Nodal oscillation and weak oscillation of elliptic equations of order 2m (Q1174940)

The author deduces a criterion for the nodal oscillation of the elliptic equation \[ Lu:=\sum^ m_{|\alpha|,|\beta|=0}(- 1)^{|\alpha|}D^{\alpha}[A_{\alpha\beta}(x)D^ \beta u]=0 \] by comparison with a special case of the equation \[ Mv:=\sum_{|\alpha|=|\beta|=m}(-1)^ mD^ \alpha[a_{\alpha\beta} (x)D^ \beta v] +a_ 0(x)v=0, \] where \(x\in\Omega\subset R^ n\) and \(\Omega\) is an unbounded open set. Here \(L\) and \(M\) are assumed to be uniformly strongly elliptic in \(\Omega\). In the process of developing the criterion (see Theorem 3.10), the author obtains a sufficient condition that weak oscillation implies nodal oscillation and uses a modification of the Gårding inequality.