Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Abstract

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 $$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>L</mml:mi><mml:mi>u</mml:mi><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:munderover><mml:msub><mml:mfenced><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>ij</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:msub></mml:mfenced><mml:msub><mml:mi>x</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:msub><mml:mo>=</mml:mo><mml:mi>∇</mml:mi><mml:mo>·</mml:mo><mml:mi>f</mml:mi><mml:mo>,</mml:mo><mml:mspace/><mml:mtext>for almost all</mml:mtext><mml:mspace/><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>Ω</mml:mi></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> where the coefficients $$a_{ij}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>a</mml:mi><mml:mrow><mml:mi>ij</mml:mi></mml:mrow></mml:msub></mml:math> belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces $$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mover><mml:mrow><mml:mi>LM</mml:mi></mml:mrow><mml:mo>~</mml:mo></mml:mover><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>φ</mml:mi></mml:mrow></mml:msubsup></mml:math> . 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.

Topics & Concepts