Higher order weighted Sobolev spaces on the real line for strongly degenerate weights. Application to variational problems in elasticity of beams

Karol Bołbotowski

Abstract

For one-dimensional interval and integrable weight function w we define via completion a weighted Sobolev space Hμw m,p of arbitrary integer order m. The weights in consideration may suffer strong degeneration so that, in general, functions u from Hμw m,p do not have weak derivatives. This contribution is focussed on studying the continuity properties of functions u at a chosen internal point x0 to which we attribute a notion of criticality of order k and with respect to the weight w. For non-critical points x0 we formulate a local embedding result that guarantees continuity of functions u or their derivatives. Conversely, we employ duality theory to show that criticality of x0 furnishes a smooth approximation of functions in Hμw m,p admitting jump-type discontinuities at x0. The work concludes with demonstration of established results in the context of variational problem in elasticity theory of beams with degenerate width distribution.

Author Karol Bołbotowski (FCE / ICE)
Karol Bołbotowski,,
- The Institute of Civil Engineering
Journal seriesJournal of Mathematical Analysis and Applications, ISSN 0022-247X, e-ISSN 1096-0813
Issue year2020
Vol488
Pages1-58
Publication size in sheets2.85
Keywords in EnglishWeighted, Sobolev space, Sobolev spaces with respect to measure, Degenerateweights, Duality theory, Elasticity of beams
ASJC Classification2603 Analysis; 2604 Applied Mathematics
DOIDOI:10.1016/j.jmaa.2020.124038
URL https://www.sciencedirect.com/science/article/pii/S0022247X20302006?via%3Dihub#!
Languageen angielski
File
1-s2.0-S0022247X20302006-main.pdf 1.14 MB
Score (nominal)70
Score sourcejournalList
ScoreMinisterial score = 70.0, 06-07-2020, ArticleFromJournal
Publication indicators Scopus Citations = 0; Scopus SNIP (Source Normalised Impact per Paper): 2018 = 1.187; WoS Impact Factor: 2018 = 1.188 (2) - 2018=1.219 (5)
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