Papers

Consider a $d$-sphere $\mathbb{S}^d$ and denote by $\Gamma(T\mathbb{S}^d)$ the set of all vector fields on $\mathbb{S}^d$. We study the Navier-Stokes equations (NSE): $$ \partial_t u + \nabla_u u + grad \pi = \mathbf{\Delta} u + \operatorname{div} f(\cdot, t);\, \operatorname{div} u=0,$$ for the vector field $u(\cdot, t)\in \Gamma(T\mathbb{S}^d)$, where $\mathbf{\Delta}$ denotes the Ebin-Marsden Laplace operator defined by $\mathbf{\Delta} u= \operatorname{div} (\nabla u + \nabla u^t)^{\sharp}$, and $\operatorname{div} f(\cdot, t)$ is the periodic external force. We investigate the Navier-Stokes equations in the framework of $L^p$-spaces over vector fields on $\mathbb{S}^d$ and prove the existence and uniqueness of a periodic solution to such equations. Moreover, exploiting the distribution of eingenvalues of Ebin-Marsden Laplace operator we show the existence of an inertial manifold for solutions around that solution.

For $n\ge3$ and $\gamma>-1$, we study normal solutions of \begin{equation*} (-\Delta)^{n/2}u=(n-1)!\,|x|^{n\gamma}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden–Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and uniqueness, up to dilation, of the radial normal solution and establish its radial nondegeneracy. Next, we carry out a detailed spectral analysis of the linearized operator and construct branches of nonradial normal solutions by the Crandall–Rabinowitz theorem. Finally, we prove that every normal solution is radial when $\gamma>0$ is sufficiently small.

We obtain the necessary and sufficient conditions on input functions for a family of positive exponent power weight $L^1$ Stein-Weiss inequalities. Earlier work by De Nápoli and Picon showed that a certain cocanceling condition was necessary and sufficient in $L^1$, but it was unknown if this condition remained necessary in the positive exponent power weight setting. This present article identifies weaker necessary conditions for positive exponent power weights, some of which were shown to be sufficient for positive, non-integer exponent power weights in the author's earlier work.

We study the convergence of minimising movement approximations to weak solutions of a fourth-order viscoelastic phase separation system with degenerate concave mobility. The underlying semi-implicit variational scheme combines a dynamic transport distance for the phase-field variable with a weighted $L^2$-distance for a transformed stress variable. We establish subsequential convergence of the scheme and show that the limiting solutions satisfy the energy-dissipation inequality. To this end, we derive a coercivity estimate for the metric slope, which yields the required space-time regularity for the De Giorgi interpolant, and construct constitutive fluxes whose dissipation is controlled by the squared metric slope. These estimates allow us to pass to the limit in the discrete energy-dissipation inequality. The weak formulation of the evolution equation for the phase-field variable is derived using the flow interchange technique. Under additional assumptions on the data, we further show that the limiting solutions satisfy an entropy-dissipation inequality

This paper is concerned with the long-time dynamics of a coupled nonlinear suspension bridge system with past history, in which the deck and the main cable interact through the nonlinear restoring force \(k[u-v]^{+}\). Building on the well-posedness and the solution semigroup established in \cite{Mukiawa2025}, we first show the exponential stability of the associated linear semigroup by means of the Gearhart–Prüss theorem together with a detailed resolvent estimate. We then combine this exponential stability with the local Lipschitz property of the nonlinearity, the fact that the nonlinearity acts only on the elastic component, the smoothing of the elastic component, and the exponential decay of the memory kernels to conclude that the solution semigroup is quasi-stable in the sense of Chueshov–Lasiecka, and therefore asymptotically smooth. Together with the dissipativity recalled from \cite{Mukiawa2025}, this leads to a compact global attractor in the natural energy space. In particular, our approach may be viewed as closing a gap in the proof of asymptotic smoothness in \cite{Mukiawa2025}, where a non-decaying initial-difference term was absorbed into an arbitrarily small constant without justification.

