Calculus of Variations and Geometric Measure Theory

G. Moscariello - G. Pascale

Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

created by pascale on 15 Mar 2023
modified on 16 Mar 2023


Published Paper

Inserted: 15 mar 2023
Last Updated: 16 mar 2023

Journal: Milan J. Math.
Year: 2021
Doi: 10.1007/s00032-021-00345-8


We consider linear elliptic systems whose prototype is \[ div \, \Lambda \left[\,\exp(-\lvert x\rvert) - \log\lvert x\rvert\,\right] I \, Du = div \, F + g \qquad \text{in}\, B. \qquad (1)\] Here $B$ denotes the unit ball of $\mathbb{R}^n$, for $n > 2$, centered in the origin, $I$ is the identity matrix, $F$ is a matrix in $W^{1, 2}(B, \mathbb{R}^{n \times n})$, $g$ is a vector in $L^2(B, \mathbb{R}^n)$ and $\Lambda$ is a positive constant. Our result reads that the gradient of the solution $u \in W_0^{1, 2}(B, \mathbb{R}^n)$ to Dirichlet problem for system $(1)$ is weakly differentiable provided the constant $\Lambda$ is not large enough.