Elliptic Systems with Almost Regular Coefficients: Singular Weight Integral Operators

Abstract

We consider the linear elliptic system of two first-order equations

$$ {{\partial }_{{\bar{z}}}}\omega + {{\mu }_{1}}\left( z \right){{\partial }_{z}}\omega + {{\mu }_{2}}\left( z \right)\overline {{{\partial }_{z}}\omega } = A\left( z \right)\omega + B\left( z \right)\bar{\omega } + F\left( z \right) $$

, where $w\left( {z,\bar{z}} \right) = u + iv$ is an unknown complex-valued function, and the related integral operators and boundary value problems. We assume that $A,B,F \in {{L}_{p}}\left( \Omega \right),p \leqslant 2$, in contrast to the regular Vekua’s theory where p > 2. We prove that in this case the solutions of the system still preserve the properties, which correspond to the regular case with respect to: the structure of zeros, Liouville’s theorem, solvability of Riemann-Hilbert boundary value problems etc.

Work partially supported by Project FCT-POCTI/34471/MAT/2000