This work is devoted to the study of blow-up phenomena for solutions of \[-\Delta u=\lambda V(x)|x|^{2N}e^v,\quad \mbox{in}\quad B_1,\quad v=0\quad \mbox{on}\quad \partial B_1. \] where $B_1$ denotes the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth function, $N\in \mathbb N$ and $\lambda>0$ is a small parameter. The main goal is to determine conditions on the potential $V$ guaranteeing the existence of families of solutions that blow up at the origin as $\lambda\to 0^+$. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at $x=0$. In particular, for $N-1\in 2^{\mathbb N}$, we construct a non-simple blow-up sequence $v_k$, corresponding to coefficient function $V_k$ whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.

Climate policy changes production costs and can reshape the geography of production. We study this spatial channel in a focused model where mobile industrial activity responds to local attractiveness, agglomeration strengthens attractiveness, and carbon policy lowers the profitability of emissions-intensive locations. The model yields an explicit carbon-policy threshold separating spatial instability from stability. When emissions intensity rises with local industrial density, spatial concentration raises aggregate emissions even with unchanged total industrial activity. Embedding these dynamics in a two-jurisdiction policy game creates a wedge between regional and collective incentives: each government values the industrial activity located within its own territory, while aggregate welfare internalizes the effect of policy on economy-wide concentration. With transferable utility, cooperation can select a policy on the stable side of the threshold while the non-cooperative policy lies on the unstable side. A numerical exercise illustrates this regime change. European regional evidence, based on industrial emissions and gross value added at NUTS2 level, documents a positive emissions-activity association and identifies the empirical quantities needed to measure relocation and concentration effects. The analysis links climate cooperation to the spatial organization of production and establishes a benchmark for policy-induced industrial relocation.

We study the asymptotic behaviour of fractional Gagliardo-type energies with a periodic, bounded and bounded away from zero coefficient depending on both variables, in the joint limit of vanishing fractional exponent $s\to0^+$ and vanishing period $\varepsilon\to0^+$. Assuming without loss of generality that $\varepsilon^{-2s}$ converges to some $\lambda\in[1,+\infty]$, we prove that the energies, rescaled by $s$, $\Gamma$-converge in the weak topology of $L^2(\mathbb{R}^d)$ to a multiple of the squared $L^2$ norm, whose constant is given by an effective coefficient $a_\lambda$ defined through a cell problem. The coefficient $a_\lambda$ results from the competition between the averaging of long-range interactions, which drives the Maz'ya–Shaposhnikova limit, and the nonlocal energy of microscopic oscillations at finite range. It interpolates between the harmonic mean ($\lambda=1$) and the arithmetic mean ($\lambda=+\infty$) of a coefficient obtained by averaging interactions “at infinity”. The proof relies on Riemann–Lebesgue-type lemmas for singular kernels, on a blow-up argument for the lower bound, and on the logarithmic-scaling properties of the energies. The case $1<p<\infty$ is also discussed.

Motivated by recent applications in generative modeling and sampling, we introduce a framework for optimal measure transport where cost captures the notion of neural network complexity. In transport-based generative models, samples from a reference distribution (e.g. Gaussian) are mapped to samples of a target distribution along ordinary or stochastic differential equations. These are implemented as deep residual networks when discretized in time, where each hidden layer approximates the associated instantaneous velocity. Thus, given a pair of target and reference measures, a natural question is to search for the most efficient neural representation that implements this transport. Our starting point is the kinetic formulation of OT, due to Benamou and Brenier. We replace the average kinetic $L^2$ energy by the Barron energy \cite{bach2017breaking, ma2022barron}, a natural norm which measures the complexity of representing a given vector field with a neural hidden layer, and which captures the adaptive properties of feature learning. This defines a metric on the space of probability measures, complementing existing Wasserstein and Stein geometries. In this work we examine the properties of this metric in the context of generative modeling. As a first application, we quantify the suboptimality of diffusion generative modeling in the Barron geometry by establishing super-polynomial score approximation lower bounds for data generated by neural network pushforwards of the Gaussian. We then investigate the benefit of adaptivity as a way to study alternative generative models. In a companion paper \cite{companionpaper} we leverage the Barron transport geometry for sampling applications, extending the scope of Stein variational gradient methods via feature adaptation.

