Annali di Matematica Pura ed Applicata最新文献

We study stable solutions to fractional semilinear equations ((-Delta )^s u = f(u)) in (Omega subset {mathbb {R}}^n), for convex nonlinearities f, and under the Dirichlet exterior condition (u=g) in ({mathbb {R}}^n {setminus } Omega) with general g. We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems. As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions (1 leqslant n leqslant 4).

This paper explores the global properties of time-independent systems of operators in the framework of time-periodic Gelfand–Shilov spaces. Our main results provide both necessary and sufficient conditions for global solvability and global hypoellipticity, based on analysis of the symbols of operators. We also present a class of time-dependent operators whose solvability and hypoellipticity are linked to the same properties of an associated time-independent system, albeit with a loss of regularity for temporal variables.

In this paper we give different estimates between Lebesgue norms of quadratic time-frequency representations. We show that, in some cases, it is not possible to have such bounds in classical (L^p) spaces, but the Lebesgue norm needs to be suitably weighted. This leads to consider weights of polynomial type, and, more generally, of ultradifferentiable type, and this, in turn, gives rise to use as functional setting the ultradifferentiable classes. As applications of such estimates we deduce uncertainty principles both of Donoho-Stark type and of local type for representations.

We study the existence of very weak solutions to a system

with a datum ({pmb {mathsf {mu }}}) being a vector-valued bounded Radon measure and ({{mathcal {A}}}:Omega times {{mathbb {R}}^{ntimes m}}rightarrow {{mathbb {R}}^{ntimes m}}) having measurable dependence on the spacial variable and Orlicz growth with respect to the second variable. We are not restricted to the superquadratic case. For the solutions and their gradients we provide regularity estimates in the generalized Marcinkiewicz scale. In addition, we show a precise sufficient condition for the solution to be a Sobolev function.

We present an effective construction of non-Kähler supersymmetric mirror pairs in the sense of Lau, Tseng and Yau (Commun. Math. Phys. 340:145–170, 2015) starting from left-invariant affine structures on Lie groups. Applying this construction we explicitly find SYZ mirror symmetric partners of all known compact 6-dimensional completely solvable solvmanifolds that admit a semi-flat type IIA structure.

Given a Euclidean submanifold (g:M^{n}rightarrow {mathbb {R}}^{n+p}), Chern and Kuiper provided inequalities between (mu ) and (nu _g), the ranks of the nullity of (M^n) and the relative nullity of g respectively. Namely, they prove that

In this work, we study the submanifolds with (nu _gne mu ). More precisely, we characterize locally the ones with (0ne (mu -nu _g)in {p,p-1,p-2}) under the hypothesis of (nu _gle n-p-1).

Keeping in view the recent developments of wavelets on locally compact Abelian groups (LCA) along with the applicability of the unifying structure of LCA groups, we present an explicit and efficient method for the construction of wavelet frames of arbitrary dilations on LCA groups. The method is exhibited via several illustrative examples.

This paper is concerned with the gradient continuity for the parabolic ((1,,p))-Laplace equation. In the supercritical case (frac{2n}{n+2}<p<infty ), where (nge 2) denotes the space dimension, this gradient regularity result has been proved recently by the author. In this paper, we would like to prove that the same regularity holds even for the subcritical case (1<ple frac{2n}{n+2}) with (nge 3), on the condition that a weak solution admits the (L^{s})-integrability with (s>frac{n(2-p)}{p}). The gradient continuity is proved, similarly to the supercritical case, once the local gradient bounds of solutions are verified. Hence, this paper mainly aims to show the local boundedness of a solution and its gradient by Moser’s iteration. The proof is completed by considering a parabolic approximate problem, verifying a comparison principle, and showing a priori gradient estimates of a bounded weak solution to the relaxed equation.

As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we introduce the equivariant CR Yamabe problem and prove some related results.

A compact manifold M together with a Riemannian metric h on its universal cover (tilde{M}) for which (pi _1(M)) acts by similarities is called a similarity structure. In the case where (pi _1(M) not subset textrm{Isom}(tilde{M}, h)) and ((tilde{M}, h)) is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial (mathbb {R}^p)-principal bundles over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure.