×

Eigenvalue inequalities for the Laplacian with mixed boundary conditions. (English) Zbl 1366.35106

On a bounded connected Lipschitz domain \(\Omega \subset {\mathbb R}^d\) denote by \( 0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots\) the eigenvalues of the (negative) Laplacian subject to a Dirichlet boundary condition on the boundary \(\partial \Omega\) and by \(0 = \mu_1 < \mu_2 \leq \mu_3 \leq \cdots\) the eigenvalues corresponding to a Neumann condition. The present paper focuses on Laplacian eigenvalues \(\lambda_k^\Gamma\) for the mixed case of a Dirichlet boundary condition on a nonempty part \(\Gamma_{\mathrm D}\) of \(\partial \Omega\) and a Neumann condition on the complement \(\Gamma_{\mathrm N}\) of \(\Gamma_{\mathrm D}\) in \(\partial \Omega\). It is assumed that \(\Gamma_{\mathrm D}\) and \(\Gamma_{\mathrm N}\) are two relatively open, non-empty subsets of \(\partial \Omega\) such that \(\Gamma_{\mathrm D} \cap \Gamma_{\mathrm N} = \emptyset\) and \(\partial \Omega \setminus (\Gamma_{\mathrm D} \cup \Gamma_{\mathrm N})\) has measure zero.
The tangential hyperplane \(T_{x'}\) exists for almost all \(x' \in \partial \Omega\). Define \(\hat \Gamma_{\mathrm N}\) to be the set of all \(x' \in \Gamma_{\mathrm N}\) such that \(T_{x'}\) exists. Define the linear subspace \[ {\mathcal S}(\Gamma_{\mathrm N}) := \bigcap_{x' \in \hat \Gamma_{\mathrm N}} T_{x'} \] of \({\mathbb R}^d\) consisting of all vectors being tangential to some \(x' \in \Gamma_{\mathrm N}\) away from a set of measure zero.
The authors prove that if \(\dim {\mathcal S}(\Gamma_{\mathrm N})\geq 1\) then \(\mu_{k+1}\leq \lambda_k^\Gamma\).
Assume that \(\Omega \subset {\mathbb R}^d\), \(d \geq 2\), is a polyhedral, convex, bounded domain. For this case the authors prove that \( \lambda_{k + \dim {\mathcal S} (\Gamma_{\mathrm D})}^\Gamma \leq \lambda_k\) holds for all \(k \in {\mathbb N}\).
The proofs are based on variational principles and proper choices of test functions.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Agranovich, M. S., Mixed problems in a Lipschitz domain for second-order strongly elliptic systems, Funct. Anal. Appl., 45, 81-98, (2011) · Zbl 1271.35024
[2] Arendt, W.; Mazzeo, R., Friedlander’s eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11, 2201-2212, (2012) · Zbl 1267.35139
[3] Band, R.; Bersudsky, M.; Fajman, D., Courant-sharp eigenvalues of Neumann 2-rep-tiles, Lett. Math. Phys., (2017), in press · Zbl 1372.35192
[4] Behrndt, J.; Rohleder, J.; Stadler, S., Eigenvalue inequalities for Schrödinger operators on unbounded Lipschitz domains, J. Spectr. Theory, (2017), in press
[5] Birman, M. Sh.; Solomjak, M. Z., Spectral theory of selfadjoint operators in Hilbert spaces, (1987), Dordrecht Holland
[6] Brown, R., The mixed problem for Laplace’s equation in a class of Lipschitz domains, Comm. Partial Differential Equations, 19, 1217-1233, (1994) · Zbl 0831.35043
[7] Dauge, M., Elliptic boundary value problems on corner domains. smoothness and asymptotics of solutions, Lecture Notes in Math., (1988), Springer-Verlag Berlin · Zbl 0668.35001
[8] Filonov, N., On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator, Algebra i Analiz, St. Petersburg Math. J., 16, 413-416, (2005), ; translation in · Zbl 1078.35081
[9] Frank, R. L.; Laptev, A., Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group, Int. Math. Res. Not. IMRN, 2889-2902, (2010) · Zbl 1198.35161
