Abstract
Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ω\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathscr {L}u{:}{=}\\sum _{i,j=1}^{n}\\left( a_{ij}(x)u_{x_{i}}\\right) _{x_{j}}=\\nabla \\cdot f,\\qquad \\hbox {for almost all }x\\in \\Omega \\end{aligned}$$\\end{document}where the coefficients a_{ij} belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces widetilde{LM}_{{x_{0}}}^{p,varphi }. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.
Highlights
Introduction and mathematical backgroundIn this note we consider the following divergence form elliptic equation n (1.1)in a bounded set ⊂ Rn, n ≥ 3.Extended author information available on the last page of the article 13 Page 2 of 20We assume that L is a linear elliptic operator and its coefficients belong to the space V M O and the vectorial field f = ( f1, f2, . . . , fn) is such that fi ∈ L M p,φ for i = 1, . . . , n, with 1 < p < ∞ and φ positive and measurable function
In the last few years have been studied several differential problems on nonstandard function spaces and, in particular, several results have been obtained on Generalized Morrey Spaces
Our main result in this paper is the study of local regularity in the Generalized Morrey Spaces L M p,φ of the first derivatives of the solutions of the equation under consideration as in the past has been done in L p−spaces and in L p,λ−spaces
Summary
Our main result in this paper is the study of local regularity in the Generalized Morrey Spaces L M p,φ of the first derivatives of the solutions of the equation under consideration as in the past has been done in L p−spaces and in L p,λ−spaces. For instance, [2] where the author obtains local regularity in the classical Lebesgue spaces L p for the first derivatives of the solutions of the equation with discontinuous coefficients. We say that f belongs to the John-Nirenberg space BMO of the functions with bounded mean oscillation if f. John-Nirenberg inequality shows that functions in B M O(Rn) are locally exponentially integrable. This implies that, for any 1 ≤ q < ∞, the functions in B M O(Rn) can be described by means of the condition:.
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