zbMATH — the first resource for mathematics

On the potential theory of some systems of coupled PDEs. (English) Zbl 1363.31004
Let \(D\) be a domain in \(\mathbb R^d\), and \(L_1\), \(L_2\) be two second order elliptic linear differential operators on \(D\) with coefficients of class \(C^\infty \). It is supposed that \(D\) is a Green domain for \(L_1\) and \(L_2\). Let \(\mu_1\) and \(\mu_2\) be two Kato measures for \(L_1\) and \(L_2\), respectively. The authors study some potential theoretical properties of harmonic and superharmonic pairs \((u,v)\) of the system (S) \(L_1u+v\mu_1=0\), \(L_2v+u\mu_2=0\) in \(D\). They prove that the minimum of two superharmonic pairs is a superharmonic pair. They study the limit of monotone sequences of superharmonic pairs. They prove the minimum principle for superharmonic pairs. They also discuss the relation between superharmonic pairs for (S) and superharmonic functions for \(L_1\) and \(L_2\).
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
31B35 Connections of harmonic functions with differential equations in higher dimensions
Full Text: DOI