Papers

In this paper, we study a class of polyharmonic Kirchhoff problems driven by a double phase operator. The reaction term has subcritical growth but does not satisfy the Ambrosetti–Rabinowitz condition. Motivated by the work of Harrabi-Hamdani-Fiscella \cite{Harrabi-Hamdani-Fiscella-2024} on m-polyharmonic Kirchhoff problems without Ambrosetti–Rabinowitz conditions, we extend their analysis to a nonhomogeneous double phase setting. We study the problem in the natural Musielak–Orlicz–Sobolev framework associated with the double phase structure. The main novelty of the paper lies in combining the nonlocal Kirchhoff term with a higher-order double phase operator under assumptions weaker than the classical Ambrosetti–Rabinowitz condition. By developing suitable modular estimates and compactness arguments, we establish the variational setting and obtain existence and multiplicity results by minimax methods.

We study finite-band optimal control of acoustic waves actuated by local clusters of subwavelength resonators. The acoustic problem reduces to a time-domain Foldy-Lax approximation capturing wave-structure interaction. Spectral analysis of the delayed transfer matrix isolates collective scattering resonances corresponding to weakly damped poles $s_\alpha^\epsilon=-\gamma_\alpha^\epsilon+i\omega_\alpha^\epsilon$ with radiation damping $\gamma_\alpha^\epsilon>0$. Projecting onto a finite band yields the coupled system $\ddot{a}+\Lambda a=C_\epsilon\eta$, $\ddot{\eta}+2\Gamma_\epsilon\dot{\eta}+K_\epsilon\eta=u$, where $a$, $\eta$, and $u$ are modal coefficients, microstructural states, and control. For a tracking functional $\mathcal{J}_{\mu}$ with regularization $\mu>0$, we prove existence and uniqueness of the optimal control and derive the adjoint system. Our main quantitative result is a resonant source-lifting estimate: if a source profile $\eta_r$ is spectrally concentrated in bands $I_\alpha$, the input $u_r=(\partial_t^2+2\Gamma_\epsilon\partial_t+K_\epsilon)\eta_r$ satisfies $\|u_r\|_{L^2(0,T)}^2 \le \sum_\alpha \left(\sup_{\nu\in I_\alpha} |(\omega_\alpha^\epsilon)^2+(\gamma_\alpha^\epsilon)^2-\nu^2 +2i\gamma_\alpha^\epsilon\nu|\right)^2 \|(\eta_r)_\alpha\|_{\mathcal{B}_T(I_\alpha)}^2$. This provides an upper bound for the optimal value function. At exact matching $\nu=\omega_\alpha^\epsilon$, the multiplier equals $2\gamma_\alpha^\epsilon\omega_\alpha^\epsilon+O((\gamma_\alpha^\epsilon)^3)$, showing clustering yields a finite resonant gain governed by the pole's real part. Finally, this attenuation enables finite-band stabilization under an explicit modal coupling condition, with a decay rate proportional to the cluster damping scale.

A variational minimum principle for linear elastodynamics of a possibly heterogeneous material without a stored energy function is developed. It involves a change of variables to dual fields, and results in a degenerate elliptic Euler-Lagrange system, even when the primal elastodynamics is hyperbolic. Uniqueness assertions for the dual dynamic and static problems and implications of the degenerate ellipticity are sketched. Some implications pertaining to heterogeneous materials and ones with indefinite elastic moduli are discussed.

We prove that every closed Riemannian $4$ or $5$-manifold $M$ contains a branched immersed closed minimal surface. That is, there exists a non-constant weakly conformal harmonic map from some closed Riemann surface into $M$. We rely on the existence of multisections in dimensions $4$ and $5$ to generate a non-trivial class of sweepouts of $M$ by mappings from a closed surface $S$ of genus at least $2$. To each sweepout in a minimizing sequence within the class, through the intermediary of quasiconformal maps of the upper half-plane, we associate a family of hyperbolic metrics on $S$ with respect to which the mappings in the sweepout have nearly equal energy and area. The harmonic replacement method of Colding and Minicozzi is then applied to obtain a min-max sequence that converges to a bubble tree of branched minimal immersions.

