Papers

We study the emergence of cuspons through wave breaking, with the Camassa–Holm equation as a principal model. For a broad open class of smooth initial data, the first singularity occurs at a unique point and has the sharp local Hölder regularity \(C^{3/5}\). We construct a continuation of the Lagrangian flow through this singular time and reconstruct from it a conservative Eulerian weak solution. We show that the first singularity does not simply persist after breaking; instead, it bifurcates immediately into an anticuspon–cuspon pair, each having the sharp local Hölder regularity \(C^{2/3}\), with opposite one-sided slope orientations. The two singular branches move apart from the original breaking point, and the portion of the profile between them reverses its local monotonicity. The proof is based on an extended inverse-slope variable that remains regular through wave breaking. At the first singular time, this variable has a nondegenerate quadratic zero, which splits immediately afterward into two simple zeros. The corresponding degeneracy of the Lagrangian flow map converts the quadratic first-breaking geometry into the \(C^{3/5}\) profile and the post-breaking simple-zero geometry into the \(C^{2/3}\) cuspon profiles. The generalized Hunter–Saxton family provides an explicit realization of the same Lagrangian mechanism and also allows us to track the subsequent interaction and annihilation of the singular branches. Thus an anticuspon–cuspon pair emerges dynamically from smooth initial data through wave breaking, rather than being imposed through a singular traveling-wave ansatz.

This paper studies the focusing Calogero–Moser derivative nonlinear Schrödinger (CMdNLS) equation under the nonzero boundary condition \(|u(t,x)|\to 1\) as \(|x|\to+\infty\). In sharp contrast to the classical paradigm for the cubic nonlinear Schrödinger equation, namely that focusing nonlinearities support only bright solitons on a zero background, we show that the focusing CMdNLS equation admits a rich family of dark multi-soliton solutions on a nonzero background, whose existence is driven by a spectral gap \([-1,0]\) in the point spectrum of the Lax operator, together with the Hardy space structure. The main results are threefold. First, we establish an explicit formula for general solutions in the Hardy–Zhidkov space \(\mathcal Z^2_+\), via a semigroup method based on new commutator estimates between the Lax–Beurling shift semigroup and Toeplitz operators. This approach circumvents the non-commutativity of the unbounded generator with the time derivative and requires no weighted hypothesis on solutions or on the domain of validity of commutator formulas. Second, the algebraic definition of dark multi-soliton potentials is shown to be equivalent to two spectral conditions on the Lax operator. Together with the conservation of a generating functional, this yields the invariance of the dark \(N\)-soliton set \(\mathcal U_N\) under the CMdNLS flow. Third, the action-angle variables on \(\mathcal U_N\) are constructed, which establishes the complete integrability of the focusing CMdNLS equation on \(\mathcal U_N\). Combining these results, we obtain an inverse spectral formula, from which it follows that the poles of dark multi-soliton solutions satisfy a complexified Calogero–Moser system. Long time asymptotics and uniform Sobolev estimates are also derived.

We study the inverse problem of determining a connected, closed Riemannian manifold of dimension at least two from wave observations. The sources are supported in an open set $S$, and the resulting waves are observed on an open receiver set $R$, which may be at a positive distance from $S$. We prove uniqueness up to isometry within the class of manifolds that satisfy a subcritical spectral lower bound: $\|\phi|_R\|_{L^2(R)} \ge ce^{-C\lambda^\theta}\|\phi\|_{L^2(M)}$ for every eigenfunction $\phi$, where $\lambda$ is the corresponding eigenvalue, $\theta\in[0,1/2)$, and the constants $c,C>0$ do not depend on $\phi$. If a bound of this type holds also on $S$, then uniqueness holds within the class of all connected, closed manifolds. By the spectral inequality, the bound always holds with $\theta=1/2$. Uniqueness nevertheless fails at this endpoint: in every dimension $n\ge 2$, we construct a pair of nonisometric connected, closed Riemannian $n$-manifolds that produce identical wave observations.