We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator \[ \mathcal L_Au=-\sum_{i=1}^N\partial_i\!\left( a_i(|\partial_i u|^{p_i})|\partial_i u|^{p_i-2} \partial_i u\right), \] where the functions $a_i$ satisfy structural growth and monotonicity conditions of $(p_i,q_i)$-type. This framework contains, among other examples, the anisotropic $\vec p$-Laplacian, the orthotropic $p\&q$-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, $r=p_1<p_N$, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave–convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz–Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration–compactness.

This paper investigates the boundary stabilization of a linear hyperbolic system with a zero characteristic speed. We address a case that cannot be stabilized by proportional boundary feedback and develop a backstepping design for the critical setting where the zero-speed subsystem has no intrinsic dissipation. To this end, we construct a Volterra transformation leading to a nonstandard integro-differential target system. Its asymptotic stability in L2 is established by both the Lyapunov-LaSalle method and a spectral semigroup argument. The backstepping kernel equations are explicitly solved, yielding an explicit boundary feedback law that asymptotically stabilizes the original system. Numerical simulations are provided to illustrate the effectiveness of the proposed control design.

We study stochastic Hamilton–Jacobi–Bellman equations arising from stochastic optimal control problems with random coefficients and possible degenerateness. The viscosity solution is reformulated by using adapted smooth random test fields that touch on stopped space–time neighborhoods almost surely. The state coefficients are predictable, with no continuity assumption on their dependence on the underlying Brownian path. Under linear growth of locally Lipschitz drift and diffusion coefficients, polynomial growth of the running and terminal cost coefficients, and a nonempty separable metric space-valued control set, we show that the value function is the unique viscosity solution in this formulation among adapted continuous fields with square-integrable random polynomial growth bounds. The comparison proof constructs adapted fields from controlled trajectories and spatially shaken conditional values, using a weak semimartingale decomposition and regularization of its positive finite-variation part. In particular, the uniqueness allows for diffusion coefficients being random, degenerate, and depending on time, state, and control, whereas earlier uniqueness results of Qiu [SIAM J. Control Optim. 56 (2018), pp. 3708–3730] and Qiu and Zhang [Ann. Appl. Probab. 33 (2023), pp. 889–930] were obtained under additional structural restrictions on the diffusion coefficient, such as determinism combined with either superparabolicity or independence of the state and controls.

In a neighborhood of the origin in $\mathbb{R}^9$, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone $C_{4,3}$, the cone over $S^4(\sqrt{4/7})\times S^3(\sqrt{3/7})$, whereas at the origin its unique tangent cone is the cone $C_{3,3}\times\mathbb{R}$. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean $\mathbb{R}^9$ with exactly one singular point at the origin, at which its unique tangent cone is $C_{4,3}$, and whose unique tangent cone at infinity is $C_{3,3}\times\mathbb{R}$.

We study the singular set of the free boundary for solutions of the thin Stefan problem in $\mathbb{R}^{n+1}\times \mathbb{R}$, which models the melting of a thin sheet of ice over water. After a suitable transformation, we reduce to studying the parabolic Signorini (thin obstacle) problem with a time-monotonicity condition. First, we prove that the free boundary is locally the graph of a differentiable melting function $t=\tau(x)$ defined on the $n$-dimensional thin space, and that the singular set coincides with the critical set of $\tau$. Next, we show that the singular set has parabolic Hausdorff dimension at most $n$. We then establish generic regularity up to dimension $n+1=4$, proving that the free boundary is regular for almost every time. Finally, under smoothness assumptions on the obstacle, we show that the singular set is contained in a $C^\infty$-hypersurface outside a set of parabolic dimension at most $n-1$. Our analysis combines ideas developed for the classical Stefan problem with several new ingredients specific to the thin setting, including frequency formulas for second blow-ups, epiperimetric inequalities, higher order expansions, and a fine stratification of the singular set.

Threshold scattering below the ground-state energy and gradient is established for the three-dimensional focusing energy-critical nonlinear Schrödinger equation. A single solution-dependent Gaussian interaction scale, mass-free angular coercivity, and a frequency-gap source estimate control the projected dynamics. These estimates yield quartic spacetime integrability and a zero-mass contradiction, without radial symmetry or an a priori $L^2$ bound.