We revisit Serrin's interior spatial regularity criterion for distributional solutions to the Navier-Stokes equations in $\mathbb R^3$ and considerably relax the hypotheses in two main directions. More precisely, we show that if $u\in{L_t^{s'}L_x^s}$ locally is a distributional solution to the Navier-Stokes equations with $\frac2{s'}+\frac3s=1$ for $s'\in[4,\infty)$, then $u\in L^q_t(C_x^\infty)$ locally for all $q\in(2,s')$. If $s'\in(2,4)$, the same conclusion holds provided that in addition $u\in L_t^4(L_x^p)$ locally, for some $p>1$. In particular, we remove any integrability hypothesis on the vorticity, and we reduce the requirement of integrability in time all the way to $L^4$ from $L^\infty$. To achieve this, we employ a new bootstrap argument, distinct from Serrin's, and we argue that a reduction of the exponent in time integrability does not follow from Serrin's original argument.

In this work, we study the initial boundary value (IBV) problems for the sine-Gordon (sG) equation in the light-cone coordinates $u_{xt}=\sin u$ in the quarter planes $x> 0$, $t>0$ and $x< 0$, $t>0$ assuming a suitable decay as $x\to +\infty$ or as $x\to -\infty$. Employing the Riemann–Hilbert (RH) problem framework, we demonstrate that these two IBV problems differ significantly with respect to the boundary data required for well-posedness. Specifically, the solution of the “right problem” ($x\ge 0$) is uniquely determined by the initial data $u(x,0)$, $x\ge 0$ alone whereas for the “left problem” ($x\le 0$), the boundary data $u(0,t)$ has to be prescribed in addition to the initial data in order to obtain a well-posed problem.

Let $\Omega\subseteq\mathbb{R}^{n}$ ($2\leq n\in\mathbb{N}$) be a domain and let $\zeta\in\{0,\infty\}$ be an isolated point of the boundary of $\Omega$ in the one-point compactification of $\mathbb{R}^{n}$ with the ideal point $\infty$. Under some further conditions, we study Fuchsian-type singularity at $\zeta$ for the Finsler $p$-Laplace equation with a potential $$-\mathrm{div}\mathcal{A}(x,\nabla u)+V|u|^{p-2}u=0\quad (1<p<\infty)\qquad \mbox{in } \Omega,$$ where $\mathcal{A}(x,\xi)\triangleq\nabla_{\xi}(H(x,\xi)^{p}/p)$ for almost all $x\in\Omega$ and all $\xi\in\mathbb{R}^{n}$, $H$ is a family of norms on $\mathbb{R}^{n}$ ($n\geq 2$) parameterized by points $x\in\Omega$, and $V$ belongs to a local Morrey space. In particular, we investigate asymptotic behaviors of positive solutions of the equation near $\zeta$ and asymptotic behaviors of their quotients.

We characterize the kernel of the linearization $R$ of the minimal surface problem about the Euclidean metric in a bounded smooth domain $\Omega\subset\mathbb{R}^n$, $n\ge2$, with the background minimal surfaces being the Euclidean planes. We show that, in the whole-space Euclidean decomposition, the kernel consists of potential fields and TT fields. For bounded domains, a similar phenomenon appears with additional boundary coupling conditions; in particular, the TT part may be coupled to a harmonic conformal component.

In this paper, we derive improved estimates for a class of fully nonlinear parabolic equations with continuous drift and admissible source terms of the form $$ \partial_{t}u - F(D^2u,x,t) + \langle B(x,t), Du\rangle = f(x, t, u^{+}, u^{-}) \quad \text{in}\quad Q_1. $$ Our analysis reveals two distinct regimes. In the first, $f=f(x,t)$ exhibits $\theta$-Hölder decay ($\theta\in(0,1)$), yielding improved gradient regularity at vanishing points via perturbative methods and geometric iteration, as well as nondegeneracy with explicit growth rates under a suitable structural condition. In the second, $f(\cdot,u^+,u^-)=(u^+)^\gamma-(u^-)^\gamma$ with $\gamma\in(0,1)$ (corresponding to an evolutionary semilinear two-phase model), we obtain enhanced regularity at branching points by combining a robust blow-up analysis with local derivative estimates for linear equations. Our results remain relevant even in linear settings with merely continuous data, linking to classical free boundary problems arising in mathematical physics and related areas.

We study the inhomogeneous form of the Jordan–von Neumann quadratic functional equation, in which the right-hand side is a prescribed function $g$ of two real variables. We prove that the existence of a $C^{2}$ solution is equivalent to $g$ being itself of class $C^{2}$ and satisfying a single three-variable cocycle identity, and we exhibit the solution as a closed-form integral expression involving the second partial derivative of $g $ along the first coordinate axis. The construction preserves regularity along the standard scale of $C^{k}$, smooth, and polynomial classes.

