Hausdorff measure and extension results for subharmonic functions, for separately subharmonic functions, for harmonic functions and for separately harmonic functions

2021 ◽  
pp. 92-104
Author(s):  
Juhani Riihentaus
Keyword(s):  
Hausdorff Measure ◽  
Harmonic Functions ◽  
Subharmonic Functions

scholarly journals On the mean value property of superharmonic and subharmonic functions

2006 ◽  
Vol 2006 ◽  
pp. 1-3
Author(s):  
Robert Dalmasso
Keyword(s):  
Harmonic Functions ◽  
Mean Value ◽  
Subharmonic Functions ◽  
Mean Value Property ◽  
The Mean

We prove a converse of the mean value property for superharmonic and subharmonic functions. The case of harmonic functions was treated by Epstein and Schiffer.

scholarly journals Integral Harnack inequality

1985 ◽  
Vol 26 (2) ◽  
pp. 115-120 ◽  
Author(s):  
Murali Rao
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Convex Domains ◽  
Dimensional Hausdorff Measure ◽  
Image Position ◽  
Continuous Boundary ◽  
Integral Version

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

scholarly journals Structure of singular sets of some classes of subharmonic functions

2021 ◽  
Vol 31 (4) ◽  
pp. 519-535
Author(s):  
B.I. Abdullaev ◽  
S.A. Imomkulov ◽  
R.A. Sharipov
Keyword(s):  
Harmonic Functions ◽  
Singular Set ◽  
Test Functions ◽  
Subharmonic Functions ◽  
Singular Sets ◽  
Basic Functions

In this paper, we survey the recent results on removable singular sets for the classes of $m$-subharmonic ($m-sh$) and strongly $m$-subharmonic ($sh_m$), as well as $\alpha$-subharmonic functions, which are applied to study the singular sets of $sh_{m}$ functions. In particular, for strongly $m$-subharmonic functions from the class $L_{loc}^{p}$, it is proved that a set is a removable singular set if it has zero $C_{q,s}$-capacity. The proof of this statement is based on the fact that the space of basic functions, supported on the set $D\backslash E$, is dense in the space of test functions defined in the set $D$ on the $L_{q}^{s}$-norm. Similar results in the case of classical (sub)harmonic functions were studied in the works by L. Carleson, E. Dolzhenko, M. Blanchet, S. Gardiner, J. Riihentaus, V. Shapiro, A. Sadullaev and Zh. Yarmetov, B. Abdullaev and S. Imomkulov.

scholarly journals Removable sets for harmonic functions in Besov spaces

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
David Kighuradze
Keyword(s):  
Hausdorff Measure ◽  
Besov Spaces ◽  
Harmonic Functions ◽  
Geometric Characterization ◽  
Compact Set ◽  
Direct Generalization ◽  
Removable Sets

AbstractIn this paper we give a direct generalization of Carleson–Ullrich's theorem in Besov spaces and geometric characterization of removable sets for harmonic functions in those spaces in terms of Hausdorff measure. In particular, for a compact set

scholarly journals REMOVABLE SETS FOR HÖLDER CONTINUOUS p(x)-HARMONIC FUNCTIONS

2012 ◽  
Vol 10 (02) ◽  
pp. 199-206 ◽  
Author(s):  
A. LYAGHFOURI
Keyword(s):  
Compact Subset ◽  
Hausdorff Measure ◽  
Harmonic Functions ◽  
Open Domain ◽  
Hölder Continuous ◽  
Closed Set ◽  
Removable Sets

We establish that a closed set E is removable for C0,α Hölder continuous p(x)-harmonic functions in a bounded open domain Ω of ℝn, n ≥ 2, provided that for each compact subset K of E, the (n - pK + α(pK - 1))-Hausdorff measure of K is zero, where pK = max x∈K p(x).

scholarly journals The Dirichlet problem for $m$-subharmonic functions on compact sets

2020 ◽  
Vol 126 (3) ◽  
pp. 497-512
Author(s):  
Per Åhag ◽  
Rafał Czyż ◽  
Lisa Hed
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Functions ◽  
Compact Set ◽  
Compact Sets ◽  
Subharmonic Functions

We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.

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