We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for general boundary and dimension $21$ for umbilic boundary. For positive scalar curvature and zero boundary mean curvature, the corresponding upper dimensions are $14$ and $20$. Together with the noncompactness examples in Part I, these results identify the transition dimensions in both boundary classes. For positive scalar curvature, we also prove compactness through dimension eight for every fixed real boundary mean curvature, and obtain higher-dimensional ranges when this curvature is near zero or sufficiently large and positive. The proof combines scalar-correction estimates for the full conformal Fermi metric expansion with a geometric formula expressing the logarithmic coefficient of the corrected energy as a negative sum of squares.

In this paper, we establish a sharp dynamical isoperimetric principle for weak mean curvature flow. We prove that, among weak mean curvature flows with fixed area of initial smooth closed hypersurfaces, the standard smoothly shrinking spherical mean curvature flow uniquely maximizes the extinction time. More precisely, for the level set flow $K_t$ starting from the boundary of a smooth bounded domain $\Omega\subset\mathbb R^{n+1}$, as well as for the integral Brakke flow $\{\mu_t\}$ with initial Radon measure $\mu_0=\mathcal H^n\llcorner\partial\Omega$, we establish the corresponding optimal extinction time estimates \begin{equation*} T_ext(\Omega),\, T^B_ext(\Omega) \leq \frac{1}{2n} \left(\frac{P( \Omega)}{|{\mathbb{S}^n}|}\right)^{\frac{2}{n}}, \end{equation*} where $P(\Omega)$ is the perimeter of $\Omega$ representing the area of $\partial \Omega$, and the equality holds if and only if $\Omega$ is a round ball and the flow is the standard multiplicity-one smoothly self-shrinking round sphere. In particular, we obtain the sharp $L^p$-estimates for the arrival time function of a smooth bounded mean convex domain. In addition, we also establish the sharp isoperimetric inequality for the parabolic measure of the space-time track filling $X$ of outward minimizing level set flow starting from the boundary of a smooth bounded domain $\Omega\subset \mathbb R^{n+1}$: \begin{equation*} \mathcal H_{\mathrm{par}}^{n+2}(X) \leq \frac{\pi} {2n(n+2)^2|{\mathbb{S}^n}|^{{\frac{2}{n}}}} P(\Omega)^{\frac{n+2}{n}}, \end{equation*} where the equality holds if and only if $\Omega$ is a round ball and the flow is standard smoothly shrinking round sphere.

In this paper, we study the long time dynamics of small perturbations of static domain walls in weighted Sobolev spaces for the one dimensional Schrödinger map with constant Dzyaloshinskii-Moriya interaction. The perturbation equation contains derivative nonlinearities, and its linearized operator has a threshold resonance. The results are twofold. (i) We prove the asymptotic stability of the domain wall modulo the translation and rotation symmetries. More precisely, the corresponding modulation parameters converge as time tends to infinity, while the radiation decays at the sharp rate $t^{-1/2}$. (ii) We prove that the radiation exhibits modified scattering, with an explicit logarithmic phase correction generated by the long range cubic interaction. The proof develops geometric reduction to overcome the loss of derivatives and exploits structural cancellations in the nonlinear interactions.