[10] Friedlander, L., Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Ration. Mech. Anal., 116, 153-160, (1991) · Zbl 0789.35124
[11] Gesztesy, F.; Mitrea, M., Nonlocal Robin Laplacians and some remarks on a paper by filonov on eigenvalue inequalities, J. Differential Equations, 247, 2871-2896, (2009) · Zbl 1181.35155
[12] Gordon, C.; Webb, D.; Wolpert, S., Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110, 1-22, (1992) · Zbl 0778.58068
[13] Grisvard, P., Alternative de Fredholm relative au problème de Dirichlet dans un polyèdre, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2, 359-388, (1975) · Zbl 0315.35034
[14] Grisvard, P., Elliptic problems in nonsmooth domains, Monogr. Stud.Math., vol. 24, (1985), Pitman (Advanced Publishing Program) Boston, MA · Zbl 0695.35060
[15] Grubb, G., The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates, J. Math. Anal. Appl., 382, 339-363, (2011) · Zbl 1223.47050
[16] Kato, T., Perturbation theory for linear operators, (1995), Springer-Verlag Berlin · Zbl 0836.47009
[17] Kelliher, J., Eigenvalues of the Stokes operator versus the Dirichlet Laplacian in the plane, Pacific J. Math., 244, 99-132, (2010) · Zbl 1185.35152
[18] Levine, H. A.; Weinberger, H. F., Inequalities between Dirichlet and Neumann eigenvalues, Arch. Ration. Mech. Anal., 94, 193-208, (1986) · Zbl 0608.35047
[19] Lotoreichik, V.; Rohleder, J., An eigenvalue inequality for Schrödinger operators with δ and \(\delta^\prime\)-interactions supported on hypersurfaces, Oper. Theory Adv. Appl., 247, 173-184, (2015) · Zbl 1346.35135
[20] Maz’ya, V.; Rossmann, J., Elliptic equations in polyhedral domains, (2010), American Mathematical Society Providence · Zbl 1196.35005
[21] McLean, W., Strongly elliptic systems and boundary integral equations, (2000), Cambridge University Press Cambridge · Zbl 0948.35001
[22] Pankrashkin, K., Eigenvalue inequalities and absence of threshold resonances for waveguide junctions, J. Math. Anal. Appl., 449, 907-925, (2017) · Zbl 1372.35199
[23] Payne, L. E., Inequalities for eigenvalues of membranes and plates, J. Ration. Mech. Anal., 4, 517-529, (1955) · Zbl 0064.34802
[24] Pólya, G., Remarks on the foregoing paper, J. Math. Phys., 31, 55-57, (1952) · Zbl 0046.32401
[25] Rayleigh, J. W.S., The theory of sound, (1945), Dover New York, reprinted:
[26] Rohleder, J., Strict inequality of Robin eigenvalues for elliptic differential operators on Lipschitz domains, J. Math. Anal. Appl., 418, 978-984, (2014) · Zbl 1304.47062
[27] Schmüdgen, K., Unbounded self-adjoint operators on Hilbert space, (2012), Springer Dordrecht · Zbl 1257.47001
[28] Seeley, R., Trace expansions for the zaremba problem, Comm. Partial Differential Equations, 27, 2403-2421, (2002) · Zbl 1055.58010
[29] Shamir, E., Regularization of mixed second-order elliptic problems, Israel J. Math., 6, 150-168, (1968) · Zbl 0157.18202
[30] Siudeja, B., Hot spots conjecture for a class of acute triangles, Math. Z., 280, 783-806, (2015) · Zbl 1335.35164
[31] Siudeja, B., On mixed Dirichlet-Neumann eigenvalues of triangles, Proc. Amer. Math. Soc., 144, 2479-2493, (2016) · Zbl 1359.35118
[32] Szegő, G., Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3, 343-356, (1954) · Zbl 0055.08802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.