Abstract
In a bounded non-simply connected planar domain Ω, with a boundary split in an interior part and an exterior part, we obtain bounds for the embedding constants of some subspaces of H 1( Ω) into L p( Ω) for any p > 1, p ≠ 2. The subspaces contain functions which vanish on the interior boundary and are constant (possibly zero) on the exterior boundary. We also evaluate the precision of the obtained bounds in the limit situation where the interior part tends to disappear and we show that it does not depend on p. Moreover, we emphasize the failure of symmetrization techniques in these functional spaces. In simple situations, a new phenomenon appears: the existence of a break even surface separating masses for which symmetrization increases/decreases the Dirichlet norm. The question whether a similar phenomenon occurs in more general situations is left open.
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References
Abdellaoui, B., Colorado, E., Peral, I.: Some critical quasilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions: relation with Sobolev and Hardy-Sobolev optimal constants. J. Math. Anal. Appl. 332(2), 1165–1188 (2007)
Alvino, A., Trombetti, G., Lions, P.-L.: On optimization problems with prescribed rearrangements. Nonlinear Anal. 13, 185–220 (1989)
Brothers, J., Ziemer, W.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)
Burchard, A.: Steiner symmetrization is continuous in W 1, p. Geom. Funct. Anal. 7, 823–860 (1997)
del Pino, M., Dolbeault, J.: Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions. J. Math. Pures Appl. 81, 847–875 (2002)
Faber, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, pp. 169–172. Verlag der Bayerischen Akademie der Wissenschaften, München (1923)
Gazzola, F., Sperone, G.: Steady Navier-Stokes equations in planar domains with obstacle and explicit bounds for unique solvability. Arch. Ration. Mech. Anal. 238(3), 1283–1347 (2020)
Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94(1), 97–100 (1925)
Lions, P.-L., Pacella, F., Tricarico, M.: Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions. Indiana Univ. Math. J. 37(2), 301–324 (1988)
Steiner, J.: Einfacher Beweise der isoperimetrischen Hauptsätze. J. Reine Angew. Math. 18, 281–296 (1838). Reprinted in Gesammelte Werke 2 (Berlin 1882), pp. 77–91 (1838)
Talenti, G.: The art of rearranging. Milan J. Math. 84(1), 105–157 (2016)
Acknowledgements
The first Author is supported by the PRIN project Direct and inverse problems for partial differential equations: theoretical aspects and applications and by the GNAMPA group of the INdAM. The second Author is supported by the Primus Research Programme PRIMUS/19/SCI/01, by the program GJ19-11707Y of the Czech National Grant Agency GAČR, and by the University Centre UNCE/SCI/023 of the Charles University in Prague.
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Gazzola, F., Sperone, G. (2021). Bounds for Sobolev Embedding Constants in Non-simply Connected Planar Domains. In: Ferone, V., Kawakami, T., Salani, P., Takahashi, F. (eds) Geometric Properties for Parabolic and Elliptic PDE's. Springer INdAM Series, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-030-73363-6_6
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DOI: https://doi.org/10.1007/978-3-030-73363-6_6
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