Abstract:In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_1u =-\mu_1v$, $L_2v =-\mu_2u$, on a domain $D$ of $\mathbb R^d$, where $\mu_1$ and $\mu_2$ are suitable measures on~$D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on~$D$ with coefficients of class~$\mathcal C^\infty$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and a convergence property for increasing sequences of solutions of~(S).
Keywords: harmonic function; superharmonic function; potential; elliptic linear differential operator; kernel; coupled PDEs system; Kato measure
DOI: DOI 10.14712/1213-7243.2015.165
AMS Subject Classification: 31B05 31B10 31B35