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.

中文翻译：

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实线上的高阶加权 Sobolev 空间用于强退化权重。在梁弹性变分问题中的应用

对于一维区间和可积权重函数 $w$，我们通过补全定义了一个任意整数阶 $m$ 的加权 Sobolev 空间 $H^{m,p}_{\mu_w}$。所考虑的权重可能会遭受强烈的退化，因此通常情况下，来自 $H^{m,p}_{\mu_w}$ 的函数 $u$ 没有弱导数。这个贡献集中在研究函数 $u$ 在选定的内部点 $x_0$ 处的连续性属性，我们将阶 $k$ 的临界性概念和权重 $w$ 归因于该点。对于非关键点 $x_0$，我们制定了一个局部嵌入结果，以保证函数 $u$ 或其导数的连续性。相反，我们采用对偶理论来证明 $x_0$ 的临界性提供了 $H^{m,p}_{\mu_w}$ 中函数的平滑逼近，承认 $x_0$ 处的跳跃型不连续性。