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Paths for subharmonic functions. (English) Zbl 0541.31001
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
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