Paths for subharmonic functions. (English) Zbl 0541.31001

Davis, Burgess; Lewis, John L.

Proc. Lond. Math. Soc., III. Ser. 48, 401-427 (1984).

The paths are investigated for non-negative subharmonic functions defined on the domains in \(R^ m\), \(m=2\), 3. Let u be a subharmonic function on the unit ball B(0,1) of \(R^ n\) and 0\(\leq u\leq 1\). Then for given \(\epsilon>0\) there exists \(r(\epsilon)>0\) with the following property: if \(| x-y|<r(\epsilon)\) and \(u(x)>\epsilon\), \(u(y)>\epsilon\), \(| x|<{1\over2}\), \(| y|<{1\over2}\), then x can be joined to y by a polygonal curve \(\gamma\) on B(0,1) with \(u\geq \epsilon /2\) on \(\gamma\) and \(| \gamma | \leq C(\epsilon)| x-y|.\) It suffices to take \(r(\epsilon)=(\epsilon /2)c/\epsilon\), \(C(\epsilon)=\exp [(2/\epsilon)^{\lambda}]\) for \(\lambda\) suitably large. The authors study questions involving paths going to the boundary of B(0,1), give a generalization for the bounded nontangentially accessible domains in the sense of Kenig and Jerison, and make some comments on the generalization to subsolutions of uniformly elliptic equations of divergent type.
\(\{\) In a letter to the editor the authors state that they have recently learned from A. Ancona that the paper of M. Brelot and G. Choquet [Ann. Inst. Fourier 3, 199-263 (1952; Zbl 0046.327), see especially p. 242] contains theorems and methods which enable easy proofs of not only the results but also the conjectures of their paper.\(\}\)

Reviewer: M.Novickii

MSC:

31A05	Harmonic, subharmonic, superharmonic functions in two dimensions
31B05	Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

subharmonic functions; nontangentially accessible domains