In this article, we study three properties of a bounded $\mathbb{C}$-convex domain $\Omega \subset \mathbb{C}^n$, $n \geq 2$, with smooth boundary, namely Gromov hyperbolicity of its Kobayashi distance, finiteness of the D'Angelo type of $\partial \Omega$, and the existence of a subelliptic estimate for the $\bar{\partial}$-Neumann problem on $(0,1)$-forms. We prove that these three properties are equivalent. The subelliptic estimates are obtained without Catlin's construction of plurisubharmonic weights. Our weights are explicit bounded transforms of the logarithm of the Bergman kernel. A single quantity controls all three properties, namely the radius of the largest disc through a point of $\Omega$ in a given complex direction. Gromov hyperbolicity forces a power bound for these radii. Finite type gives the power $1/M$, where $M$ is the maximal type, and this power cannot be improved. A subelliptic estimate gives a power bound through the canonical solution operator applied to normalized Bergman kernels. The analytic estimates need no boundary regularity. On a bounded pseudoconvex domain, a bounded weight whose induced Hessian on $(0,q)$-forms is bounded below by a negative power of the boundary distance gives a Sobolev estimate for the extension by zero of $(0,q)$-forms, and also eigenvalue bounds for the $\bar{\partial}$-Neumann operator. On smooth $\mathbb{C}$-convex domains of finite type, we show that the directional expansion exponent, the normal expansion exponent, and the supremum of the subelliptic gains are all equal to $1/M$. Next, we prove that on a Lipschitz $\mathbb{C}$-convex domain whose Kobayashi distance is Gromov hyperbolic, the Euclidean boundary and the Gromov boundary are bi-Hölder equivalent. Finally, we give a Hardy-type criterion for Gromov hyperbolicity of collar metrics.

This is a continuation of our previous paper on toric gravitational instantons. We develop a systematic theory of compact simply-connected Einstein four-manifolds with toric symmetry. Applications include (1) The construction of simply connected positive Einstein four-manifolds with arbitrarily large $b_2$. Our method is based on local gluing combined with a global virtual counting argument. The key idea involves a particular design of rod structures to enable a gluing construction and restrict possible degenerations. (2) A topological characterization of algebraically special toric Einstein metrics. This uses special curvature identities for $W^+$ in the toric setting. (3) Uniqueness, up to scaling and isometry, on toric four-manifolds admitting algebraically special Einstein metrics. The proofs uses a variational study of the Einstein–Hilbert functional. (4) Diffeomorphism classification of compact toric Einstein four-manifolds in terms of an improved Hitchin-Thorpe inequality $\chi\geq3|\tau|$. In particular, our construction recovers all the classical toric Einstein metrics, including the Page and Chen–LeBrun–Weber metrics, starting from the round sphere through moduli spaces of conical Einstein metrics.

We consider the inverse problem of recovering a potential $q$, depending on space and time, in the wave equation $(\Box_g+q)u=0$ on a Lorentzian manifold. We prove Lipschitz stability for a formally determined version of this problem under a null-cone foliation assumption. As a consequence, we obtain uniqueness in the Lorentzian Calderón problem for $q$ under this assumption. Our approach is based on a modification of the Bukhgeim–Klibanov method using distorted plane waves.

It is well known that the compressible Euler equations with planar shock initial data may admit infinitely many entropy weak solutions, as shown by Chiodaroli, De Lellis and Kreml [Comm. Pure Appl. Math. (2015)]. In this paper, we introduce a suitable notion of global-in-time weak solutions to the two-dimensional (2D) compressible Navier-Stokes equations with density-dependent viscosities in $\mathbb{R} \times \mathbb{T}$. Within this solution framework, we establish the first vanishing viscosity limit selection criterion for the physically relevant planar shock solution to the 2D compressible Euler equations. More precisely, we prove the uniqueness and stability of the 2D Euler planar shock solution within a class of inviscid limits of Navier-Stokes solutions emanating from near-planar initial data while the shock direction perturbation can be arbitrarily large. Starting from the relative $\kappa$-entropy for 2D compressible Navier-Stokes equations in $\mathbb{T}^2$ by Bresch, Noble and Vila [ESAIM Proc. Surveys (2017)], we first establish the relative $\frac12$-entropy between 2D weak solution of compressible Navier-Stokes equations in $\mathbb{R} \times \mathbb{T}$ and one-dimensional (1D) strong solution for averaged initial data in $\mathbb{R}$. Then we achieve the desired vanishing viscosity limit selection criterion by successfully bridging the fundamental structural gap between 2D Eulerian formulation of the original Navier-Stokes system in the relative $\frac12$-entropy and 1D vanishing viscosity limit to a shock solution of Euler equations in Lagrangian coordinates by Kang and Vasseur [Invent. Math. (2021)]. The framework developed herein can be applied to the three-dimensional limit selection criterion in $\mathbb{R} \times \mathbb{T}^2$, as well as to more complex planar Riemann solutions and interacting wave patterns.

