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Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces
- Source :
- Analysis and Mathematical Physics. 11
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
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$$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$Lu:=∑i,j=1naij(x)uxixj=∇·f,for almost allx∈Ωwhere the coefficients$$a_{ij}$$aijbelong to the Central (that is, Local) Sarason class CVMO andfis assumed to be in some Modified Local Generalized Morrey Spaces$$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$LM~{x0}p,φ. 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.
- Subjects :
- Physics
Pure mathematics
Algebra and Number Theory
Operator (physics)
010102 general mathematics
Second order equation
VMO
01 natural sciences
Omega
Divergence
010101 applied mathematics
Elliptic curve
Integral Operators
Elliptic Equations
Morrey-Type Spaces
Nabla symbol
0101 mathematics
Singular integral operators
Mathematical Physics
Analysis
Subjects
Details
- ISSN :
- 1664235X and 16642368
- Volume :
- 11
- Database :
- OpenAIRE
- Journal :
- Analysis and Mathematical Physics
- Accession number :
- edsair.doi.dedup.....de4b7a76fd1b347ff0bb7b5f24d80368