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

This is the authors' abstract. We don't add key points for this paper.

For one-dimensional interval and integrable weight function w we define via completion a weighted Sobolev space Hμwm,p of arbitrary integer order m. The weights in consideration may suffer strong degeneration so that, in general, functions u from Hμwm,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 Hμwm,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.