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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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##### References:
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