Estimates for Green's matrices of elliptic systems by \(L^ p\) theory
Consider elliptic systems of the form \[ - D_\alpha(A^{\alpha\beta}_{ij}(x) D_\beta u^j)= D_\alpha f^\alpha_i+ F_i\quad\text{in} \quad \Omega, \] where \(\Omega\) is a bounded \(C^1\) domain in \(\mathbb{R}^n\) and \(A^{\alpha \beta}_{ij}\in C^0(\overline\Omega)\). The authors prove the existence of a Green's matrix and establish optimal decay properties by deriving optimal estimates for the Green's function and its gradient in weak Lebesgue spaces through \(L^p\) estimates and interpolation theorems. Standard interior regularity estimates and scaling lead to pointwise estimates. A sharp comparison with the corresponding Green's matrix for systems with constant coefficients is also obtained.
scientific article; zbMATH DE number 866167