The Mean Value Theorem and the Subharmonic Functions on the Stratified Set
DOI:
https://doi.org/10.52575/2687-0959-2025-57-4-266-271Keywords:
Stratified Set, Stratified Measure, Laplacian, Mean Value TheoremAbstract
In this paper, an analogue of the mean value theorem for subharmonic functions is presented in the following setting: instead of a domain in R<i>d</i>, a stratified set Ω is considered, and instead of the classical Laplacian, a "stratified"one is used. Previously, a similar result was obtained for the so-called soft Laplacian, whose properties are as close as possible to the classical one. Here, we present a result—an analogue of the mean value theorem—that holds for all stratified Laplacians. The mean value theorem plays an important role in discussing the qualitative properties of subharmonic functions on stratified sets and in addressing the solvability of the Dirichlet problem on them.
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Покорный Ю.В., Пенкин О.М., Прядиев В.Л., Боровских А.В., Лазарев К.П., Шабров С.А. Дифференциальные уравнения на геометрических графах. М.:Физматлит; 2005. 272с.
Oshchepkova S.N., Penkin O.M. The mean-value theorem for elliptic operators on stratified sets. Mathematical Notes. 2007;81(3):365-372 https://doi.org/10.1134/S0001434607030108
Oshchepkova S.N., Penkin O.M., Savasteev D.V. Strong maximum principle for an elliptic operator on a stratified set. Mathematical Notes. 2012;92(2):249–259 https://doi.org/10.1134/S0001434612070267
Adamowicz T., Gaczkowski M., Gorka P. Harmonic functions on metric measure spaces. Revista Matematica Complutense. 2019;32:141-186 https://doi.org/10.1007/s13163-018-0272-7
Ambrosio L., Gigli N., Savare G. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Inventiones Mathematicae. 2014;195:289-391 https://doi.org/10.1007/s00222-013-0456-1
Dairbekov N.S., Penkin O.M., Savasteev D.V. On Removable Singularities of Harmonic Functions on a Stratified Set. Doklady Mathematics. 2024;110:297-300 https://doi.org/10.1134/S1064562424601379
Dairbekov N.S., Penkin O.M., Savasteev D.V. Harnack’s Inequality for Harmonic Functions on Stratified Sets. Siberian Mathematical Journal. 2023;64(5):1137–1144 https://doi.org/10.1134/S0037446623050063
References
Pokornyi YV., Penkin OM., Priadiev VL., Borovskikh AV., Lazarev KP., Shabrov SA. Differentcial’nye uravneniia na geometricheskikh grafakh [Differential equations on geometric graphes]. Moscow: Fizmatlit; 2005. 272p (In Russ.).
Oshchepkova SN., Penkin OM. The mean-value theorem for elliptic operators on stratified sets. Mathematical Notes. 2007;81(3):365-372 https://doi.org/10.1134/S0001434607030108
Oshchepkova SN., Penkin OM., Savasteev DV. Strong maximum principle for an elliptic operator on a stratified set. Mathematical Notes. 2012;92(2):249-259 https://doi.org/10.1134/S0001434612070267
Adamowicz T., Gaczkowski M., Gorka P. Harmonic functions on metric measure spaces. Revista Matematica Complutense. 2019;32:141-186 https://doi.org/10.1007/s13163-018-0272-7
Ambrosio L., Gigli N., Savare G. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Inventiones Mathematicae. 2014;195:289-391 https://doi.org/10.1007/s00222-013-0456-1
Dairbekov NS., Penkin OM., Savasteev DV. On Removable Singularities of Harmonic Functions on a Stratified Set. Doklady Mathematics. 2024;110:297-300 https://doi.org/10.1134/S1064562424601379
Dairbekov NS., Penkin OM., Savasteev DV. Harnack’s Inequality for Harmonic Functions on Stratified Sets. Siberian Mathematical Journal. 2023;64(5):1137–1144 https://doi.org/10.1134/S0037446623050063
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