We study endpoint mapping properties of low-energy wave operators for Schrödinger operators $H=-\Delta+V$ on $\mathbb R^n$, $n\ge3$. In dimension four, we prove that a zero-energy resonance prevents $L^1$ boundedness, whether or not zero is also an eigenvalue, under $|V(x)|\lesssim\langle x\rangle^{-\beta}$ with $\beta>10$. For a zero-energy eigenvalue without a resonance, we obtain a complete low-energy $L^p$ classification in every dimension $n\ge3$ under $\beta>n+4$. In particular, $L^\infty$ boundedness is equivalent to the vanishing of the zeroth, first, and harmonic second moments of $V\psi$ for every zero-energy eigenfunction $\psi$. The proof identifies the finite-rank obstruction and shows that the remaining eigenvalue correction cannot cancel its critical asymptotic profiles. The same conclusions hold for the full wave operators when the corresponding high-energy bounds are available.

We summarize recent developments in the theory of the Benjamin-Ono equation, a well-known long-wave asymptotic model for internal waves in deep water, focusing on results obtained by means of an explicit formula for the solution of the Cauchy problem in terms of given initial data. We give a full description of this formula, and provide simple examples, with the aim of bringing it into the realm of applied mathematics. We also show how the formula simplifies further upon restriction to rational initial data. We then show how the formula and its rational restriction explain in great detail, and in a strikingly simple fashion, the asymptotic behavior of solutions of the Benjamin-Ono equation in the small-dispersion and long-time limits. In the setting of long-time asymptotics, we prove some new results related to soliton resolution yielding pointwise convergence with a concrete decay rate under the assumption of rational initial data. The first half of the paper is a stand-alone general survey and user's guide, while the appendices constituting the second half are intended for readers who want to explore the topics in greater depth, and it includes the proofs of our new results.

The Navier-Stokes/Mean Curvature Flow system describes a two-phase flow of incompressible, viscous and immiscible fluids separated by a sharp interface, whose evolution is governed by a convective mean curvature flow equation, coupled to a two-phase Navier-Stokes equation accounting for surface tension. Among others, this model describes the dynamics of dry foams and it also arises as a sharp interface limit of Navier-Stokes/Allen-Cahn systems with non-vanishing mobility. Despite this wide range of applications, global-in-time existence of any kind of weak solutions remains a challenging open problem. In this work, we introduce a novel notion of varifold weak solution in two and three ambient dimensions, based on a sharp energy dissipation principle à la De Giorgi, and we show for the first time the unconditional global-in-time existence of such a weak solution. We also show, by means of a relative entropy approach, that any classical solution to the Navier-Stokes/Mean Curvature Flow system is unique in the class of our new weak varifold solutions.

We prove that the positive first Dirichlet eigenfunction of every smooth domain with convex boundary in $\mathbb{CP}^n$ invariant under the standard $U(1)\times U(n-1)$-action is strictly log-concave for $n=2,3,4$.

In this paper, we construct explicit formulas for the heat kernel of the Hodge Laplacian restricted to an invariant subspace of $1$-forms on the real hyperbolic space $\mathbb{H}^{n}$. Using warped product decompositions, the problem is reduced to component operators on specific subspaces, yielding precise closed-form expressions for both odd and even dimensions. We further establish Millson-type recurrence relations that connect heat kernels across differing dimensions. Based on these explicit representations, we derive sharp uniform estimates and long-time asymptotic behavior, extending classical scalar heat kernel bounds to $1$-forms on real hyperbolic spaces.

We study ground states, criticality, and variational compactness for genuinely multipolar $L^p$-Hardy weights, $1<p<n$. A ground-state representation across the poles gives a complete criticality classification of the full potential and characterizes finite-energy attainment. Removing a geometric interaction term produces a variational problem governed by concentration at the poles and infinity. A strict concentration gap yields compactness, positive extremizers, and criticality of the optimal operator; for $p\ge2$, it is also necessary for attainment. In the subquadratic range, the optimal truncated coefficient is below one and attained throughout the full-potential critical set. For admissible parameters with $1<p<2$ and $(n-p)/p<s\le m(n-p)/p$, arbitrary fixed pole configurations exhibit negative directions, attained critical states, and coercive subcritical regimes outside this set along positive-weight paths towards the simplex vertices. A single-pole degeneration formula and continuity in the weights establish this coexistence. In the superquadratic range, an interaction threshold determines a non-attained infinity-concentration interval. Under strict separation from pole concentration, its endpoint is critical with infinite energy and a compact phase follows. We prove separation for every bipolar configuration and in several multipolar regimes, including a vertex neighborhood uniform in the exponent. Near parameters strictly inside the quadratic attained region, the optimal coefficient crosses one continuously through $p=2$. The proofs combine ground-state remainders, capacity, weighted compactness, and critical Hardy barriers.

