Annali di Matematica Pura ed Applicata最新文献

The goal of this paper is to obtain, via the periodic unfolding method, the homogenized limit of a stationary diffusion model describing a composite made by a hosting medium containing a periodic array of inclusions of size (varepsilon ). The thermal potentials of the two phases are connected through suitable imperfect contact conditions imposed on the interface separating the two materials. Despite the fact that the limit model, obtained as (varepsilon rightarrow 0), is governed by a standard Dirichlet problem for an elliptic equation, the construction of the homogenized matrix and of the limit source term deserves a deep investigation. We propose this microscopic model inspired by the result of a concentration procedure performed in a simplified flat geometry, where we have two different bulk materials separated by a thin layer of another material with thickness of the order (eta ). The thin layer presents an inner interface with imperfect contact conditions of non-local type and we deal with the concentration, as (eta rightarrow 0), of such a layer. The final interface conditions thus obtained are exactly the interface conditions we impose in the above mentioned microscopic model set in a more general geometry.

We prove the weak Harnack inequality for the functions u which belong to the corresponding De Giorgi classes (DG^{-}(Omega )) under the additional assumption that (uin L^{s}_{loc}(Omega )) with some (s> 0). In particular, our result covers new cases of functionals with a variable exponent or double-phase functionals under the non-logarithmic condition.

We introduce a new set of relations in the tautological Chow rings of the moduli space of stable curves of genus g. These relations are obtained by computing the Poincaré-dual class of empty loci in the Hodge bundle in terms of the standard generators of the tautological rings. In particular, we use these relations to obtain a new expression for the Chern classes of the Hodge bundle. We prove that the ((g-i))th Chern class of the Hodge bundle, can be expressed as a linear combination of tautological classes constructed from stable graphs with at most i loops. In particular, the top Chern class can be expressed with trees. This property was expected as a consequence of the DR/DZ equivalence conjecture by Buryak–Guéré–Rossi.

We continue the study of approximate propagation of smallness in parabolic periodic homogenization in [SIAM J. Math. Anal., 53(5):5835–5852, 2021], where by using the asymptotic behaviors of fundamental solutions and the Lagrange interpolation technique, we obtained the approximate two-sphere one-cylinder inequality in parabolic homogenization. In this paper, we consider the parabolic equations with suitable lower-order terms in homogenization, and the approximate two-sphere one-cylinder inequality continues to hold. The difficulty is to handle the more “worse" junior coefficients in parabolic homogenization. The results obtained in the paper can be easily extended to the elliptic case, which are also new in elliptic homogenization.

We consider the critical points of Steklov eigenfunctions on a compact, smooth n-dimensional Riemannian manifold M with boundary (partial M). For generic metrics on M we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to (partial M), and the Euler characteristic of M. In dimension 2 this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of M and of the number of sign changes of u on (partial M). In the case of the second Steklov eigenfunction on a genus 0 surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on M Steklov eigenfunctions are Morse functions in M.

We prove antisymmetric maximum principles and Hopf-type lemmas for linear problems described by the Logarithmic Laplacian. As application, we prove the symmetry of solutions for semilinear problems in symmetric sets, and a rigidity result for the parallel surface problem for the Logarithmic Laplacian.

We establish Hölder and Harnack estimates for weak solutions of a class of elliptic nonlocal equations that are modeled on integro-differential operators with kernels of measure. The approach is of De Giorgi-type, as developed by DiBenedetto, Gianazza and Vespri in a local setting. Our results generalize the work by Dyda and Kassmann (Anal PDE 13(2):317-370, 2020).

A partial regularity theorem is presented for minimisers of (k^{textrm{th}})-order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an N-function satisfying the (Delta _2) and (nabla _2) conditions, assuming no quantitative estimates on the second derivative of the integrand; this is new even in the (k=1) case. These results will also be extended to the case of strong local minimisers.

We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in (L^1_{loc}((0,T];BV(mathbb {T}^d;mathbb {R}^d))cap L^2((0,T) times mathbb {T}^d;mathbb {R}^d))), there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw in [13], for which there are infinitely many distinct bounded solutions to the advection equation.

In this paper, we consider fractional Sobolev spaces equipped with weights being powers of the distance to the boundary of the domain. We prove the versions of Bourgain–Brezis–Mironescu and Maz’ya–Shaposhnikova asymptotic formulae for weighted fractional Gagliardo seminorms. For (p>1) we also provide a nonlocal characterization of classical weighted Sobolev spaces with power weights.