Advertisement

Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

  • Sebastian BechtelEmail author
  • Moritz Egert
Article
  • 9 Downloads

Abstract

A full interpolation theory for Sobolev functions with smoothness between 0 and 1 and vanishing trace on a part of the boundary of an open set is established. Geometric assumptions are mostly of measure theoretic nature and reach beyond Lipschitz regular domains. Previous results were limited to regular geometric configurations or Hilbertian Sobolev spaces. Sets with porous boundary and their characteristic multipliers on smoothness spaces play a major role in the arguments.

Keywords

Interpolation of Banach spaces (fractional) Sobolev spaces Traces and extensions of Sobolev functions Porous sets Measure density conditions Hardy’s inequality 

Mathematics Subject Classification

Primary: 46B70 Secondary: 46E35 

Notes

Acknowledgements

Both authors are grateful to Joachim Rehberg for many fruitful discussions on and around the topic. The first named author thanks his Ph.D. advisor Robert Haller-Dintelmann for his support and the Laboratoire de Mathématiques d’Orsay for hospitality during a stay in March 2018 where this project got started.

References

  1. 1.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314. Springer, Berlin (1996)CrossRefGoogle Scholar
  2. 2.
    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96. Birkhäuser, Basel (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Auscher, P., Badr, N., Haller-Dintelmann, R., Rehberg, J.: The square root problem for second order divergence form operators with mixed boundary conditions on \({L}^p\). J. Evol. Equ. 15(1), 165–208 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Axelsson, A., Keith, S., McIntosh, A.: The Kato square root problem for mixed boundary value problems. J. Lond. Math. Soc. 74, 113–130 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bechtel, S.: The Kato Square Root Property for Mixed Boundary Conditions. Master’s thesis, TU Darmstadt, (2017). http://www3.mathematik.tu-darmstadt.de/index.php?id=3305
  6. 6.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)zbMATHGoogle Scholar
  7. 7.
    Bonifacius, L., Neitzel, I.: Second order optimality conditions for optimal control of quasilinear parabolic equations. Math. Control Relat. Fields 8(1), 1–34 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brewster, K., Mitrea, D., Mitrea, I., Mitrea, M.: Extending Sobolev functions with partially vanishing traces from locally \((\varepsilon,\delta )\)-domains and applications to mixed boundary problems. J. Funct. Anal. 266(7), 4314–4421 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cwikel, M.: Complex interpolation spaces, a discrete definition and reiteration. Indiana Univ. Math. J. 27(6), 1005–1009 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Denk, R., Hieber, M., Prüss, J.: \({\cal{R}}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Am. Math. Soc. 166, 788 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Disser, K.: Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions. Analysis 35(4), 309–317 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Disser, K., Rehberg, J.: The 3D transient semiconductor equations with gradient-dependent and interfacial recombination. arXiv preprint, available at https://arxiv.org/abs/1805.01348
  13. 13.
    Disser, K., Meyries, M., Rehberg, J.: A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces. J. Math. Anal. Appl. 430(2), 1102–1123 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dyda, B., Vähäkangas, A.: A framework for fractional Hardy inequalities. Ann. Acad. Sci. Fenn. Math. 39(2), 675–689 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Egert, M., Tolksdorf, P.: Characterizations of Sobolev functions that vanish on a part of the boundary. Discret. Contin. Dyn. Syst. Ser. S 10(4), 729–743 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Egert, M., Haller-Dintelmann, R., Tolksdorf, P.: The Kato Square Root Problem for mixed boundary conditions. J. Funct. Anal. 267(5), 1419–1461 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)Google Scholar
  18. 18.
    Griepentrog, J.A., Gröger, K., Kaiser, H.-C., Rehberg, J.: Interpolation for function spaces related to mixed boundary value problems. Math. Nachr. 241, 110–120 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grisvard, P.: Équations différentielles abstraites. Ann. Sci. Écol. Norm. Supér. 2, 311–395 (1969)CrossRefzbMATHGoogle Scholar