It was known that the Gaussian measures with monomial weights satisfy the sharp pointwise curvature-dimension condition $CD(1, \infty)$, and consequently they allow us to derive the sharp constant $1$ for the Poincaré inequality only with partial monomial weights. In this paper, we first establish the Gaussian measures with full monomial weights satisfy the integrated curvature-dimension condition $ICD(2, \infty)$ which is sufficient for us to prove the sharp Poincaré inequalities with optimal constant $2$ and the stability of the Brascamp–Lieb inequality for full monomial Gaussian measures. Several of these results are extended to homogeneous Gaussian measures on convex cones. Using the sharp Poincaré inequality and the hypercontractivity, we obtain an improved Beckner inequality that recover the sharp Poincaré and logarithmic Sobolev constants at the endpoints. By developing a Laguerre spectral decomposition of the associated Ornstein–Uhlenbeck type generator, we derive an exact identity for the Poincaré deficit and obtain sharp gradient stability for the Poincaré inequality with full monomial weights. We further establish $L^2$ and weighted gradient stability estimates for the Brascamp–Lieb inequality with explicit constants when all monomial exponents exceed $1$. The $L^2$ estimate extends to log-concave homogeneous weights under a subharmonicity assumption. Finally, we obtain an improved integrated curvature-dimension bound under a scale-invariant Hessian condition and identify the sharp constant for the Poincaré inequality in the radial class.

Entropy-regularized optimal transport (EOT) has become a key tool in optimal transport, providing both theoretical insight and efficient computation via the Sinkhorn algorithm. We study the first-order \(\Gamma\)-expansion of EOT as the regularization parameter \(\varepsilon\) vanishes, by rewriting EOT as the minimization of a relative entropy with respect to a Gibbs measure \(\gibbs\) associated with the duality gap energy. Using a mild extension of the Laplace method, we prove the convergence of the Gibbs measures and derive this \(\Gamma\)-expansion for non-degenerate ground costs, under suitable assumptions on the duality gap, the contact set and the marginals. This yields at the same time an entropic selection principle and an expansion of the EOT cost up to \(o(\varepsilon)\) for this class of ground costs, generalizing existing results.

Let $n\ge2$, $0<s<1$, and $p,q>0$. We prove a Liouville theorem for the fractional Lane–Emden system \[ (-\Delta)^s u=v^p,\qquad (-\Delta)^s v=u^q, \qquad u, v>0 \qquad\text{in }\R^n. \] More precisely, we show that the system does not have a strictly positive entire solution whenever \[ \frac1{p+1}+\frac1{q+1}>\frac{n-2s}{n}. \] No radial symmetry, global boundedness, finite-energy assumption, or prescribed decay at infinity is required. Our proof adapts the localized potential-kernel virial and source-layer interval-pressure method recently introduced for the classical Hénon–Lane–Emden system. The basic reason the method also works in the fractional setting is simple: the Riesz kernel associated with $(-\Delta)^s$ has the same logarithmic derivative structure as the Newton kernel, with $n-2$ replaced by $n-2s$. However, two genuinely nonlocal points require special treatments. When $2s\le1$, the individual kernel-gradient terms are not locally absolutely integrable, so the two equations must be combined before the diagonal limit is taken. When $2s>n-1$, the kernel restricted to a line is not integrable at infinity, and the interval comparison must therefore be formulated using nonnegative extended integrals. After these issues are resolved, a localized virial identity, a one-dimensional interval-pair inequality, and an averaged source-layer estimate yield a universal local energy bound. The subcritical scaling then forces the same energy to diverge, giving the desired contradiction.

