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Poincaré Trace Inequalities in \(\textit{BV}({\mathbb {B}}^n)\) with Non-standard Normalization

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Abstract

Extremal functions are exhibited in Poincaré trace inequalities for functions of bounded variation in the unit ball \({\mathbb {B}}^n\) of the n-dimensional Euclidean space \({{\mathbb {R}}}^n\). Trial functions are subject to either a vanishing mean value condition, or a vanishing median condition in the whole of \({\mathbb {B}}^n\), instead of just on \(\partial {\mathbb {B}}^n\), as customary. The extremals in question take a different form, depending on the constraint imposed. In particular, under the median constraint, unusually shaped extremal functions appear. A key step in our approach is a characterization of the sharp constant in the relevant trace inequalities in any admissible domain \(\Omega \subset {{\mathbb {R}}}^n\), in terms of an isoperimetric inequality for subsets of \(\Omega \).

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References

  1. Alvino, A., Ferone, A., Volpicelli, R.: Sharp Hardy inequalities in the half-space with trace remainder term. Nonlinear Anal. 75, 5466–5472 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andreu, F., Mazón, J.M., Rossi, J.D.: The best constant for the Sobolev trace embedding from \(W^{1,1}(\Omega )\) into \(L^1(\partial \Omega )\). Nonlinear Anal. 59, 1125–1145 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Anzellotti, G., Giaquinta, M.: Funzioni \(BV\) e tracce. Rend. Sem. Mat. Univ. Padova 60, 1–21 (1978)

    MathSciNet  MATH  Google Scholar 

  4. Belloni, M., Kawohl, B.: A symmetry problem related to Wirtinger’s and Poincaré’s inequality. J. Differ. Equ. 156, 211–218 (1999)

    Article  MATH  Google Scholar 

  5. Berchio, E., Gazzola, F., Pierotti, D.: Gelfand type elliptic problems under Steklov boundary conditions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27, 315–335 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bouchez, V., Van Schaftingen, J.: Extremal functions in Poincaré–Sobolev inequalities for functions of bounded variation. In: Nonlinear Elliptic Partial Differential Equations American Mathematical Society Contemporary Mathematics No. 540, pp. 47–58 (2011)

  7. Brasco, L., Franzina, G.: An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities. Nonlinear Differ. Equ. Appl. (NoDEA) 20, 1795–1830 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezis, H., Van Schaftingen, J.: Circulation integrals and critical Sobolev spaces: problems of sharp constants. Perspectives in partial differential equations, harmonic analysis and applications. In: Proceedings of Symposia in Pure Mathematics, vol. 79. American Mathematical Society, Providence, pp. 33–47 (2008)

  9. Brock, F.: An isoperimetric inequality for eigenvalues of the Stekloff problem. Z. Angew. Math. Mech. 81, 69–71 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Burago, Yu.D., Maz’ya, V.G.: Some questions of potential theory and function theory for domains with non-regular boundaries. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 3, 1–152 (1967) (Russian); English translation: Seminars in Mathematics, V.A. Steklov Mathematical Institute, Leningrad, Consultants Bureau, New York 3, 1–68 (1969)

  11. Cianchi, A.: A sharp form of Poincaré type inequalities on balls and spheres. Z. Angew. Math. Phys. (ZAMP) 40, 558–569 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cianchi, A.: Moser–Trudinger trace inequalities. Adv. Math. 217, 2005–2044 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cianchi, A.: A sharp trace inequality for functions of bounded variation in the ball. Proc. R. Soc. Edinb. 142A, 1179–1191 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cianchi, A., Pick, L.: Sobolev embeddings into spaces of Campanato, Morrey, and Hölder type. J. Math. Anal. Appl. 282, 128–150 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cianchi, A., Ferone, V., Nitsch, C., Trombetti, C.: Balls minimize trace constants in BV. J. Reine Angew. Math. (Crelle J.) 725, 41–61 (2017)

