Mean value formulas, Weyl's lemma and Liouville theorems for \(\Delta^ 2\) and Stokes' system (Q1205493)

The mean value property, Harnack's inequality, Weyl's lemma and Liouville's theorems for harmonic functions in \(\mathbb{R}^ n\) are extended to biharmonic functions and to the Stokes' homogeneous system, defined also in \(\mathbb{R}^ n\). The Stokes' system is \(-\Delta{\mathbf u}+\nabla p=0\) in \(G\) and \(\text{div }{\mathbf u}=0\) in \(G\), where \(G\subset\mathbb{R}^ n\) is an open set, \({\mathbf u}=(u_ 1,u_ 2,\dots,u_ n)\in C^ 2(G)^ n\) and \(p\in C^ 1(G)\). The corresponding weak solution of the homogeneous Stokes' system is the pair \(({\mathbf u},p)\in L_{\text{loc}}^ 1(G)^{n+1}\) which satisfies \[ \int_ G {\mathbf u}\Delta\varphi dy+ \int_ G p \text{div }\varphi dy=0, \] for all \(\varphi\in C_ 0^ \infty(G)^ n\) and \(\int_ G{\mathbf u}\Delta\psi dy=0\) for all \(\psi\in C_ 0^ \infty(G)\). The author proves the theorems extending the quoted properties to the analyzed cases and discusses their similarities and differences with those valid for the harmonic functions.