L p theory of second-order elliptic operators with discontinuous coefficients

A study was conducted to demonstrate Lp theory of second-order elliptic operators with discontinuous coefficients. The existence and uniqueness of a weak solution in the class with first derivatives from Lp was analyzed for a divergence-form elliptic equation in R2 with discontinuous piecewise constant coefficients and a divergence right-hand side. The study aimed at calculating the dimensions of the kernel and cokernel of the corresponding elliptic operator for all p ∈ (1, ∞). It was necessary to investigate auxiliary eigenvalue problems corresponding to singular points such as a corner point and a node to analyze the dimensions of the kernel and cokernel of L.