  20. 20.
    Gröger, K.: A \(W^{1, p}\)-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283(4), 679–687 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hajłasz, P., Koskela, P., Tuominen, H.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254(5), 1217–1234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Haller-Dintelmann, R., Rehberg, J.: Maximal parabolic regularity for divergence operators including mixed boundary conditions. J. Differ. Equ. 247(5), 1354–1396 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Haller-Dintelmann, R., Jonsson, A., Knees, D., Rehberg, J.: Elliptic and parabolic regularity for second-order divergence operators with mixed boundary conditions. Math. Methods Appl. Sci. 39(17), 5007–5026 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Heinonen, J.: Lectures on Analysis on Metric Spaces Universitext. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  25. 25.
    Janson, S., Nilsson, P., Peetre, J.: Notes on Wolff’s note on interpolation spaces. Proc. Lond. Math. Soc. 48(2), 283–299 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Jawerth, B., Frazier, M.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jonsson, A., Wallin, H.: Function spaces on subsets of \({{\mathbb{R}}}^n\). Math. Rep. 2(1), (1984)Google Scholar
  28. 28.
    Lehrbäck, J.: Weighted Hardy inequalities and the size of the boundary. Manuscr. Math. 127(2), 249–273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lehrbäck, J., Tuominen, H.: A note on the dimensions of Assouad and Aikawa. J. Math. Soc. Jpn. 65(2), 343–356 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Luukkainen, J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc. 35(1), 23–76 (1998)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Meinlschmidt, H., Meyer, C., Rehberg, J.: Optimal control of the thermistor problem in three spatial dimensions, Part 1: Existence of optimal solutions. SIAM J. Control Optim. 55(5), 2876–2904 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Meinlschmidt, H., Meyer, C., Rehberg, J.: Optimal control of the thermistor problem in three spatial dimensions, Part 2: Optimality conditions. SIAM J. Control Optim. 55(4), 2368–2392 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer Monographs in Mathematics. Springer, Berlin (2012)Google Scholar
  34. 34.
    Rychkov, V.S.: Linear extension operators for restrictions of function spaces to irregular open sets. Stud. Math. 140(2), 141–162 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Seeley, R.: Interpolation in \(L^{p}\) with boundary conditions. Stud. Math. 44, 44–60 (1972)CrossRefzbMATHGoogle Scholar
  36. 36.
    Shvartsman, P.: Local approximations and intrinsic characterization of spaces of smooth functions on regular subsets of \({{\mathbb{R}}}^n\). Math. Nachr. 279(11), 1212–1241 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sickel, W.: Pointwise multipliers of Lizorkin–Triebel spaces. The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol. 110. Birkhäuser, Basel (1999)Google Scholar
  38. 38.
    Simon, J.: Sobolev, Besov and Nikol’skii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval. Ann. Mater. Pura Appl. 157, 117–148 (1990)CrossRefzbMATHGoogle Scholar
  39. 39.
    ter Elst, A.F.M., Rehberg, J.: Hölder estimates for second-order operators on domains with rough boundary. Adv. Differ. Equ. 20(3–4), 299–360 (2015)zbMATHGoogle Scholar
  40. 40.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland, Amsterdam (1978)Google Scholar
  41. 41.
    Triebel, H.: A note on function spaces in rough domains. Tr. Mat. Inst. Steklova 293, 346–351 (2016)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Väisälä, J.: Porous sets and quasisymmetric maps. Trans. Am. Math. Soc. 299(2), 525–533 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Wolff, T.H.: A note on interpolation spaces. Lecture Notes in Mathematics. Harmonic analysis (Minneapolis, Minn., 1981), vol. 908. Springer, Berlin (1982)Google Scholar
  44. 44.
    Yeh, J.: Real Analysis. World Scientific Publishing, Hackensack (2006)CrossRefzbMATHGoogle Scholar
  45. 45.
    Zhou, Y.: Fractional Sobolev extension and imbedding. Trans. Am. Math. Soc. 367(2), 959–979 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Fachbereich Mathematik, Technische Universität DarmstadtDarmstadtGermany
  2. 2.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

Personalised recommendations