We consider $D$-equivariant solutions to the harmonic map heat flow from $B^2$ to $S^2$ under general time-dependent smooth boundary data and prove that there is no finite-time blow-up when $D \geq 2$.

The CR Paneitz operator is closely related to several important problems in CR geometry. In this paper, we study the CR Paneitz operator on non-embeddable three-dimensional tori. Under mild assumptions, we show that it possesses infinitely many negative eigenvalues. We also provide concrete examples satisfying the assumptions.

Although various boundary integral formulations are available for the Helmholtz transmission problem, the weakly singular Burton-Miller (BM) equation is promising because it is well-suited for the Nyström discretization. Moreover, unlike other formulations such as the PMCHWT or Müller equations, its fictitious eigenvalues do not coincide with eigenvalues of a different transmission problem. This paper rigorously shows that the weakly singular BM equation is well-posed.

A soliton gas theory for the derivative nonlinear Schrödinger equation is developed, focusing on the Gerdjikov-Ivanov equation and using its gauge equivalence with the Kaup-Newell and Chen-Lee-Liu equations. Starting from the reflectionless inverse scattering problem, we formulate pure N-soliton solutions through a meromorphic Riemann-Hilbert problem and pass to the continuum limit as the discrete spectrum condenses on the planar spectral domains. Under a suitable scaling of the norming constants, the limit yields a compactly supported dbar-problem. For domains admitting a Schwarz function, in particular elliptic domains, this problem reduces to a Riemann-Hilbert problem on the associated mother body. For the elliptic soliton gas, the large-x and long-time asymptotic behaviors are established: the solution decays as x->+infty, approaches an elliptic finite-gap background as x->-infty, and exhibits stratified long-time sectors described by one-, two-, and three-phase Riemann theta functions. A distinctive feature of derivative nonlinear Schrodinger equation is the z->-z symmetry, which induces a quotient reduction of the effective Abelian geometry via lambda=z^2. We further derive a kinetic equation for the effective velocity of the elliptic soliton gas and an Its-Izergin-Korepin-Slavnov type Fredholm determinant representation of the continuum tau-function.

We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton–Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.

We establish Bareket's conjecture from 1977 for convex domains in all dimensions in the regime of weak boundary coupling. In other words, we consider the Laplace operator, subject to negative boundary conditions, and show that the ball maximises the first eigenvalue among all bounded convex domains of fixed volume, provided that the boundary parameter is sufficiently close to zero. The smallness depends on the volume and dimension only. The proof relies on a comparison with spherical shells with combined Neumann–Robin boundary conditions obtained via the method of parallel coordinates, which we manage to extend to all dimensions, and on a careful analysis of the corresponding radial problem.

In this paper, we study quantitative homogenization for viscosity solutions of multi-scale semilinear second order partial differential equations (PDEs) on convex domains with Neumann boundary conditions. To this aim we use the probabilistic approach by studying the quantitative homogenization of backward stochastic differential equations (SDEs) associated with slow-fast systems of reflected SDEs.

We establish the higher differentiability of the local minimizers to a class of non autonomous convex integral functionals satisfying anisotropic subquadratic growth conditions, that include, as a particular case, those with orthotropic structure. The result is obtained under a gap bound on the exponents \(p_i\), that guarantees the local boundedness of the minimizers and under a suitable Sobolev assumption on the map that measures the oscillation of the energy density with respect to the $x$ variable, that is independent on the dimension.

We study global hypoellipticity and global solvability for Vekua-type operators associated with diagonal left-invariant operators on compact Lie groups. The conjugation term produces, on the Fourier side, a family of coupled systems whose determinants govern both regularity and solvability. Under a natural non-self-duality assumption, we obtain complete Diophantine-type characterizations of global hypoellipticity and global solvability, the latter on the natural space of admissible data. We then show how the theory must be modified in the self-dual setting by carrying out a complete analysis of the model case \(\mathbb S^3\simeq SU(2)\), where conjugation acts inside each representation block. The resulting criteria exhibit two Fourier-side mechanisms for Vekua-type operators on compact Lie groups: coupling between distinct conjugate representations and coupling inside self-dual representation blocks. We also present examples on product groups illustrating how these mechanisms may coexist and how global solvability may hold even when global hypoellipticity fails.

