CyberSec Research

We discuss the validity of Minkowski integral identities for hypersurfaces inside a cone, intersecting the boundary of the cone orthogonally. In doing so we correct a formula provided in [3]. Then we study rigidity results for constant mean curvature graphs proving the precise statement of a result given in [9] and [10]. Finally we provide an integral estimate for stable constant mean curvature hypersurfaces in cones.

Based on the Lie symmetry method, we investigate a Feynman-Kac formula for the classical geometric mean reversion process, which effectively describing the dynamics of short-term interest rates. The Lie algebra of infinitesimal symmetries and the corresponding one-parameter symmetry groups of the equation are obtained. An optimal system of invariant solutions are constructed by a derived optimal system of one-dimensional subalgebras. Because of taking into account a supply response to price rises, this equation provides for a more realistic assumption than the geometric Brownian motion in many investment scenarios.

For a bounded open set $\Omega \subset \mathbb{R}^n$ with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue $\lambda_1(\Omega)$ of the Laplacian is at least that of the unit ball $B$. We prove that the deficit $\lambda_1(\Omega)- \lambda_1(B)$ in the Faber-Krahn inequality controls the square of the distance between the resolvent operator $(-\Delta_\Omega)^{-1}$ for the Dirichlet Laplacian on $\Omega$ and the resolvent operator on the nearest unit ball $B(x_\Omega)$. The distance is measured by the operator norm from $C^{0,\alpha}$ to $L^2$. As a main application, we show that the Faber-Krahn deficit $\lambda_1(\Omega)- \lambda_1(B)$ controls the squared $L^2$ norm between $k$th eigenfunctions on $\Omega$ and $B(x_\Omega)$ for every $k \in \mathbb{N}.$ In both of these main theorems, the quadratic power is optimal.

We study the Dirichlet problem for Monge-Amp\`ere equation in bounded convex polytopes. We give sharp conditions for the existence of global $C^2$ and $C^{2,\alpha}$ convex solutions provided that a global $C^2$, convex subsolution exists.

In the first part of this work, we investigate the Cauchy problem for the $d$-dimensional incompressible Oldroyd-B model with dissipation in the stress tensor equation. By developing a weighted Chemin-Lerner framework combined with a refined energy argument, we prove the existence and uniqueness of global solutions for the system under a mild constraint on the initial velocity field, while allowing a broad class of large initial data for the stress tensor. Notably, our analysis accommodates general divergence-free initial stress tensors ( $\mathrm{div}\tau_0=0$) and significantly relaxes the requirements on initial velocities compared to classical fluid models. This stands in sharp contrast to the finite-time singularity formation observed in the incompressible Euler equations, even for small initial data, thereby highlighting the intrinsic stabilizing role of the stress tensor in polymeric fluid dynamics. The second part of this paper focuses on the small-data regime. Through a systematic exploitation of the perturbative structure of the system, we establish global well-posedness and quantify the long-time behavior of solutions in Sobolev spaces $H^3(\mathbb{T}^d)$. Specifically, we derive exponential decay rates for perturbations, demonstrating how the dissipative mechanisms inherent to the Oldroyd-B model govern the asymptotic stability of the system.

Magnetic helicity is a quantity that underpins many theories of magnetic relaxation in electrically conducting fluids, both laminar and turbulent. Although much theoretical effort has been expended on magnetic fields that are everywhere tangent to their domain boundaries, many applications, both in astrophysics and laboratories, actually involve magnetic fields that are line-tied to the boundary, i.e. with a non-trivial normal component on the boundary. This modification of the boundary condition requires a modification of magnetic helicity, whose suitable replacement is called relative magnetic helicity. In this work, we investigate rigorously the behaviour of relative magnetic helicity under turbulent relaxation. In particular, we specify the normal component of the magnetic field on the boundary and consider the \emph{ideal limit} of resistivity tending to zero in order to model the turbulent evolution in the sense of Onsager's theory of turbulence. We show that relative magnetic helicity is conserved in this distinguished limit and that, for constant viscosity, the magnetic field can relax asymptotically to a magnetohydrostatic equilibrium.

In this paper, we characterize the geometry of solutions to one-phase inhomogeneous fully nonlinear Stefan problem with flat free boundaries under a new nondegeneracy assumption. This continues the study of regularity of flat free boundaries for the linear inhomogeneous Stefan problem started in [9], as well as justifies the definition of flatness assumed in [15].

In this paper, we consider the mass-critical focusing nonlinear Schr\"odinger on bounded two-dimensional domains with Dirichlet boundary conditions. In the absence of control, it is well-known that free solutions starting from initial data sufficiently large can blow-up. More precisely, given a finite number of points, there exists particular profiles blowing up exactly at these points at the blow-up time. For pertubations of these profiles, we show that, with the help of an appropriate nonlinear feedback law located in an open set containing the blow-up points, the blow-up can be prevented from happening. More specifically, we construct a small-time control acting just before the blow-up time. The solution may then be extended globally in time. This is the first result of control for blow-up profiles for nonlinear Schr\"odinger type equations. Assuming further a geometrical control condition on the support of the control, we are able to prove a null-controllability result for such blow-up profiles. Finally, we discuss possible extensions to three-dimensional domains.

