On the behavior of harmonic functions in the neighborhood of an irregular boundary point

1954 ◽  
Vol 4 (1) ◽  
pp. 209-221 ◽  
Author(s):  
Marcel Brelot
Keyword(s):  
Boundary Point ◽  
Harmonic Functions ◽  
Irregular Boundary

scholarly journals Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

2021 ◽  
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Irregular Boundary ◽  
Complete Metric Spaces ◽  
Boundary Points ◽  
Local Properties ◽  
Harmonic Measures

AbstractThe trichotomy between regular, semiregular, and strongly irregular boundary points for $$p$$ p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $$p$$ p -harmonic measures, removability, and semibarriers.

Estimates for solutions of the Dirichlet problem for the biharmonic equation in a neighbourhood of an irregular boundary point and in a neighbourhood of infinity. Saint-Venant's principle

1983 ◽  
Vol 93 (3-4) ◽  
pp. 327-343 ◽  
Author(s):  
Y. A. Kondratiev ◽  
O. A. Oleinik
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Boundary Point ◽  
Biharmonic Equation ◽  
Maximum Modulus ◽  
Integral Inequalities ◽  
Energy Estimates ◽  
Irregular Boundary ◽  
Continuity Of Solutions

SynopsisIn this paper energy estimates for solutions of the Dirichlet problem for the biharmonicequation, expressing Saint-Venant's principle in elasticity, are proved. From these integral inequalities, estimates for the maximum modulus of solutions and the gradient of solutions with homogeneous Diriehlet's boundary conditions in a neighbourhood of an irregular boundary point or in a neighbourhood of infinity are derived. These estimates characterize the continuity of solutions and their gradients at these points.

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

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