This article is the third in a series investigating ill-posedness in critical Sobolev spaces for Camassa-Holm-type equations. Following results for the $b$-Novikov and Fokas-Olver-Rosenau-Qiao equations, we study the hyperelastic rod equation at the critical Sobolev regularity $H^{3/2}(\mathbb R)$. We prove ill-posedness at this endpoint, complementing the local well-posedness theory in $H^s(\mathbb R)$ for $s>3/2$ and the recently established norm-inflation result for $1<s<3/2$. The nature of the critical ill-posedness depends on the parameter $\gamma$. For $0<\gamma\leq3$, we construct arbitrarily small smooth initial data whose corresponding solutions develop finite-time blow-up, with the maximal lifespan tending to zero with the size of the initial data. For $\gamma<0$ or $\gamma>3$, we construct solutions with arbitrarily small $H^{3/2}$ initial data whose $H^{3/2}$ norm becomes arbitrarily large in arbitrarily short time.

We introduce a conservative two-exponent dispersive family, with $p>1$ the nonlinear-flux exponent and $r>0$ the gradient-dispersion exponent, that separates the mechanisms governing compactification, free-boundary regularity and linearised spectral structure while retaining an explicit travelling-wave theory. The family reduces to the absolute Rosenau-Hyman equation when $r=1$. The deformation leaves the amplitude-speed law unchanged, but makes the width amplitude dependent except at $r=1$. Two independent transition curves emerge: $pr=1$ separates infinite-tail solitary waves from compactons, whereas $r(p-1)=2$ separates compactons with a regular free boundary from edge-cusped compactons with divergent slope. The latter remain distributional solutions, with no singular measures at either the free boundary or the crest. For $r\ge1$, the linearisation factors through a singular Sturm-Liouville operator whose natural Friedrichs realisation is solvable: after a compacton-adapted change of variable, the eigenvalue equation is of Jacobi-hypergeometric type, with two explicit spectral sequences, exactly one negative eigenvalue and a one-dimensional translation kernel. Numerical experiments connect these exact structures to approximation and dynamics. Measured Fourier tails and projection errors agree with the rates predicted by the free-boundary exponent, while padded Fourier-Galerkin time integration propagates the cusped waves coherently with small phase drift. A structure-adapted Jacobi-Galerkin diagnostic of the full generator finds no detectable real part in the tested truncated point spectra. The family therefore provides an explicit testbed in which geometric singularity, weak admissibility, spectral structure and numerical regularity of compactons can be studied within a single degenerate dispersive model.

We establish a Pogorelov type interior $C^2$ estimate for the real twisted Monge-Ampere equation without assuming uniform ellipticity. This equation is fully nonlinear and elliptic, but is neither concave nor a function of the eigenvalues of the Hessian, so the standard techniques for interior second-derivative estimates do not directly apply. Building on the partial Legendre transform formulation of Streets and Warren, we derive a differential identity for the associated positive-definite matrix $W(D^2u)$ and use some linear algebra to control the third-order terms arising in the maximum principle argument. As an application, we prove a Liouville theorem for entire split convex solutions with quadratic growth, extending the rigidity theorem of Streets and Warren by replacing their uniform ellipticity hypothesis with suitable quadratic growth assumptions.

We derive the half-space KPZ equation as a continuum limit from a general class of Ginzburg-Landau SPDEs under weak nonlinearity scaling. To our knowledge, this is the first half-space universality result for the KPZ equation beyond integrable models (and their perturbations).

