Annali di Matematica Pura ed Applicata最新文献

We propose the novel spectral properties of the Neumann Laplacian in a two-dimensional bounded domain perturbed by an infinite number of compact sets with zero Lebesgue measure, so-called curved holes. These holes consist of segments or parts of curves enclosed in small spheres such that the diameters of holes tend to zero as the number of holes approaches infinity. Specifically, we rigorously demonstrate that the spectrum of the Neumann Laplacian on the perturbed domain converges to that of the original operator on the domain without holes under specific geometric assumptions and an appropriate selection of hole sizes. Furthermore, we derive sophisticated estimates on the convergence rate in terms of operator norms and estimate the Hausdorff distance between the spectra of the Laplacians.

Let G be the semidirect product (N rtimes mathbb {R}), where N is a stratified Lie group and (mathbb {R}) acts on N via automorphic dilations. Homogeneous left-invariant sub-Laplacians on N and (mathbb {R}) can be lifted to G, and their sum (Delta ) is a left-invariant sub-Laplacian on G. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin–Hörmander type was proved for (Delta ), showing that an operator of the form (F(Delta )) is of weak type (1, 1) and bounded on (L^p(G)) for all (p in (1,infty )) provided F satisfies a scale-invariant smoothness condition of order (s > (Q+1)/2), where Q is the homogeneous dimension of N. Here we show that, if N is a group of Heisenberg type, or more generally a direct product of Métivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold (s>(d+1)/2), where d is the topological dimension of N. The proof is based on lifting to G weighted Plancherel estimates on N and exploits a relation between the functional calculi for (Delta ) and analogous operators on semidirect extensions of Bessel–Kingman hypergroups.

We start providing a quantitative stability theorem for the rigidity of an overdetermined problem involving harmonic functions in a punctured domain. Our approach is inspired by and based on the proof of rigidity established in Enciso and Peralta-Salas (Nonlinear Anal 70(2):1080–1086, 2009), and reveals essential differences with respect to the stability results obtained in the literature for the classical overdetermined Serrin problem. Secondly, we provide new weighted Poincaré-type inequalities for vector fields. These are crucial tools for the study of the quantitative stability issue initiated in Poggesi (Soap bubbles and convex cones: optimal quantitative rigidity, 2022. arXiv:2211.09429) concerning a class of rigidity results involving mixed boundary value problems. Finally, we provide a mean value-type property and an associated weighted Poincaré-type inequality for harmonic functions in cones. A duality relation between this new mean value property and a partially overdetermined boundary value problem is discussed, providing an extension of a classical result obtained in Payne and Schaefer (Math Methods Appl Sci 11(6):805–819, 1989).

We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves (R^{bullet }phi _{!}mathcal {O}) for the structure sheaf (mathcal {O}) on the total space of a holomorphic fiber bundle (phi ) has canonical topology structures. Using the standard Čech argument we prove a density lemma for QDFS-topology on this stalks. In particular, we obtain a vanishing result for holomorphic fiber bundles with Stein fibers. Using Künnet formulas, properties of an inductive topology (with respect to the pair of spaces) on the stalks of the sheaf (R^{1}phi _{!}mathcal {O}) and a cohomological criterion for the Hartogs phenomenon we obtain the main result on the Hartogs phenomenon for the total space of holomorphic fiber bundles with (1, 0)-compactifiable fibers.

Let (Omega ) be a bounded, smooth domain of ({mathbb {R}}^{N},) (Nge 2.) For (1<p<N) and (0<q(p)<p^{*}:=frac{Np}{N-p}), let

We prove that if (lim _{prightarrow 1^{+}}q(p)=1,) then (lim _{prightarrow 1^{+}}lambda _{p,q(p)}=h(Omega )), where (h(Omega )) denotes the Cheeger constant of (Omega .) Moreover, we study the behavior of the positive solutions (w_{p,q(p)}) to the Lane–Emden equation (-{text {div}} (left| nabla wright| ^{p-2}nabla w)=left| wright| ^{q-2}w,) as (prightarrow 1^{+}.)

The generalized Zermelo navigation problem looks for the shortest time paths in an environment, modeled by a Finsler manifold (M, F), under the influence of wind or current, represented by a vector field W. The main objective of this paper is to investigate the relationship between the isoparametric functions on the manifold M with and without the presence of the vector field W. Our work generalizes results in (Dong and He in Differ Geom Appl 68:101581, 2020; He et al. in Acta Math Sinica Engl Ser 36:1049–1060, 2020; He et al. in Differ Geom Appl 84:101937, 2022; Ming et al. in Pub Math Debr 97:449–474, 2020; Xu et al. in Isoparametric hypersurfaces induced by navigation in Lorentz Finsler geometry, 2021). For the positive-definite cases, we also compare the mean curvatures in the manifold. Overall, we follow a coordinate-free approach.

In this study, the stability problem of a laminated beam with only structural damping is analyzed. The results obtained in this study improve the analysis of the problem by investigating stability without introducing additional dissipation. This is accomplished by considering only the usual assumption of equal wave velocities as the stability criterion.

We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups which admit invariant SNP Weyl connections.

It is well-known that an integrally closed domain D can be expressed as the intersection of its valuation overrings but, if D is not a Prüfer domain, most of the valuation overrings of D cannot be seen as localizations of D. The Kronecker function ring of D is a classical construction of a Prüfer domain which is an overring of D[t], and its localizations at prime ideals are of the form V(t) where V runs through the valuation overrings of D. This fact can be generalized to arbitrary integral domains by expressing them as intersections of overrings which admit a unique minimal overring. In this article we first continue the study of rings admitting a unique minimal overring extending known results obtained in the 1970s and constructing examples where the integral closure is very far from being a valuation domain. Then we extend the definition of Kronecker function ring to the non-integrally closed setting by studying intersections of Nagata rings of the form A(t) for A an integral domain admitting a unique minimal overring.

We give a very general splitting type theorem for biholomorphic maps close to identity in the context of smoothly bounded pseudoconvex domains (Theorem 1.4). As a particular case, in the context of worm domains, we essentially reprove the splitting type result (Theorem 1.3) from Bracci et al. (Math Z 292:879–893, 2019) (by a different method). We also discuss some properties of the Nebenhülle of worm domains.