We consider the Dirichlet problem for the Lorentz–Minkowski mean curvature operator in a radial setting, $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda u + \mu g(u) \quad \text{in } \mathcal B_R, \qquad u = 0 \quad \text{on } \partial\mathcal B_R, $$ on a ball of radius $R$ centered at the origin $\mathcal B_R \subset \mathbb{R}^N$, where $\lambda, \mu \geq 0$ are parameters, and the nonlinearity $g$ satisfies suitable growth assumptions at zero. We first prove an a priori dichotomy for the asymptotic behaviour, as $\mu \to +\infty$, of solutions with a prescribed number of zeros: either they converge uniformly to $0$, or they converge to a piecewise affine limit function $u_\infty$, with $|u_\infty'|=1$ a.e., having the same nodal properties. We then show that, when the number of zeros is appropriately prescribed, both alternatives are actually realized. More precisely, building on the multiplicity result in [A. Boscaggin, F. Colasuonno, and R. Ziegele, Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics, arXiv:2607.15956 (2026)], we refine the shooting construction therein to obtain a family of small solutions converging to zero and, through a backward shooting argument, a family of large solutions converging to a saturated piecewise affine profile.

For every $1<p<\infty$, the Hilbert transform on $L^p(\mathbb{R};\mathcal{S}_p)$ and the ordered Hilbert Schur multiplier on $\mathcal{S}_p$ have norm $C_p=\cot(\pi/(2p^*))$, where $p^*=\max\{p,p/(p-1)\}$. The scalar Hilbert transform and the Schur multiplier have the same canonical completely bounded norm. This proves the Gohberg-Krupnik sharp Volterra conjecture. A simultaneous finite-dimensional realization also resolves the Riesz-Titchmarsh conjecture and Laeng's Conjecture 5.7 for all shifts. The norm identities extend to finite families of real Hilbert-transform pencils with $\ell^p$ direct-sum ranges. The proof uses a spectral-defect identity with two nonnegative remainders, based on Heinävaara's tracial joint spectral measure. It classifies trace inequalities for finite signed real ridge sums and identifies the least homogeneous superharmonic majorant as the pointwise least additive eigenvalue correction. The associated angular cone is characterized by nonnegative potentials and realized by weak-* limits of weighted angular measures of finite nilpotent matrices.

We prove a sharp Szegö-Weinberger-type inequality for balanced doubly connected domains in $\mathbb S^2$, $\mathbb R^2$ and $\mathbb H^2$. More precisely, the first positive Neumann eigenvalue is bounded above by that of the corresponding rotationally symmetric annulus, determined by the area of the domain and the area of the smallest component of the complement. Equality holds only for the rotationally symmetric annulus. We also show that the balancing assumption is essential: if we drop it, the inequality fails in general.

\noindent Let \(\Om\subset\R^2\) be a smooth bounded simply connected domain. We prove that all positive solutions of \(-\Delta u=\theta ue^{u^2}\), with \(\theta>0\) and zero Dirichlet boundary data, have Dirichlet energy bounded by a constant depending only on \(\Om\). Consequently, the Trudinger–Moser functional has no nonnegative constrained critical points at sufficiently large prescribed energies. The proof combines a weighted Pohozaev identity obtained after conformal change of variables, concentration analysis of normalized measures, and local radial comparison. The comparison hypotheses are obtained from local integral estimates, without an a priori energy bound. An estimate for the exterior Green potential then excludes sequences of unbounded energy.

We consider a thermodynamically consistent diffuse-interface model for binary mixtures of incompressible viscous Newtonian fluids with a soluble surfactant. This hydrodynamic system generalizes the Abels-Garcke-Grün model for two-phase flows with unmatched densities by incorporating an additional Cahn-Hilliard equation for the surfactant concentration. We study the strong well-posedness of the initial-boundary value problem with physically relevant singular potentials, subject to a no-slip boundary condition for the fluid velocity and homogeneous Neumann boundary conditions for the phase-field variables and the corresponding chemical potentials. First, we establish the existence of local-in-time strong solutions in three dimensions and global-in-time strong solutions in two dimensions. In the two-dimensional case, we further show that, under natural growth assumptions on the singular potentials, strong solutions remain strictly separated from the pure phases, thereby guaranteeing uniqueness. In the three-dimensional case, we establish local-in-time uniqueness of strong solutions, provided that the initial data are strictly separated from the pure phases. Our results extend previous work on strong solutions to the Abels-Garcke-Grün model from the special case of constant mobility to the more general scenario with concentration-dependent mobilities and interactions with a surfactant.