The Kerr stability conjecture has been proved in the slowly rotating case, i.e., $|a|\ll m$, in the sequence of works \cite{KS-GCM1} \cite{KS-GCM2} \cite{KS:Kerr} by Sergiu Klainerman and the author, \cite{GKS22} by Elena Giorgi, Sergiu Klainerman and the author, and \cite{Shen} by Dawei Shen, and extended in \cite{Sze}, by the author, to the full subextremal range $|a|<m$, thereby completing the proof of the Kerr stability conjecture. The proof in \cite{Sze} crucially relies on the two companion papers \cite{MaSz24} \cite{MaSz26} by Siyuan Ma and the author, in which we prove energy-Morawetz estimates respectively for the scalar wave equation and for Teukolsky equations on perturbations of Kerr with $|a|<m$. In addition, two results in \cite{Sze}, concerning the derivation of the Teukolsky wave-transport system in perturbations of Kerr and energy-Morawetz estimates for Teukolsky in the setting of \cite{Sze}, are stated without proofs. The goal of the present paper companion paper to \cite{Sze} is to provide the proof of these two results.

We prove local well-posedness in the Hadamard sense for the three-dimensional compressible Euler equations governing a liquid with a free surface and no surface tension. Under the Taylor sign condition and the boundary compatibility conditions, we establish existence, uniqueness, and strong continuous dependence in a hybrid Sobolev class for every $s>3$. The velocity and interface have $H^s$ regularity, while the enthalpy and velocity divergence belong to $H^{s+1/2}$ and $H^{s-1/2}$, respectively. This separation reflects the coupling of a nondegenerate acoustic equation with the free-surface and vorticity dynamics. The proof uses regularization operators that smooth the moving domain and extend the nonlinear compatibility hierarchy, together with $L^2$ and partial $H^2$ distance estimates for solutions on different domains. Regular solutions are constructed by a regularized forward Euler scheme. High-order energy estimates and frequency envelopes then yield strong convergence of smooth approximations and continuity of the data-to-solution map. Integer Sobolev indices are treated using a Lions–Magenes endpoint compatibility condition.

We consider the functional $I_{\delta,\Psi_1,\Psi_2}:X \to \mathbb{R}$ defined for any $z=(u,v) \in X$ as \begin{align*} I_{\delta,\Psi_1,\Psi_2}(z) & = \int_{\Omega} \Psi_1(\nabla u) \, dx + \int_{\Omega} \Psi_2(\nabla v ) \, dx & - \int_{\Omega} H(\delta,x,u(x),v(x)) \,dx, \end{align*} where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\Psi_1, \Psi_2: \mathbb{R}^N \to \mathbb{R}$ are convex functions satisfying suitable conditions for $1 < p, q < N$, and the nonlinearity $H$ is allowed to exhibit both subcritical and critical growth and may depend on a parameter $\delta \in I \subseteq \mathbb{R}$. Here, the functional space setting is given by the product space $X :=W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$ equipped with the norm $\|z\|= \|u\|_{1,p} + \|v\|_{1,q}$ for any $z=(u,v)\in X$, where $\| \cdot \|_{1,s}$ is the usual norm in $W^{1,s}_0(\Omega)$. In this paper we prove that $I_{\delta,\Psi_1,\Psi_2}'$ is of class $(S)_+$ and we infer that each isolated critical point of $I_{\delta,\Psi_1,\Psi_2}$ has critical groups of finite type and a Poincaré-Hopf formula holds.

We consider the wave equation on a manifold of dimension d>1 with smooth strictly convex boundary, with Neumann boundary condition. We construct a sharp local in time parametrix for the Neumann wave equation near glancing, extending the classical Melrose-Taylor construction for the Dirichlet problem to the Neumann boundary condition. Our construction is based on the microlocal framework and the parametrix developed in [10] for the Dirichlet problem, together with the Melrose-Taylor argument for solving the transport equations under more general, affine-type boundary conditions. Once the Neumann parametrix is constructed, the dispersive and Strichartz estimates follow the analysis developed in [10] for the Dirichlet problem. We therefore only recall the main ingredients of that argument. In particular, the fixed time decay rate for the Green function exhibits the same loss of 1/4 with respect to the boundaryless case, associated with swallowtail type singularities in the wave front set, and this decay is optimal. Moreover, the corresponding Strichartz estimates are obtained by balancing lossy long time estimates at a given incidence with short time ones with no loss: for d=3, this heuristically means that, on average, the decay loss is only 1/6 .

