Narrow Results

In 1954, Hirzebruch reported a problem posed by Kodaira and Spencer: on compact almost complex manifolds, is the dimension $h_{\bar{\partial }}^{p,q}$ of the kernel of the Dolbeault Laplacian independent of the choice of almost Hermitian metric? In this paper, we review recent progresses on the original problem and we introduce a similar one: on compact almost complex manifolds, find a generalization of Bott–Chern and Aeppli numbers which is metric-independent. We find a solution to our problem valid on almost Kähler $4$-manifolds.

We prove unique solvability and corresponding homogeneous L^p estimates for the Poisson problem associated to the uniformly elliptic operator , provided the coefficients are bounded and uniformly continuous, and admit a (non-zero) limit as |x| goes to infinity. Some important consequences are also derived.

We prove precise $L^p$–$L^q$ estimates for the semigroup generated by

In this paper, we study the periodic homogenization of the stationary heat equation in a domain with two connected components, separated by an oscillating interface defined on prefractal Koch type curves. The problem depends both on the parameter $n$, which is the index of the prefractal iteration, and $𝜀$, that defines the periodic structure of the composite material. First, we study the limit as $n$ goes to infinity, giving rise to a limit problem defined on a domain with fractal interface. Then, we compute the limit as $𝜀$ vanishes, showing that the homogenized problem is strictly dependent on the amplitude of the oscillations and the parameter appearing in the transmission condition. Finally, we discuss about the commutative nature of the limits in $𝜀$ and $n$.

Let $L=-\mathrm{\text{div}}(A\nabla )$ be a second-order divergence form elliptic operator and $A$ an accretive, $n\times n$ matrix with bounded measurable complex coefficients in ${ℝ}^n.$ In this paper, we establish $L^p$ theory for the commutators generated by the fractional differential operators related to $L$ and bounded mean oscillation (BMO)–Sobolev functions.

In this paper, by using variational methods, we study the following elliptic problem

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions provide the exact expressions. Then, the accuracy of the scheme is assessed by computing the fundamental solutions of a generally anisotropic magneto-electro-elastic material.

In this paper we derive a maximum-type principle for a certain class of differential operators from some earlier results related to an eigenvalue problem involving the p(·)-Laplacian.

The article contains an overview of our recent results about initial value problems with fractal filling space attractors and about fully discrete models for nonlinear anomalous diffusions on synchronized space-time grids.