A generalization of a Phragmén-Lindelöf-type theorem for elliptic linear equations
Let L denote a strongly elliptic second order differential operator (with certain mild restriction of continuity and boundedness of the coefficients) in an unbounded domain D contained in a strip in \(R^ n | x_ n|<h\leq 1\), \(-\infty<x_ i<\infty\), \(i=1,...,n-1\). A theorem of P-L type asserts that a subsolution u\((Lu\leq 0)\) that is \(\leq 0\) on the boundary \(\partial D\) of D and continuous on the closure of D satisfies either (1) \(u\leq 0\) everywhere in D or (2) u(x) increases exponentially as \(| x| \to \infty\) in D. The present paper studies the case where an exceptional set \(E\subset \partial D\) is allowed and it is only assumed that \(u\leq 0\) on \(\partial D\backslash E\), but a similar alternative still holds. The condition imposed upon the size of the set E is expressed in terms of the s- capacity of E: \(Cap_ s(E)=\sup \mu(E),\) where the supremum is over measures \(\mu\) carried by E such that \[ \sup_{x\not\in E}\int_{E}| x-y|^{-s}\quad d\mu(y)\leq 1. \] In most of the work one has \(s=n-2\).
strongly elliptic second order differential operator
scientific article; zbMATH DE number 3867809