  16. Davila, J., Dupaigne, L., Montenegro, M.: The extremal solution of a boundary reaction problem. Commun. Pure Appl. Anal. 7, 795–817 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Della Pietra, F., Gavitone, N.: Symmetrization for Neumann anisotropic problems and related questions. Adv. Nonlinear Stud. 12, 219–235 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Dem’yanov, A.V., Nazarov, A.I.: On the existence of an extremal function in Sobolev embedding theorems with a limit exponent. Algebra i Analiz 17, 105–140 (2005) (in Russian); English translation: St. Petersburgh Math. J. 17, 773–796 (2006)

  19. Escobar, J.F.: Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J. 37, 687–698 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Escobar, J.F.: An isoperimetric inequality and the first Steklov eigenvalue. J. Funct. Anal. 165, 101–116 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Esposito, L., Nitsch, C., Trombetti, C.: Best constants in Poincaré inequalities for convex domains. J. Convex Anal. 20, 253–264 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Esposito, L., Ferone, V., Kawohl, B., Nitsch, C., Trombetti, C.: The longest shortest fence and sharp Poincaré–Sobolev inequalities. Arch. Ration. Mech. Anal. 206, 821–851 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)

    MATH  Google Scholar 

  24. Ferone, V., Nitsch, C., Trombetti, C.: A remark on optimal weighted Poincaré inequalities for convex domains. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 23, 467–475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Girão, P., Weth, T.: The shape of extremal functions for Poincaré–Sobolev-type inequalities in a ball. J. Funct. Anal. 237, 194–233 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984)

    Book  MATH  Google Scholar 

  27. Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  28. Leckband, M.: A rearrangement based proof for the existence of extremal functions for the Sobolev–Poincaré inequality on \(B^n\). J. Math. Anal. Appl. 363, 690–696 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Maggi, F., Villani, C.: Balls have the worst best Sobolev constants. J. Geom. Anal. 15, 83–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Maggi, F., Villani, C.: Balls have the worst best Sobolev inequalities II. Variants and extensions. Calc. Var. Partial Differ. Equ. 31, 47–74 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Maz’ya, V.G.: Classes of regions and imbedding theorems for function spaces. Dokl. Akad. Nauk. SSSR 133, 527–530 (1960) (in Russian); English translation: Soviet Math. Dokl. 1, 882–885 (1960)

  32. Maz’ya, V.G.: Classes of sets and imbedding theorems for function spaces. Dissertation MGU, Moscow (1962) (in Russian)

  33. Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Heidelberg (2011)

    MATH  Google Scholar 

  34. Nazarov, A.I., Repin, S.I.: Exact constants in Poincaré type inequalities for functions with zero mean boundary traces. Math. Methods Appl. Sci. 38, 3195–3207 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nazaret, B.: Best constant in Sobolev trace inequalities on the half-space. Nonlinear Anal. 65, 1977–1985 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rossi, J.D.: First variations of the best Sobolev trace constant with respect to the domain. Can. Math. Bull. 51, 140–145 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Weinstock, R.: Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3, 745–753 (1954)

    MathSciNet  MATH  Google Scholar 

  38. Ziemer, W.P.: Weakly Differentiable Functions. Springer, New York (1989)

    Book  MATH  Google Scholar 

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Acknowledgements

This research was partly supported by the research project of MIUR (Italian Ministry of Education, University and Research) Prin 2012, No. 2012TC7588, “Elliptic and parabolic partial differential equations: geometric aspects, related inequalities, and applications,” and by GNAMPA of the Italian INdAM (National Institute of High Mathematics).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/a, 50137, Firenze, Italy

    Andrea Cianchi

  2. Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II”, Complesso Monte S.Angelo, Via Cintia, 80126, Napoli, Italy

    Vincenzo Ferone

  3. Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Complesso Monte S.Angelo, Via Cintia, 80126, Napoli, Italy

    Carlo Nitsch & Cristina Trombetti

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  1. Andrea Cianchi
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  2. Vincenzo Ferone
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  3. Carlo Nitsch
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  4. Cristina Trombetti
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Correspondence to Cristina Trombetti.

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Cianchi, A., Ferone, V., Nitsch, C. et al. Poincaré Trace Inequalities in \(\textit{BV}({\mathbb {B}}^n)\) with Non-standard Normalization. J Geom Anal 28, 3522–3552 (2018). https://doi.org/10.1007/s12220-017-9968-z

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