Publications 2020


8.  Maximum principles for some quasilinear elliptic systems

Leonardi, Salvatore and Leonetti, Francesco and Pignotti, Cristina and Rocha, Eugénio and Staicu, Vasile

Nonlinear Analysis: Theory, Methods and Applications


We give maximum principles for solutions u:Ω→ℝ N to a class of quasilinear elliptic systems whose prototype is [Formula presented]where α∈{1,…,N} is the equation index and Ω is an open, bounded subset of ℝ n . We assume that coefficients [Formula presented] are measurable with respect to x, continuous with respect to y∈ℝ N , bounded and elliptic. In vectorial problems, when trying to bound the solution by means of the boundary data, we need to bypass De Giorgi's counterexample by means of some additional structure assumptions on the coefficients [Formula presented]. In this paper, we assume that off-diagonal coefficients [Formula presented], α≠β, have support in some staircase set along the diagonal in the y α ,y β plane | doi | Peer Reviewed

7.  Variable exponent Besov-Morrey spaces

Almeida, Alexandre and Caetano, António

Journal of Fourier Analysis and Applications

Springer Verlag

In this paper we introduce Besov-Morrey spaces with all indices variable and study some fundamental properties. This includes a description in terms of Peetre maximal functions and atomic and molecular decompositions. This new scale of non-standard function spaces requires the introduction of variable exponent mixed Morrey-sequence spaces, which in turn are defined within the framework of semimodular spaces. In particular, we obtain a convolution inequality involving special radial kernels, which proves to be a key tool in this work. | doi | Peer Reviewed

6.  Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

Almeida, Alexandre

Nonlinear Analysis


We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators. | doi | Peer Reviewed

5.  On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

Alabalik, Aysegul Ç. and Almeida, Alexandre and Samko, Stefan

Banach Journal of Mathematical Analysis


We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of C∞0(Rn) in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy–Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators. | doi | Peer Reviewed

4.  Decompositions with atoms and molecules for variable exponent Triebel-Lizorkin-Morrey spaces

Caetano, António and Kempka, Henning

Constructive Approximation

Springer Verlag

We continue the study of the variable exponent Morreyfied Triebel-Lizorkin spaces introduced in a previous paper. Here we give characterizations by means of atoms and molecules. We also show that in some cases the number of zero moments needed for molecules, in order that an infinite linear combination of them (with coefficients in a natural sequence space) converges in the space of tempered distributions, is much smaller than what is usually required. We also establish a Sobolev type theorem for related sequence spaces, which might have independent interest. | doi | Peer Reviewed

3.  Variable exponent Triebel-Lizorkin-Morrey spaces

Caetano, António and Kempka, Henning

Journal of Mathematical Analysis and Applications


We introduce variable exponent versions of Morreyfied Triebel-Lizorkin spaces. To that end, we prove an important convolution inequality which is a replacement for the Hardy-Littlewood maximal inequality in the fully variable setting. Using it we obtain characterizations by means of Peetre maximal functions and use them to show the independence of the introduced spaces from the admissible system used. | doi | Peer Reviewed

2.  New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type

Castro, Luis P. and Guerra, Rita C. and Tuan, Nguyen Minh

Mathematical Methods in the Applied Sciences


We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier-type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener-Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier-type series are used to produce the solution formula of such equations and a Shannon-type sampling formula is also obtained. | doi | Peer Reviewed

1.  Mixed impedance boundary value problems for the Laplace–Beltrami equation

Castro, Luis and Duduchava, Roland and Speck, Frank-Olme

Journal of Integral Equations and Applications

Rocky Mountain Mathematics Consortium

This work is devoted to the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem (MIND~BVP) for the Laplace-Beltrami equation on a compact smooth surface $mathcal{C}$ with smooth boundary. We prove, using the Lax-Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting $mathbb{H}^1(mathcal{C})$ when considering positive constants in the impedance condition. The main purpose is to consider the MIND~BVP in a nonclassical setting of the Bessel potential space $mathbb{H}^s_p(mathcal{C})$, for $s> 1/p$, $11/p$ and $1 | doi | Peer Reviewed
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