August  2017, 10(4): 729-743. doi: 10.3934/dcdss.2017037

Characterizations of Sobolev functions that vanish on a part of the boundary

1. 

Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

2. 

Fachbereich Mathematik, Technische Universität, Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany

* Corresponding author: Moritz Egert

Received  July 2016 Revised  September 2016 Published  April 2017

Fund Project: The first author was supported by a public grant as part of the FMJH. The second author was supported by "Studienstiftung des deutschen Volkes"

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with a Sobolev extension property around the complement of a closed part $D$ of its boundary. We prove that a function $u \in {\rm{W}}^{1,p}(\Omega)$ vanishes on $D$ in the sense of an interior trace if and only if it can be approximated within ${\rm{W}}^{1,p}(\Omega)$ by smooth functions with support away from $D$. We also review several other equivalent characterizations, so to draw a rather complete picture of these Sobolev functions vanishing on a part of the boundary.

Citation: Moritz Egert, Patrick Tolksdorf. Characterizations of Sobolev functions that vanish on a part of the boundary. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 729-743. doi: 10.3934/dcdss.2017037
References:
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D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, vol. 314, Springer, Berlin, 1996. Google Scholar

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K. BrewsterD. MitreaI. Mitrea and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally (ε, δ)-domains and applications to mixed boundary problems, J. Funct. Anal., 266 (2014), 4314-4421.  doi: 10.1016/j.jfa.2014.02.001.  Google Scholar

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M. EgertR. Haller-Dintelmann and J. Rehberg, Hardy's inequality for functions vanishing on a part of the boundary, Potential Anal., 43 (2015), 49-78.  doi: 10.1007/s11118-015-9463-8.  Google Scholar

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A. F. M. ter Elst and J. Rehberg, Hölder estimates for second-order operators on domains with rough boundary, Adv. Differential Equations, 20 (2015), 299-360.   Google Scholar

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L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton FL, 1992. Google Scholar

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H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, vol. 153, Springer, New York, 1969. Google Scholar

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M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations Ⅰ. Results in Mathematics and Related Areas. 3rd Series, vol. 37, Springer-Verlag, Berlin, 1998. Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001. Google Scholar

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P. Haj laszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234.  doi: 10.1016/j.jfa.2007.11.020.  Google Scholar

[10]

R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, J. Differential Equations, 247 (2009), 1354-1396.  doi: 10.1016/j.jde.2009.06.001.  Google Scholar

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A. Jonsson and H. Wallin, Function spaces on subsets of $\mathbb{R}^N$, Math. Rep., (1984), 2.   Google Scholar

[12]

D. Swanson and W. P. Ziemer, Sobolev functions whose inner trace at the boundary is zero, Ark. Mat., 37 (1999), 373-380.  doi: 10.1007/BF02412221.  Google Scholar

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J. Yeh, Real Analysis, World Scientific Publishing, Hackensack NJ, 2006. Google Scholar

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W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, vol. 120, Springer, New York, 1989. Google Scholar

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften, vol. 314, Springer, Berlin, 1996. Google Scholar

[2]

K. BrewsterD. MitreaI. Mitrea and M. Mitrea, Extending Sobolev functions with partially vanishing traces from locally (ε, δ)-domains and applications to mixed boundary problems, J. Funct. Anal., 266 (2014), 4314-4421.  doi: 10.1016/j.jfa.2014.02.001.  Google Scholar

[3]

M. EgertR. Haller-Dintelmann and J. Rehberg, Hardy's inequality for functions vanishing on a part of the boundary, Potential Anal., 43 (2015), 49-78.  doi: 10.1007/s11118-015-9463-8.  Google Scholar

[4]

A. F. M. ter Elst and J. Rehberg, Hölder estimates for second-order operators on domains with rough boundary, Adv. Differential Equations, 20 (2015), 299-360.   Google Scholar

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton FL, 1992. Google Scholar

[6]

H. Federer, Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, vol. 153, Springer, New York, 1969. Google Scholar

[7]

M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations Ⅰ. Results in Mathematics and Related Areas. 3rd Series, vol. 37, Springer-Verlag, Berlin, 1998. Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, Berlin, 2001. Google Scholar

[9]

P. Haj laszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234.  doi: 10.1016/j.jfa.2007.11.020.  Google Scholar

[10]

R. Haller-Dintelmann and J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, J. Differential Equations, 247 (2009), 1354-1396.  doi: 10.1016/j.jde.2009.06.001.  Google Scholar

[11]

A. Jonsson and H. Wallin, Function spaces on subsets of $\mathbb{R}^N$, Math. Rep., (1984), 2.   Google Scholar

[12]

D. Swanson and W. P. Ziemer, Sobolev functions whose inner trace at the boundary is zero, Ark. Mat., 37 (1999), 373-380.  doi: 10.1007/BF02412221.  Google Scholar

[13]

J. Yeh, Real Analysis, World Scientific Publishing, Hackensack NJ, 2006. Google Scholar

[14]

W. P. Ziemer, Weakly Differentiable Functions, Graduate Texts in Mathematics, vol. 120, Springer, New York, 1989. Google Scholar

Figure 1.  The dyadic 'skeleton' of $\Omega$ is obtained from the square $[0,1] \times [1,2]$ by iteratively attaching a total number of $2^j$ disjoint squares of side length $2^{-j}$ at the bottom of the existing construction. The domain $\Omega$ is then constructed by blowing up the line segments to appropriately sized open rectangles
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