Hölder ratio of solutions of elliptic equations with a point on the boundary

Let u be the classical solution to a Dirichlet problem for a uniformly second order elliptic equation in a smooth bounded domain Ω ⊂ Rn (n 2) and let w(x, y) = (u(x) − u(y))/|x − y|λ be the Hölder ratio of u with exponent λ ∈ (0, 1) and x ∈ Ω, y ∈ ∂Ω. Conditions on λ are found such that w does not admit absolute maxima or minima in Ω × ∂Ω. An example shows that these conditions necessarily depend on the boundary data. Finally it is proved that, if the Hölder ratio is replaced by another suitable ratio, called pseudo-Hölder ratio, then the maximum principle always holds.  2002 Elsevier Science (USA). All rights reserved.