Higher order weighted Sobolev spaces on the real line for strongly degenerate weights. Application to variational problems in elasticity of beams
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.
|Journal series||Journal of Mathematical Analysis and Applications, ISSN 0022-247X, e-ISSN 1096-0813|
|Publication size in sheets||2.85|
|Keywords in English||Weighted, Sobolev space, Sobolev spaces with respect to measure, Degenerateweights, Duality theory, Elasticity of beams|
|Score||= 70.0, 06-07-2020, ArticleFromJournal|
|Publication indicators||= 0; : 2018 = 1.187; : 2018 = 1.188 (2) - 2018=1.219 (5)|
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