We propose a characterization of a $p$-Laplace higher eigenvalue based on the inverse iteration method with balancing the Rayleigh quotients of the positive and negative parts of solutions to consecutive $p$-Poisson equations. The approach relies on the second eigenvalue's minimax properties, but the actual limiting eigenvalue depends on the choice of initial function. The well-posedness and convergence of the iterative scheme are proved. Moreover, we provide the corresponding numerical computations. As auxiliary results, which also have an independent interest, we provide several properties of certain $p$-Poisson problems.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-freqency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions 7 and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov's Lemma for ODE that is to our knowledge new.

How individual dispersal patterns and human intervention behaviours affect the spread of infectious diseases constitutes a central problem in epidemiological research. This paper develops an impulsive nonlocal faecal-oral model with free boundaries, where pulses are introduced to capture a periodic spraying of disinfectant, and nonlocal diffusion describes the long-range dispersal of individuals, and free boundaries represent moving infected fronts. We first check that the model has a unique nonnegative global classical solution. Then, the principal eigenvalue, which depends on the infected region, the impulse intensity, and the kernel functions for nonlocal diffusion, is examined by using the theory of resolvent positive operators and their perturbations. Based on this value, this paper obtains that the diseases are either vanishing or spreading, and provides criteria for determining when vanishing and spreading occur. At the end, a numerical example is presented in order to corroborate the theoretical findings and to gain further understanding of the effect of the pulse intervention. This work shows that the pulsed intervention is beneficial in combating the diseases, but the effect of the nonlocal diffusion depends on the choice of the kernel functions.

We investigate the asymptotic spectral distribution of the twisted Laplacian associated with a real harmonic 1-form on a compact hyperbolic surface. In particular, we establish a sublinear lower bound on the number of eigenvalues in a sufficiently large strip determined by the pressure of the harmonic 1-form. Furthermore, following an observation by Anantharaman \cite{nalinideviation}, we show that quantum unique ergodicity fails to hold for certain twisted Laplacians.

In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set where the solutions become identically equal to their maximum. We prove uniqueness, sharp regularity, and non-degeneracy bounds for solutions under suitable assumptions on the reaction term. Of independent interest, our methods allow us to prove that the free boundaries of a broad class of semilinear equations have locally finite $H^{n-1}$ measure.

Understanding how spectral quantities localize on manifolds is a central theme in geometric spectral theory and index theory. Within this framework, the BFK formula, obtained by Burghelea, Friedlander and Kappeler in 1992, describes how the zeta-regularized determinant of an elliptic operator decomposes as the underlying manifold is cut into pieces. In this paper, we present a novel proof of this result. Inspired by work of Br\"uning and Lesch on the eta invariant of Dirac operators, we derive the BFK formula by interpolating continuously between boundary conditions and understanding the variation of the determinant along this deformation.

We use De Giorgi-Nash-Moser iteration scheme to establish that weak solutions to a coupled system of elliptic equations with critical growth on the boundary are in $L^\infty(\Omega)$. Moreover, we provide an explicit $L^\infty(\Omega)$- estimate of weak solutions with subcritical growth on the boundary, in terms of powers of $H^1(\Omega)$-norms, by combining the elliptic regularity of weak solutions with Gagliardo--Nirenberg interpolation inequality.

In our previous work [Van de Moortel, The breakdown of weak null singularities, Duke Mathematical Journal 172 (15), 2957-3012, 2023], we showed that dynamical black holes formed in charged spherical collapse generically feature both a null weakly singular Cauchy horizon and a stronger (presumably spacelike) singularity, confirming a longstanding conjecture in the physics literature. However, this previous result, based on a contradiction argument, did not provide quantitative estimates on the stronger singularity. In this study, we adopt a new approach by analyzing local initial data inside the black hole that are consistent with a breakdown of the Cauchy horizon. We prove that the remaining portion is spacelike and obtain sharp spacetime estimates near the null-spacelike transition. Notably, we show that the Kasner exponents of the spacelike portion are positive, in contrast to the well-known Oppenheimer-Snyder model of gravitational collapse. Moreover, these exponents degenerate to (1,0,0) towards the null-spacelike transition. Our result provides the first quantitative instances of a null-spacelike singularity transition inside a black hole. In our companion paper, we moreover apply our analysis to carry out the construction of a large class of asymptotically flat one or two-ended black holes featuring coexisting null and spacelike singularities.

The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-\mu \rho^2$ (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any $\mu \in \big[0,\frac{1}{3}\big)$. This range of $\mu$ is sharp; for $\mu \ge \frac{1}{3}$, the logistic damping effect suppresses the blowup as shown in [Kang-Stevens, 2016] and [Tello-Winkler, 2007]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted $L^2$ method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted $L^2$ space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation.

We look for traveling waves of the semi-discrete conservation law $4\dot u_j +u_{j+1}^2-u_{j-1}^2 = 0$, using variational principles related to concepts of ``hidden convexity'' appearing in recent studies of various PDE (partial differential equations). We analyze and numerically compute with two variational formulations related to dual convex optimization problems constrained by either the differential-difference equation (DDE) or nonlinear integral equation (NIE) that wave profiles should satisfy. We prove existence theorems conditional on the existence of extrema that satisfy a strict convexity criterion, and numerically exhibit a variety of localized, periodic and non-periodic wave phenomena.