Abstract
In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Dirichlet boundary conditions imposed on a subset which is scaling to a point; by a blow-up analysis for the capacitary potentials, we detect the vanishing order of the Sobolev capacity of such shrinking Dirichlet boundary portion.
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References
Abatangelo, L., Felli, V., Hillairet, L., Léna, C.: Spectral stability under removal of small capacity sets and applications to Aharonov–Bohm operators. J. Spectr. Theory 9(2), 379–427 (2019)
Abatangelo, L., Felli, V., Léna, C.: Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications. ESAIM Control Optim. Calc. Var. 26, 39, 47 (2020)
Abatangelo, L., Felli, V., Noris, B.: On simple eigenvalues of the fractional Laplacian under removal of small fractional capacity sets. Commun. Contemp. Math. 22(8), 1950071, 32 (2020)
Adolfsson, V., Escauriaza, L.: \(C^{1,\alpha }\) domains and unique continuation at the boundary. Commun. Pure Appl. Math. 50(10), 935–969 (1997)
Arendt, W., Warma, M.: The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 19(4), 341–363 (2003)
Bernard, J.-M.E.: Density results in Sobolev spaces whose elements vanish on a part of the boundary. Chin. Ann. Math. Ser. B 32(6), 823–846 (2011)
Bers, L.: Local behavior of solutions of general linear elliptic equations. Commun. Pure Appl. Math. 8, 473–496 (1955)
Colorado, E., Peral, I.: Semilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions. J. Funct. Anal. 199(2), 468–507 (2003)
Courtois, G.: Spectrum of manifolds with holes. J. Funct. Anal. 134(1), 194–221 (1995)
Daners, D.: Dirichlet problems on varying domains. J. Differ. Equ. 188(2), 591–624 (2003)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions, revised ed. Textbooks in Mathematics. CRC Press, Boca Raton (2015)
Felli, V., Ferrero, A.: Almgren-type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations. Proc. R. Soc. Edinb. Sect. A 143(5), 957–1019 (2013)
Gadyl’shin, R.: Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition. Math. Notes 52(4), 1020–1029 (1992)
Gadyl’shin, R.R.: The splitting of a multiple eigenvalue in a boundary value problem for a membrane clamped to a small section of the boundary. Sibirsk. Mat. Zh. 34(3), 43–61 (1993). 221, 226
Helffer, B.: Spectral Theory and Its Applications, vol. 139 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2013)
Leonori, T., Medina, M., Peral, I., Primo, A., Soria, F.: Principal eigenvalue of mixed problem for the fractional Laplacian: moving the boundary conditions. J. Differ. Equ. 265(2), 593–619 (2018)
Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, vol. 51 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)
Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa (3) 13, 115–162 (1959)
Robbiano, L.: Dimension des zéros d’une solution faible d’un opérateur elliptique. J. Math. Pures Appl. (9) 67(4), 339–357 (1988)
Stampacchia, G.: Contributi alla regolarizzazione delle soluzioni dei problemi al contorno per equazioni del secondo ordine ellitiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 12, 223–245 (1958)
Wolff, T.H.: A property of measures in \({{\mathbb{R}}}^N\) and an application to unique continuation. Geom. Funct. Anal. 2(2), 225–284 (1992)
Acknowledgements
The authors acknowledge the support of INdAM and CNRS-PICS project n. PICS08262 entitled “VALeurs propres d’un opérateur Aharonov-Bohm avec pôLE variable-VALABLE”. V. Felli is partially supported by the PRIN 2015 grant “Variational methods, with applications to problems in mathematical physics and geometry”. B. Noris was partially supported by the INdAM-GNAMPA Project 2019 “Il modello di Born-Infeld per l’elettromagnetismo nonlineare: esistenza, regolarità e molteplicità di soluzioni”.
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Communicated by A. Malchiodi.
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Felli, V., Noris, B. & Ognibene, R. Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Dirichlet region. Calc. Var. 60, 12 (2021). https://doi.org/10.1007/s00526-020-01878-3
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DOI: https://doi.org/10.1007/s00526-020-01878-3