Mathematische Annalen最新文献

This article studies the canonical Hilbert energy (H^{s/2}(M)) on a Riemannian manifold for (sin (0,2)), with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type ({mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+int _M F(v) , dV), with (Fge 0), is given, which includes in particular the case of nonlocal s-minimal surfaces. Finally, we prove some estimates for the Caffarelli–Silvestre extension problem, which are of general interest. This work is motivated by Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form

When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution u, which is called s-polynomial. To prove (C_s^{2+alpha })-regularity and higher regularity of the solution u, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity.

We study the stabilization and the well-posedness of solutions of the quintic wave equation with locally distributed damping. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we approximate weak solutions for regular solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures.

We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of (mathbb {Z})-valued, linearly independent homology concordance homomorphisms which vanish for knots coming from (S^3). This shows that the homology concordance group modulo knots coming from (S^3) contains an infinite-rank summand. The techniques used here generalize the classification program established in previous papers regarding the local equivalence group of knot Floer complexes over (mathbb {F}[U, V]/(UV)). Our results extend this approach to complexes defined over a broader class of rings.

In this paper, we investigate the Hörmander type theorems for the multi-linear and multi-parameter Fourier multipliers. When the multipliers are characterized by (L^u)-based Sobolev norms for (1<ule 2), our results on the smoothness assumptions are sharp in the multi-parameter and bilinear case. In the multi-parameter and multi-linear case, our results are almost sharp. Moreover, even in the one-parameter and multi-linear case, our results improve earlier ones in the literature.

In his seminal 1981 study D. Jerison showed the remarkable negative phenomenon that there exist, in general, no Schauder estimates near the characteristic boundary in the Heisenberg group (mathbb {H}^{n}.) On the positive side, by adapting tools from Fourier and microlocal analysis, he developed a Schauder theory at a non-characteristic portion of the boundary, based on the non-isotropic Folland–Stein Hölder classes. On the other hand, the 1976 celebrated work of Rothschild and Stein on their lifting theorem established the central position of stratified nilpotent Lie groups (nowadays known as Carnot groups) in the analysis of Hörmander operators but, to present date, there exists no known counterpart of Jerison’s results in these sub-Riemannian ambients. In this paper we fill this gap. We prove optimal (Gamma ^{k,alpha }) ((kge 2)) Schauder estimates near a (C^{k,alpha }) non-characteristic portion of the boundary for (Gamma ^{k-2, alpha }) perturbations of horizontal Laplacians in Carnot groups.

Gromov’s (open) question whether the closed convex hull of finitely many points in a complete ({{,textrm{CAT},}}(0)) space is compact naturally extends to weaker notions of non-positive curvature in metric spaces. In this article, we consider metric spaces admitting a conical geodesic bicombing, and show that the question has a negative answer in this setting. Specifically, for each (n>1), we construct a complete metric space X admitting a conical geodesic bicombing, which is the closed convex hull of n points and is not compact. The space X moreover has the universal property that for any n points (A={x_1,ldots ,x_n}subset Y) in a complete ({{,textrm{CAT},}}(0)) space Y there exists a Lipschitz map (f:Xrightarrow Y) such that the convex hull of (A) is contained in f(X).

We construct symplectic surface bundles over surfaces with positive signatures for all but 19 possible pairs of fiber and base genera. Meanwhile, we determine the commutator lengths of a few new mapping classes.

A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by McMullen can be used to obtain a characterization of continuous, epi-translation invariant, and n-epi-homogeneous valuations on convex functions, which was previously established by Colesanti, Ludwig, and Mussnig. Following an approach by Goodey and Weil, a new characterization of 1-epi-homogeneous valuations is obtained.