The Monge-Ampère-type equation for forms of positive degree was introduced by Dinew and Popovici to prove the qualitative part of an analytic version of Demailly's transcendental Morse inequality for higher cohomology classes, conditional on the solvability of this nonlinear PDE. In this paper, we show that this can be proven unconditionally. We first show that the originally proposed Laplacian trace condition, $\Lambda^{m-2}\Delta u = 0$, is analytically too rigid to permit solutions and hence introduce a general gauge-fixing that leads to the $(a,b)$ Monge-Ampère-type equation. By deriving a priori estimates, we establish the solvability of this class of equations for different parametric values $(a,b)$. As a geometric application, for $b=0$ this framework reduces to the classical complex Monge-Ampère equation, which yields the qualitative Demailly's inequality for higher-degree forms. Further, we prove that the uniqueness of these solutions can be significantly strengthened under the kernel condition $\bar{\partial}^* u = 0$.

We study sequential testing and Bayesian quickest detection for diffusion observations whose drift changes between two alternatives while the signal-to-noise ratio may depend on the current observation. In this setting the posterior probability is generally not a closed one-dimensional Markov statistic: the natural sufficient state is the augmented process consisting of the posterior (or likelihood ratio) and the observed diffusion. We formulate both testing and quickest detection within this common filtering framework and identify the corresponding degenerate free-boundary problems. The main contribution is a delay-penalty comparison principle. For a common terminal false-alarm or terminal decision cost, a pointwise larger running delay penalty increases the value of continuation, shrinks the continuation region, and yields earlier stopping. When the stopping set has a one-sided posterior representation, this gives an order relation for the optimal alarm boundaries. The result applies to linear delay costs and to nonlinear marginal delay penalties after the appropriate Markovian augmentation, and is illustrated by a constant signal-to-noise Shiryaev example in which the alarm threshold is computed numerically and shown to be monotone in the delay cost. The framework clarifies how state-dependent information and nonlinear delay costs jointly affect the geometry of sequential testing and quickest-detection rules.

We consider finite-time singularity formation for classical solutions to the following parameterized nonlinear wave equation: \[ u_{tt}=c(u)^2u_{xx}+\lambda c(u)c'(u)(u_x)^2, \] where \(\lambda\in[0,2]\) is a parameter. In previous works, it is known that finite-time blow-up solutions exist for \(0<\lambda\le1\) and for \(\lambda=2\). The cases \(\lambda=1\) and \(\lambda=2\) are known as the variational wave equation and the \(p\)-system, respectively, and they possess symmetric structures or conservation laws. For \(0<\lambda<1\), the blow-up construction is based on the fact that, after decomposing the wave into two Riemann variables, one component can be kept sufficiently small and hence does not prevent the other component from blowing up. In the present paper, we treat the remaining intermediate case \(1<\lambda<2\). This case is essentially different from the previous one, since both Riemann variables may grow. The key idea is to introduce suitable supremum functions for the Riemann variables and to derive a comparison principle through right Dini derivatives. This allows us to control the relative size of the two components and to obtain a Riccati-type differential inequality for the dominant supremum function.

We develop an operator-theoretic framework for finite-dimensional, regime-dependent Volterra equations with completely monotone memory kernels, dissipative network coupling, and Hawkes-type self-excitation. For each fixed regime we construct the associated Volterra resolvent family and prove global well-posedness, continuity across regime switches, and explicit a priori bounds. The main stability result is sharp in the commuting case: after simultaneous diagonalization of the network Laplacian and the excitation operator, each mode obeys a scalar characteristic equation, and global asymptotic stability holds exactly when every modal branching ratio lies below the intensity damping threshold. We also give a norm-based sufficient condition for noncommuting operators and a Perron–Frobenius spectral criterion for nonnegative intensity blocks, showing when norm estimates are conservative. Beyond mean stability, we prove a pathwise finite-range power law for burst amplitudes generated by residence in a Hurwitz but nonnormal regime: under a cone-alignment event, the survival exponent is the ratio of the regime exit rate to a cone-corrected finite-time growth rate bounded above by the logarithmic norm of a fixed Markovian realization in the chosen Euclidean metric. A complementary idealized-feedback result shows how a logarithmic-norm contraction caps the amplification band. Finally, we derive the deterministic intensity block as a mean-field limit of a relaxing long-memory Hawkes system with regimes. Numerical experiments on modal equations, a small-world network, and a switched nonnormal ODE validate the sharp threshold and the finite-range amplification mechanism without using the closed-form tail formula as input.

