends the thresholds known for separated models. Regular unobserved regions away from the circles do not change this time, provided all other singularities are observed. For singular circles of constant bracket step greater than two, local analyticity implies failure of observability at every time for observation sets at positive distance. We also construct quasimodes concentrated near simple singular circles, with exponentially small errors. Their superpositions give a lower bound for the Schrödinger observation time.

The focusing coupled Kundu-Eckhaus (CKE) equation extends the coupled nonlinear Schrödinger (CNLS) equation by adding quintic and derivative nonlinearities. We study the Cauchy problem for the focusing CKE equation with Schwartz initial data under the assumption of no discrete spectrum and no spectral singularities on the real axis. Based on the gauge equivalence between the focusing CKE and CNLS equations, we analyze the associated $3\times3$ matrix Riemann-Hilbert problem by the Deift-Zhou nonlinear steepest descent method. By establishing joint error estimates for the CNLS solution and the gauge phase, we obtain a uniform long-time asymptotic formula for the dispersive radiation of the CKE equation with an $O(t^{-1}\log t)$ remainder in bounded velocity cones.

We study the one-dimensional periodic nonlinear Schrödinger equation with a power-type nonlinear damping, both in the focusing and the defocusing regime. For a sufficiently small damping parameter and initial data close in relative $H^1$ norm to a nonzero spatially homogeneous state, we prove global existence and asymptotic convergence, after a constant phase adjustment, to an explicit homogeneous solution. The \(L^2\) and \(H^1\) norms follow the explicit adiabatic decay law, with the same leading coefficient as the homogeneous solution. We obtain explicit convergence rates, including a \(t^{-1/2}\) bound for the relative spatial perturbation. For every nonconstant initial datum in a small neighborhood of the homogeneous solution, the kinetic energy has a strictly positive leading coefficient and decays one power of time faster than the squared \(L^2\) norm.

We show that the spectrum of the Dirichlet-biLaplacian can be quantitatively bounded from below in terms of the inradius only, for a general open set in dimensions $2$ and $3$. For the same dimensions, we also extend this result to fractional powers $s$ of the Laplacian, with $N/2<s<2$. The resulting estimate comes with a constant having the correct asymptotic behavior as $s$ approaches the two endpoints $N/2$ and $2$. The proofs exploit capacitary techniques in the vein of Maz'ya.

We study the asymptotic behaviour of the nonlocal perimeter of tubular neighbourhoods. For a closed $C^{1,1}$ submanifold of Euclidean space, we show that a suitable rescaling of the nonlocal perimeter of its tube converges to an explicit constant times the Hausdorff measure of the submanifold. The constant is the fractional perimeter of the unit ball in the codimension of the submanifold. For finite point sets, we obtain an exact tube formula in terms of a generalised hypergeometric function. In contrast with the classical Weyl formula, this expansion has infinitely many terms, contains both noninteger and integer powers of the tubular radius, and has coefficients that depend on the relative positions of the points rather than only on their intrinsic geometry. We also establish a point-set expansion for general analytic kernels.

We study absorption at the origin in radial pressureless Euler flow in $\mathbb{R}^n$, $n\geq2$. For rotation-invariant nonnegative locally finite initial mass and bounded radial initial velocity vanishing at the origin, we prove global existence and uniqueness in a full-space radial entropy class with a one-sided Lipschitz bound and an initial kinetic-energy trace. Absorption is permanent, including retention of any initial origin atom. For finite positive mass with finite second moment, a Lagrangian projection representation characterizes the absorbed mass and its left limits at positive times through extreme minimizers of a free-position primitive. We derive a radial scalar momentum balance with a nonnegative Stieltjes boundary impulse measure. The minimum value of the same primitive yields a weighted Stieltjes identity equating accumulated time-weighted impulse with the integral of initial radii over absorbed labels, valid also for jumps and singular continuous absorption; the boundary impulse measure and the absorption measure are mutually absolutely continuous on $(0,\infty)$. For such finite-mass data with common total mass, strong $L^2$ convergence of initial quantiles and Lagrangian velocities, together with convergence of initial origin masses, yields convergence of absorbed mass at continuity times of the limiting absorption and local weak convergence of absorption measures on $(0,\infty)$. Finally, even with a fixed smooth initial radial probability density, every probability measure compactly supported in $(0,\infty)$ can be realized as the absorption measure by choosing a bounded radial initial velocity; in particular, purely singular continuous absorption occurs.