In the present paper, we extend the results in the sequence of works \cite{KS-GCM1} \cite{KS-GCM2} \cite{KS:Kerr} by Sergiu Klainerman and the author, \cite{GKS22} by Elena Giorgi, Sergiu Klainerman and the author, and \cite{Shen} by Dawei Shen from the slowly rotating regime $|a|\ll m$ to the full subextremal range $|a|<m$, thereby completing the proof of the Kerr stability conjecture. The restriction $|a|\ll m$ enters in fact only through the estimates for wave equations and for the Bianchi system established in \cite{GKS22}. To remove this restriction, we crucially rely on the two companion papers \cite{MaSz24} \cite{MaSz26} by Siyuan Ma and the author, in which we prove energy-Morawetz estimates respectively for the inhomogeneous scalar wave equation and for inhomogeneous Teukolsky equations on perturbations of Kerr with $|a|<m$. Additionally, some of the results of the present paper are proved in our companion paper \cite{Sze}.

We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

In this article, we analyze a recently introduced kinetic model based on Hegselmann-Krause-type interaction dynamics describing consensus-based image segmentation in the mean-field regime, where particle density depends on spatial coordinates and the normalized gray level (feature) value. The model is formulated as an evolutive nonlocal partial differential equation featuring a non-local spatial drift term involving a bounded confidence interaction kernel, depending on spatial and feature proximity, and a spatial diffusion term degenerating in the feature variable representing aleatoric uncertainties. Establishing well-posedness on a bounded domain, essential for practical image processing, presents significant analytical challenges: the lack of uniform control of space derivatives due to kernels with jump discontinuities and degenerate spatial diffusion, complex non-standard boundary terms under homogeneous Neumann conditions, and a lack of compactness arising from the absence of derivatives with respect to the feature variable. To address these difficulties, we introduce a three-level regularization scheme. Specifically, we smooth the interaction kernel, apply structural boundary regularization to the non-local interaction term to simplify boundary conditions, and incorporate artificial diffusion in spatial and feature variables to resolve degeneracy. We prove the well-posedness and regularity of the regularized system via a finite difference scheme and study the asymptotic limit as regularization parameters vanish, proving the existence of weak/distributional solutions of the original model. This approach advances the mathematical theory of kinetic models for image segmentation by extending the framework of distributional solutions to bounded domains and non-smooth interaction kernels with jump discontinuities.

We study the problem of optimally placing a large number of small absorbing traps to minimize the mean first-passage time (MFPT) of particles diffusing in a heterogeneous medium with space-dependent diffusivity $D(x)$ and prescribed initial particle distribution $\omega(x)$. In two and three dimensions we identify two distinct regimes, depending on how strongly the trap ensemble depletes the particles. In the weak trapping regime (fewer traps) the optimal trap density $\rho$ is the arithmetic average $\rho = \frac12(\mu+\omega)$ of the normalized inverse diffusivity $\mu \propto 1/D$ and the initial particle distribution $\omega$. In the strong trapping regime (more traps), it is proportional to their geometric average, $\rho \propto \sqrt{\mu \omega}$. We revisit the classical Green's function approach and show that it applies to weak trapping only; in 2D this requires the trap size to be exponentially small in the number of traps, so that most applications of interest fall in the strong trapping regime instead. We therefore develop a homogenization approach that captures both regimes. A special choice of initial particle distribution is $\omega \propto 1/D$, which corresponds to the equilibrium particle distribution in the absence of traps according to the Itô interpretation. In this case we show that the optimal trap distribution is the stationary particle distribution itself, regardless of the trapping strength, and this makes the MFPT constant throughout the domain; the same trap distribution also minimizes the worst-case capture time. Finally, in one dimension we find instead $\rho\propto\left(\mu \omega\right) ^{1/3}$, and a similar analysis covers space-dependent drift and thin domains of variable cross-section. Direct numerical optimization confirms the analytical results.

