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A density result for Sobolev functions and functions of higher order bounded variation with additional integrability constraints. (English) Zbl 1379.46029
Let \(\Omega\subset\mathbb{R}^n\) be an open set, \(m\in \mathbb{N}\) and \(1\leq p< \infty\). By \(W^{m,p}(\Omega)\) the Sobolev space of all functions \(u\) defined on \(\Omega\) is denoted whose distributional derivatives up to order \(m\) are represented by \(p\)-integrable functions; and by \(BV^m(\Omega)\) the space of all functions \(u\) on \(\Omega\) is denoted whose distributional gradients \(\nabla^k u\) up to order \(m-1\) are represented by 1-integrable tensor-valued functions and whose \(m\)-th distributional gradient \(\nabla^m u\) is a tensor-valued Radon measure with finite total variation.
The main results of the paper are the following theorems.
Theorem 1. Let \(1\leq p< q< \infty\), \(\Omega\subset\mathbb{R}^n\) be open and bounded with Lipschitz boundary, \(D \Subset\Omega\) be an open precompact subset with minimally smooth boundary and \(u\in W^{m,p}(\Omega)\cap L^q(\Omega\setminus D)\). Then there is a sequence of smooth functions \(\varphi_k\in C^\infty(\overline{\Omega})\) such that \[ \|u-\varphi_k\|_{W^{m,p}(\Omega)} +\|u-\varphi_k\|_{L^q(\Omega\setminus D)}\rightarrow 0\;\;\;\;(k\rightarrow\infty). \]
Theorem 2. Let \(1< q< \infty\), \(\Omega\subset\mathbb{R}^n\) be open and bounded with \(C^1\)-boundary, \(D \Subset\Omega\) be an open precompact subset with \(C^1\)-boundary which is star shaped with respect to a point \(x_0\in D\) and \(u\in BV^{m}(\Omega)\cap L^q(\Omega\setminus D)\). Then there is a sequence of smooth functions \(\varphi_k\in C^\infty(\overline{\Omega})\) such that \[ \|u-\varphi_k\|_{W^{m-1,1}(\Omega)} +\|u-\varphi_k\|_{L^q(\Omega\setminus D)} + \Big||\nabla^m u|(\Omega) -\int_\Omega|\nabla^m \varphi_k|dx\Big|+ \] \[ \Big|\sqrt{1+|\nabla^m u|^2}(\Omega) - \int_\Omega \sqrt{1+|\nabla^m \varphi_k|^2}dx \Big| \rightarrow 0\;\;\;\;(k\rightarrow\infty). \]
For the case \(p=1\) and \(m=1\) similar results were previously obtained in the work by M. Fuchs and C. Tietz [J. Math. Sci., New York 210, No. 4, 458–475 (2015; Zbl 1331.49014); translation from Probl. Mat. Anal. 81, 107–120 (2015)].

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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