Pseudo Harmonic Measures and the Dirichlet Problem

1971 ◽  
Vol 23 (6) ◽  
pp. 1116-1120
Author(s):  
Maynard Arsove ◽  
Heinz Leutwiler
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Harmonic Functions ◽  
General Setting ◽  
Ideal Boundary ◽  
Jordan Curves ◽  
Boundary Properties ◽  
Harmonic Measures ◽  
Carry Over ◽  
Boundary Sets

For the case of plane regions bounded by finitely many disjoint Jordan curves, the solution of the Dirichlet problem can be expressed in terms of the classical harmonic measure of boundary arcs. At an appropriate stage in the development it is, in fact, useful to observe that the existence of such harmonic measures is equivalent to solvability of the Dirichlet problem (although one subsequently proves that all such regions are Dirichlet regions). We propose here to carry over this order of ideas to a quite general setting, in which arbitrary regions and ideal boundary structures are allowed. The counterparts of the classical harmonic measures of arcs are then harmonic functions with analogous boundary properties, but they no longer appear as measures in the boundary sets, in general. We shall refer to them as “pseudo harmonic measures”. Our main result shows how pseudo harmonic measures can be used to solve the Dirichlet problem.

scholarly journals Stability estimates for discrete harmonic functions on product domains

2013 ◽  
Vol 7 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Maru Guadie
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Harmonic Measure ◽  
Harmonic Functions ◽  
Discrete Analog ◽  
Stability Estimates ◽  
Product Domains ◽  
Discrete Harmonic Functions

We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm?n-Lindel?f theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.

Curvature of Level Curves of Harmonic Functions

1983 ◽  
Vol 26 (4) ◽  
pp. 399-405 ◽  
Author(s):  
Marvin Ortel ◽  
Walter Schneider
Keyword(s):  
Harmonic Function ◽  
Harmonic Measure ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Outer Boundary ◽  
Level Curves ◽  
Jordan Curves ◽  
Open Set ◽  
Doubly Connected Domain ◽  
Minimal Curvature

AbstractIf H is an arbitrary harmonic function defined on an open set Ω⊂ℂ, then the curvature of the level curves of H can be strictly maximal or strictly minimal at a point of Ω. However, if Ω is a doubly connected domain bounded by analytic convex Jordan curves, and if H is harmonic measure of Ω with respect to the outer boundary of Ω, then the minimal curvature of the level curves of H is attained on the boundary of Ω.

scholarly journals Some remarks concerning harmonic functions on homogeneous graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Anders Karlsson
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Cayley Graphs ◽  
Group Cohomology ◽  
Radial Variation ◽  
International Audience

International audience We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

scholarly journals Infinite dimension of solutions of the Dirichlet problem

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Vladimir Ryazanov
Keyword(s):  
Dirichlet Problem ◽  
Unit Disk ◽  
Harmonic Functions ◽  
Infinite Dimension ◽  
Boundary Limits ◽  
Nontangential Boundary

AbstractIt is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.

Conditional measure and flip invariance of Bowen-Margulis and harmonic measures on manifolds of negative curvature

1995 ◽  
Vol 15 (4) ◽  
pp. 807-811 ◽  
Author(s):  
Chengbo Yue
Keyword(s):  
Harmonic Measure ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Conditional Measure ◽  
Rigidity Result ◽  
Harmonic Measures

AbstractKifer and Ledrappier have asked whether the harmonic measures {νx} on manifolds of negative curvature are equivalent to the conditional measures of the harmonic measure v of the geodesic flow associated with the fibration {SxM}x∈M. We settle this question with a rigidity result. We also clear up the same problem concerning the Patterson-Sullivan measure and the Bowen–Margulis measure.

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