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

## TL;DR

This paper develops higher order weighted Sobolev spaces on the real line with strongly degenerate weights, analyzing their continuity properties and applying these results to variational problems in beam elasticity.

## Contribution

It introduces a new framework for weighted Sobolev spaces with degenerate weights and studies their local continuity and approximation properties, with applications to elasticity theory.

## Key findings

- Continuity of functions at non-critical points is guaranteed.
- Critical points allow for jump discontinuities and smooth approximation.
- Application to variational problems in beam elasticity with degenerate width.

## Abstract

For one-dimensional interval and integrable weight function $w$ we define via completion a weighted Sobolev space $H^{m,p}_{\mu_w}$ of arbitrary integer order $m$. The weights in consideration may suffer strong degeneration so that, in general, functions $u$ from $H^{m,p}_{\mu_w}$ do not have weak derivatives. This contribution is focussed on studying the continuity properties of functions $u$ at a chosen internal point $x_0$ to which we attribute a notion of criticality of order $k$ and with respect to the weight $w$. For non-critical points $x_0$ 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 $x_0$ furnishes a smooth approximation of functions in $H^{m,p}_{\mu_w}$ admitting jump-type discontinuities at $x_0$. The work concludes with demonstration of established results in the context of variational problem in elasticity theory of beams with degenerate width distribution.