We show that the space of vacuum initial data sets on a closed manifold often has many non-trivial homotopy groups. The starting point is a result of the second named author, which constructs non-trivial elements in the homotopy groups of the space of initial data sets satisfying the strict dominant energy condition, together with a later result in joint work with Bernd Ammann showing that these elements often persist when the strictness assumption is dropped. In this work, we use a parametrized version of the conformal method to show that these elements may also be represented by maps into the space of vacuum initial data sets. This requires two results that may be of independent interest: metrics admitting conformal Killing vectors can be removed from the space of metrics without changing its weak homotopy type, and over the remaining metrics the York decomposition can be carried out in families, the TT-tensors forming a trivial Hilbert bundle. To our knowledge, this is the first result on the global topology of this space.

We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension $N\ge22$ and with nonumbilic boundary in every $N\ge15$. For positive scalar curvature and minimal boundary, the corresponding ranges are $N\ge21$ and $N\ge15$. For every $N\ge9$, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is $N(N-1)$ and the prescribed boundary mean curvature is below a negative threshold depending on $N$. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.

We study the simultaneous low Mach number limit of a compressible Navier–Stokes–Euler two-phase system in $\mathbb R^3$, in which the pressure terms in both phases are scaled by $\varepsilon^{-2}$. For sufficiently small $H^3$-perturbations around the constant equilibrium, we establish the global well-posedness of the scaled compressible system and derive global energy-dissipation estimates that are uniform with respect to the Mach number $\varepsilon$. A key feature of the analysis is the degenerate dissipation structure: viscosity acts only on the Navier–Stokes phase, while the drag coupling transfers dissipation to the inviscid Euler phase through the relaxation mode. We then prove the global well-posedness and large-time decay of the limiting incompressible two-phase system, and show that the relative velocity $u-\omega$ decays faster than the full velocity pair. Finally, for the well-prepared initial data, we introduce pressure-corrected acoustic variables to remove the singular pressure mismatch and establish a global-in-time $H^2$-error. As a result, the scaled compressible solutions converge uniformly in time to the corresponding solution of the limiting incompressible system.

We consider a strongly dispersive Boussinesq-type system arising as a model for the propagation of weakly nonlinear water waves. The system is characterized by a nonlinearity parameter $0<\epsilon\le 1$ and a shallow water parameter $0<\mu\le 1$. In the Boussinesq regime $\mu \sim \epsilon$, Saut, Wang and Xu \cite{SWX2017} used a hyperbolic approach to prove the well-posedness of the system on a time scale of order $1/\epsilon$, in both one and two dimensions, under a non-cavitation condition. In this paper, we establish well-posedness on a time scale of order $ ( \mu^{ 1/4} \epsilon^{-1} h_0 )^{4/3}$ in one dimension and of order $(\mu^{ 1/4}\epsilon^{-1} h_0)^{2-}$ in two dimensions, assuming a non-cavitation condition with a parameter $h_0>0$. Our results show quantitatively how the lifespan depends on the balance between nonlinearity and dispersion in the regimes $\epsilon\lesssim \mu$ and $\mu\ll\epsilon$. They also make explicit how the existence time deteriorates as $h_0\to0$, that is, as the wave approaches cavitation. Our proofs combine dispersive and Strichartz estimates with energy estimates.

