Annali di Matematica Pura ed Applicata最新文献

We prove a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems whose prototype is

with parameters (p>1) and (q>0) and (Omega subset {mathbb {R}}^n). In this paper, we are concerned with the ranges (q>1) and (p>frac{n(q+1)}{n+q+1}). A key ingredient in the proof is an intrinsic geometry that takes both the solution u and its spatial gradient Du into account.

Euler’s elastica is defined by a critical point of the total squared curvature under the fixed length constraint, and its (L^p)-counterpart is called p-elastica. In this paper we completely classify all p-elasticae in the plane and obtain their explicit formulae as well as optimal regularity. To this end we introduce new types of p-elliptic functions which streamline the whole argument and result. As an application we also classify all closed planar p-elasticae.

We establish the asymptotic expansion of the fundamental solutions with precise error estimates for second-order parabolic operators

in the case (ell ne 2,) where the spatial and temporal variables oscillate on non-self-similar scales and do not homogenize simultaneously. To achieve the goal, we explore the direct quantitative two-scale expansions for the aforementioned operators, which should be of some independent interests in quantitative homogenization of parabolic operators involving multiple scales. In the self-similar case (ell =2), similar results have been obtained in Geng and Shen (Anal PDE 13(1): 147–170, 2020).

We compute the relaxed Cartesian area for a general 0-homogeneous map of bounded variation, with respect to the strict BV-convergence. In particular, we show that the relaxed area is finite for this class of maps and we provide an integral representation formula.

Let X be a K3 surface which doubly covers an Enriques surface S. If (nin {mathbb {N}}) is an odd number, then the Hilbert scheme of n-points (X^{[n]}) admits a natural quotient (S_{[n]}). This quotient is an Enriques manifold in the sense of Oguiso and Schröer. In this paper we construct slope stable sheaves on (S_{[n]}) and study some of their properties.

This work is devoted to the study of the obstacle problem associated to the Kolmogorov–Fokker–Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems.

We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature K satisfies

These surfaces are referred to as rotational Ricci surfaces. As an application, we show that there is a one-parameter family of such surfaces meeting the boundary of the unit Euclidean three-ball orthogonally. In addition, we show that this family interpolates a vertical geodesic and the critical catenoid.

We study the analytic and Gevrey regularity for a “sum of squares” operator closely connected to the Schrödinger equation with minimal coupling. We however assume that the (magnetic) vector potential has some degree of homogeneity and that the Hörmander bracket condition is satisfied. It is shown that the local analytic/Gevrey regularity of the solution is related to the multiplicities of the zeroes of the Lie bracket of the vector fields.

Abstract

We investigate a few aspects of the notion of Levi core, recently introduced by the authors in Dall’Ara, Mongodi (J l’École Polytech Math 10:1047-1095, 2023): a basic finiteness question, the connection with Kohn’s algorithm, and with Catlin’s property (P).

In this paper, we will describe some general properties regarding surfaces with Prym-canonical hyperplane sections, determining also important conditions on the geometric genera of the possible singularities that such a surface can have. Moreover, we will construct new examples of this type of surfaces.