Harnack's inequality for second order linear ordinary differential inequalities
Harnack's inequality, maximum principles
Harnack type inequalities for nonnegative solutions of second order linear ordinary differential inequalities of the form \( L u \leq f(x), x \in I :=(A, B)\) where, for \(u \in C^2(I)\), NEWLINENEWLINE\[NEWLINEL u := u'' + p(x) u'(x) + q(x) u(x).NEWLINE\]NEWLINE NEWLINENEWLINEThe coefficients \(p, q\) and \(f\) are assumed to be continuous real valued functions on \(I\). To make the paper self contained, simpler proofs of known results from the literature are also given. The following main result is established for the nonnegative solutions of \(L u \leq 0\) on \(I\).NEWLINENEWLINETheorem. Given \([a,b] \subseteq I,\) for any nonnegative solution of \(L u \leq 0\) on \(I\), there is a positive constant \(C\) that does not depend on \(u\), such that NEWLINENEWLINE\[NEWLINE\max_{a \leq x \leq b}\; u(x) \leq C \;\min_{a \leq x \leq b} \; u(x) .NEWLINE\]NEWLINE NEWLINEThe same cannot be established if \(L u \geq 0\). Another inequality of Harnack type is established for nonnegative solutions of nonhomogeneous equations \(L_0 u = f\) on \(I\), where NEWLINENEWLINE\[NEWLINEL_0 u = u'' + p u ' - q^- u \text{ and } q^-(x) = \max\{-q(x), 0\}, f \geq 0.NEWLINE\]NEWLINE NEWLINEFinally, a Harnack-type inequality for nonnegative solutions of \( L u \leq f\) with no sign restrictions on \(f\) is derived.
scientific article; zbMATH DE number 6553287