American Institute of Mathematical Sciences

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:
 [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. Brewster, D. Mitrea, I. 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. Egert, R. 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 lasz, P. 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

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. Brewster, D. Mitrea, I. 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. Egert, R. 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 lasz, P. 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
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
 [1] Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems & Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43 [2] Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018 [3] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [4] Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205 [5] Igor E. Verbitsky. The Hessian Sobolev inequality and its extensions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6165-6179. doi: 10.3934/dcds.2015.35.6165 [6] Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041 [7] Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683 [8] Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241 [9] Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427 [10] Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171 [11] Stathis Filippas, Luisa Moschini, Achilles Tertikas. Trace Hardy--Sobolev--Maz'ya inequalities for the half fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (2) : 373-382. doi: 10.3934/cpaa.2015.14.373 [12] Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure & Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359 [13] Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019227 [14] Jian Liu, Sihem Mesnager, Lusheng Chen. Variation on correlation immune Boolean and vectorial functions. Advances in Mathematics of Communications, 2016, 10 (4) : 895-919. doi: 10.3934/amc.2016048 [15] Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383 [16] Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. [17] Pablo L. De Nápoli, Irene Drelichman, Ricardo G. Durán. Improved Caffarelli-Kohn-Nirenberg and trace inequalities for radial functions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1629-1642. doi: 10.3934/cpaa.2012.11.1629 [18] Hirobumi Mizuno, Iwao Sato. L-functions and the Selberg trace formulas for semiregular bipartite graphs. Conference Publications, 2003, 2003 (Special) : 638-646. doi: 10.3934/proc.2003.2003.638 [19] Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 [20] Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265

2018 Impact Factor: 0.545