We prove the Riemannian positive mass theorem in all dimensions for asymptotically flat $L^\infty$-metrics with subcritical singular sets. More precisely, we consider complete asymptotically flat manifolds whose metrics are smooth away from a compact singular set of Minkowski dimension less than $n-3+\frac{2}{n}$, and whose scalar curvature is nonnegative on the regular set. We show that the ADM mass of each asymptotically flat end is nonnegative, and that the mass vanishes in some end only in the Euclidean case. For the rigidity statement, we require additionally that the Minkowski dimension of the singular set is not larger than $n-3+\frac{1}{n-1}$. This gives an asymptotically flat analogue of Schoen's codimension-three conjecture for positive scalar curvature. The proof combines a density theorem for singular asymptotically flat metrics, capacity estimates across the singular set, conformal blow-up inspired by Bi-Hao-He-Shi-Zhu [3], and a $\mu$-bubble dimension-descent argument adapted from Brendle-Wang [6].

In this paper we show that bounded entropy solutions of genuinely nonlinear $2\times 2$ systems of hyperbolic conservation laws admit a strong trace on Lipschitz curves. The main argument relies on a new half-space Liouville-type theorem for isentropic solutions with constant normal traces, providing the first extension to general genuinely nonlinear systems of the strong trace properties available for scalar conservation laws.

We study the fractional porous medium equation on connected infinite graphs with no local finiteness assumption. We introduce a notion of weak dual solution adapted to the discrete setting, and establish existence results for nonnegative initial data belonging to a weighted space defined through the fractional Green function, extending beyond the classical $\ell^1$ framework. Our approach relies on weighted estimates and on a detailed analysis of the associated fractional Green function. In the particular case of infinite trees with standard weights, we establish comparison principles and derive estimates for the fractional Green function, which lead to quantitative smoothing effects for solutions.

In this work, we study the positive mass theorem under critical low regularity assumptions using Ricci flow smoothing. We show that asymptotically flat manifolds $(M^n,g)$ of regularity $L^\infty\cap W^{1,n}$ with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De~Cecco-Palmieri. This extends the recent work of Hafemann to the critical regularity case. Our approach is based on showing that Riemannian metrics of regularity $L^\infty\cap W^{1,n}$, whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense. In contrast, Cecchini-Frenck-Zeidler constructed examples of metrics which are in $L^\infty\cap W^{1,p}$ for all $2<p<n$, and whose distributional scalar curvature is bounded from below, that cannot be approximated by smooth metrics with the same scalar curvature lower bound. In this sense, our result is optimal.

On an asymptotically Schwarzschild smooth metric measure space modelled on the Euclidean half-space, we define a mass-type quantity similar to the classical ADM mass in General Relativity and show its geometric invariance properties. Such quantity has a close relation with the fractional Yamabe problem and the relevant Green's function.

We study the sharp stability constant in the logarithmic Sobolev inequality, defined as the infimum of the logarithmic Sobolev deficit divided by the squared $L^2$-distance from the manifold of Gaussian optimizers. For every $N\geq 1$, first we prove that this infimum is attained in the Euclidean formulation of the inequality. Then, we show that every such optimizer decays exponentially fast at infinity, and therefore gives rise to an optimizer in the Gaussian formulation, where the same sharp constant appears. The proof is based on an extension of the general strategy introduced by Bianchi and Egnell for the Sobolev inequality and recently developed by König to prove the existence of optimizers for the corresponding sharp stability constant. The main difference in the logarithmic Sobolev setting is the analysis of minimizing sequences approaching the manifold of optimizers, since the entropy term $u^2\log u^2$ does not allow for a direct global second-order expansion. This is overcome by a perturbative estimate near the Gaussian manifold, which provides the compactness threshold needed to rule out the loss of compactness.

We prove an essential spectral gap for 2D anisotropic quantum open baker's map. This extends the 1D results of Dyatlov–Jin 2017 and the isotropic 2D results of Cohen 2025a. The key ingredient is the anisotropic discrete fractal uncertainty principle (FUP) associated with a 2D anisotropic fractal set called the Bedford–McMullen carpet. We also study the relation between our anisotropic discrete FUP and its continuous counterpart in the spirit of Dyatlov–Jin 2018 and Cohen 2025a. In particular, we prove {continuous FUP} for 2D {anisotropic porous} sets, extending the (high-dimensional) isotropic results of Cohen 2025b. To the best of our knowledge, the anisotropic (line) porosity condition – a variant of Cohen's line porosity and stronger than ball porosity – appears to be new to the literature.