The role of positive boundary data in generalized clamped plate equations (Q1392754)

Let \(u(x)\) be a solution to the problem \((-\partial_\nu)^ju=0\) on \(\partial B\) for \(j=1,\ldots , m-3\), \((-\partial_\nu)^{m-2}u=\psi\) on \(\partial B\) and \((-\partial_\nu)^{m-1}u=\phi\) on \(\partial B\), where \(m\geq 2\) and \(B\) is the unit ball in \(\mathbb{R}^n\). The authors prove that if \(\psi(x)\geq 0\) and if \(\phi(x)\geq s\psi(x)\) on \(\partial B\) with \(s\geq (n-2-m)(m-1)/2\) then \(u(x)\geq 0\) in \(B\). Furthermore, whenever \(s>(n-2-m)(m-1)/2\), they extend the previous positiveness result to equations having suitable lower order terms.