Journal of the London Mathematical Society-Second Series最新文献

We investigate the regularity of solutions to linear elliptic equations in both divergence and non-divergence forms, particularly when the principal coefficients have Dini mean oscillation. We show that if a solution $�u$ to a divergence-form equation satisfies $��D�u�\left(�x^o�\right)�=�0�$ at a point, then the second derivative $��D^2�u��\left(�x^o�\right)��$ exists and satisfies sharp continuity estimates. As a consequence, we obtain “$�C^{�2�,�\alpha �}$ regularity” at critical points when the coefficients of $�L$ are $�C^{\alpha }$. This result refines a theorem of Teixeira (Math. Ann. 358 (2014), no. 1–2, 241–256) in the linear setting, where both linear and nonlinear equations were considered. We also establish an analogous result for equations in the non-divergence form.

For the piecewise linear Liénard system $��\dot{x}�=�F��\left(�x�\right)��-�y�$, $��\dot{y}�=�x�$, where $��F�\left(�x�\right)�$ is a continuous piecewise linear function with $�n$ fold points, the question of how many limit cycles that such a system can have has been a classical and open problem in differential equations and dynamical systems. Nowadays, we only know the answer for the case $��n�=�1�$. For the cases $��n�⩾�2�$, the problem still remains open. This paper aims to give an affirmative answer for the case $��n�=�2�$ of this open problem, that is, the maximum number of limit cycles of the continuous piecewise linear function with 2 fold points is 2. This system exhibits abundantly interesting and rich dynamics, including the generalized Hopf bifurcation, boundary equilibrium bifurcation, grazing limit cycle bifurcation, and double limit cycle bifurcation, which may have potential interdisciplinary applications.

We study the large-time behavior of finite-energy weak solutions for the Vlasov–Navier–Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable-density case. In both cases, large-time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han-Kwan et al. [16] dedicated to the homogeneous, three-dimensional case, where an additional smallness condition on the velocity was required. In the nonhomogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.

We consider a generalization of the Kauffman bracket skein algebra of a surface that is generated by loops and arcs between marked points on the interior or boundary, up to skein relations defined by Muller and Roger-Yang. We compute the center of the Muller–Roger–Yang skein algebra and show that this algebra is almost Azumaya when the quantum parameter $�q$ is a primitive $�n$th root of unity with odd $�n$. We also discuss the implications on the representation theory of the Muller–Roger–Yang generalized skein algebra.

We compare the marked length spectra of isometric actions of groups with non-positively curved features. Inspired by the recent works of Butt, we study approximate versions of marked length spectrum rigidity. We show that for pairs of metrics, the supremum of the quotient of their marked length spectra is approximately determined by their marked length spectra restricted to an appropriate finite set of conjugacy classes. Applying this to fundamental groups of closed negatively curved Riemannian manifolds allows us to refine Butt's result. Our results, however, apply in greater generality and do not require the acting group to be hyperbolic. For example, we are able to compare the marked length spectra associated to mapping class groups acting on their Cayley graphs or on the curve graph.

In a recent paper, Brugallé and Jaramillo-Puentes showed that the coefficients of small codegree of the tropical refined invariant are polynomial in the Newton polygon. This raised the question of the existence of universal polynomials giving these coefficients, that is, polynomials depending only on the genus and the codegree, and with variables the combinatorial data of the Newton polygon. In this paper, we show that such universal polynomials exist for rational enumeration, and we give an explicit formula. The proof relies on the manipulation of floor diagrams.

We show that analytic analogs of Brunn–Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saorín Gómez. By restricting to a smaller set of admissible functions, we then introduce a family of variational functionals and establish Wulff-type inequalities for these quantities. In addition, we derive inequalities for the corresponding family of mixed functionals, thereby generalizing an earlier Aleksandrov–Fenchel-type inequality by Klartag and recovering a special case of a Pólya–Szegő-type inequality by Klimov, which was also recently investigated by Bianchi, Cianchi, and Gronchi.

We show that Wise's power alternative is stable under certain group constructions, use this to prove the power alternative for new classes of groups and recover known results from a unified perspective. For groups acting on trees, we introduce a dynamical condition that allows us to deduce the power alternative for the group from the power alternative for its stabilisers of points. As an application, we reduce the power alternative for Artin groups to the power alternative for free-of-infinity Artin groups, under some conditions on their parabolic subgroups. We also introduce a uniform version of the power alternative and prove it, among other things, for a large family of two-dimensional Artin groups. As a corollary, we deduce that these Artin groups have uniform exponential growth. Finally, we prove that the power alternative is stable under taking relatively hyperbolic groups. We apply this to show that various examples, including all free-by-$�Z$ groups and a natural subclass of hierarchically hyperbolic groups, satisfy the uniform power alternative.

In this article, we show that generally almost regular flows, introduced by Bamler and Kleiner, in closed 3-manifolds will either go extinct in finite time or flow to a collection of smooth embedded minimal surfaces, possibly with multiplicity. Using a perturbative argument, then we construct piecewise almost regular flows that either go extinct in finite time or flow to a stable minimal surface, possibly with multiplicity. We apply these results to construct minimal surfaces in 3-manifolds in a variety of circumstances, mainly novel from the point of the view that the arguments are via parabolic methods.