In this paper, we establish symmetry results for solutions of overdetermined problems arising in optimal transportation. These problems involve the Monge-Ampère equation with a Dirichlet boundary condition $u=0$ on $\partial \Omega$ and the natural boundary condition $Du(\Omega)=\Omega^{*}$. We show that symmetry holds when the target is the unit ball $B$. For more general target domains, including the cases $\Omega^*=\Omega$ and arbitrary $\Omega^{*}$, symmetry is retained under an additional volume constraint on $\Omega$ and a boundary condition on $|Du|$. Finally, we introduce a curvature-type overdetermined problem for the Monge-Ampère equation and obtain ellipsoidal or spherical symmetry under an additional integral normalization condition. Our proofs rely on distinct techniques in different contexts: optimal transport and convex analysis for some cases, integral identities and isoperimetric inequalities for others, and P-function methods for the remaining cases. As a byproduct, in the $\tau=2$ case, a new maximum principle for the $P$-function $\phi(x)=\sum_{k,l=1}^{n}{\frac{\partial{S_{\tau}(D^2{u})}}{\partial{u_{kl}}}u_{k}u_{l}}-2\binom{n-1}{\tau-1}\int_{0}^{u}{f^{\frac{\tau}{n}}(t)\,dt}$ is established for arbitrary positive and nondecreasing $f$, which is of independent interest.

We study the reversible reaction $1+2\rightleftarrows3+4$ in a dilute gas described by a Boltzmann system. We prove that the outgoing flux of one product species determines the forward collision kernel throughout the domain on a prescribed admissible set of relative velocities and outgoing directions. The elastic collision laws are known, and the reverse kernel is fixed by a prescribed reciprocity relation. The measurements use two synchronized incoming packets with fixed relative velocity and a common velocity tending to infinity. Under bounded angularly integrated collision rates, the short transit time allows us to isolate the contribution of one forward reaction, with estimates uniform as the packets concentrate. Suitable tests of the product flux yield the spatial line integrals of the kernel and a Fourier reconstruction, without differentiating the nonlinear boundary map. We also estimate the effects of finite beam speed, packet width, detector resolution, and measurement noise on the recovered line integrals.

The known proofs of the gradient estimate for complex Hessian equations rely on the second order estimate of Hou-Ma-Wu and on a Liouville theorem of Dinew-Kolodziej. In this paper, we give a direct proof of the gradient estimate for concave fully nonlinear elliptic equations on compact Hermitian manifolds, which uses neither the Hou-Ma-Wu estimate nor the Liouville theorem. Instead, the proof combines a stability estimate with a comparison argument against nearby smooth admissible functions. We also give an independent proof of the complex Hessian estimate on compact Kähler manifolds, based on Dirichlet Green's functions, under additional assumptions on the operator.

This paper constructs explicit closed-form Green's functions for the Helmholtz equation on annular and spherical-shell domains with two independent Robin impedances on the inner and outer boundaries. Graf's and the hyperspherical addition theorems reduce each angular mode to an explicit $2\times2$ linear system, and the resonance spectrum is governed by a characteristic determinant bilinear in the two impedances. This bilinearity has a direct inverse-problem consequence: two resonant frequencies of one non-radial angular mode generate at most two candidate impedance pairs via an explicit quadratic equation, a third resonance selects the physical pair, and an exact reflection-symmetry obstruction identifies where the radial spherical mode cannot recover the impedances. Spectrally, we prove the branches positive, simple and strictly increasing in both impedances; derive first-order asymptotics at the four corners of the impedance plane, with coefficients given by boundary masses and normal derivatives of the limiting eigenfunctions, and an explicit mixed second-order coefficient at the Dirichlet–Dirichlet corner, which for the radial mode evaluates in closed form to $2\pi/(R_2-R_1)^3$; establish a low-frequency resonance-free band, with a second-order threshold expansion explicit in dimension three and a rational approximation accurate over the whole impedance range; and prove the universal high-frequency spacing law with a shell-curvature correction. Jacobian-based sensitivity and conditioning criteria are included. The kernels and spectra are computable to machine precision, all asymptotic regimes are confirmed numerically, and the kernels provide reference solutions for finite-element and boundary-element validation.

We determine the exact number, up to rotations, of nonconstant positive $C^2$ solutions to the planar isotropic $L_p$ dual Minkowski problem for every $(p, q) \in \mathbb{R}^2$. We obtain a new parametrization of the associated period integral, characterize the regions where the period is strictly increasing or strictly decreasing, and prove that in the remaining nonmonotone region it has a unique nondegenerate maximum. As a consequence, we have a complete classification.