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

- 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.Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314. Springer, Berlin (1996)CrossRefGoogle Scholar
- 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.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.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.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.Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)zbMATHGoogle Scholar
- 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.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.Cwikel, M.: Complex interpolation spaces, a discrete definition and reiteration. Indiana Univ. Math. J.
**27**(6), 1005–1009 (1978)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Disser, K.: Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions. Analysis
**35**(4), 309–317 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Dyda, B., Vähäkangas, A.: A framework for fractional Hardy inequalities. Ann. Acad. Sci. Fenn. Math.
**39**(2), 675–689 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)Google Scholar
- 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.Grisvard, P.: Équations différentielles abstraites. Ann. Sci. Écol. Norm. Supér.
**2**, 311–395 (1969)CrossRefzbMATHGoogle Scholar - 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.Hajłasz, P., Koskela, P., Tuominen, H.: Sobolev embeddings, extensions and measure density condition. J. Funct. Anal.
**254**(5), 1217–1234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Heinonen, J.: Lectures on Analysis on Metric Spaces Universitext. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
- 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.Jawerth, B., Frazier, M.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal.
**93**(1), 34–170 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Jonsson, A., Wallin, H.: Function spaces on subsets of \({{\mathbb{R}}}^n\). Math. Rep.
**2**(1), (1984)Google Scholar - 28.Lehrbäck, J.: Weighted Hardy inequalities and the size of the boundary. Manuscr. Math.
**127**(2), 249–273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Luukkainen, J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc.
**35**(1), 23–76 (1998)MathSciNetzbMATHGoogle Scholar - 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.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.Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer Monographs in Mathematics. Springer, Berlin (2012)Google Scholar
- 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.Seeley, R.: Interpolation in \(L^{p}\) with boundary conditions. Stud. Math.
**44**, 44–60 (1972)CrossRefzbMATHGoogle Scholar - 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.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.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.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.Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland, Amsterdam (1978)Google Scholar
- 41.Triebel, H.: A note on function spaces in rough domains. Tr. Mat. Inst. Steklova
**293**, 346–351 (2016)MathSciNetzbMATHGoogle Scholar - 42.Väisälä, J.: Porous sets and quasisymmetric maps. Trans. Am. Math. Soc.
**299**(2), 525–533 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Yeh, J.: Real Analysis. World Scientific Publishing, Hackensack (2006)CrossRefzbMATHGoogle Scholar
- 45.Zhou, Y.: Fractional Sobolev extension and imbedding. Trans. Am. Math. Soc.
**367**(2), 959–979 (2015)MathSciNetCrossRefzbMATHGoogle Scholar