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C. Pommerenke (1992)
Boundary Behaviour of Conformal Maps
T. Broadbent (1957)
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Cel l ina
DEMONSTRATIO MATHEMATICAVol. XXXVINo 22003Lucio R. BerroneCHARACTERIZATION OF DOMAINS THROUGHFAMILIES OF MEASURESAbstract. Let Q be a plane domain limited by a regular Jordan curve F. For every(L) measurable subset E of T and every point z € CI, consider the probability P{E\ z)that a Brownian particle starting its motion at z hits the boundary T (by the first time)in a point belonging to E. Now, let C be a constant such that 0 < C < |T| and considerthe optimization problem(1)s u p { P ( £ ; z ) : | £ | = C}(| • | denotes the Lebesgue measure on the boundary T). What are the domains Q suchthat single arcs of the boundary are optimal subsets for (1) for every z 6 fi and everyo < c < in?For a plane domain Cl which is starlike with respect to an interior point O, the internalvisual angle ©(O; E) of a measurable subset of the boundary E C <90 is defined to be thesingle under which E is observed from O. Posing the optimization problem(2)s u p { © ( 0 ; E ) : | E | = C},it is asked for the convex domains CI
Demonstratio Mathematica – de Gruyter
Published: Apr 1, 2003
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