Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian
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Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian Ayman Kachmar1 · Mikael P. Sundqvist2 Received: 20 December 2019 / Accepted: 13 May 2020 / © The Author(s) 2020
Abstract We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to −∞. Keywords Magnetic Laplacian · Robin boundary condition · Eigenvalues · Diamagnetic inequalities Mathematics Subject Classification (2010) Primary 35P15 · 47A10 · 47F05
1 Introduction 1.1 Magnetic Robin Laplacian We denote by = {x ∈ R2 : |x| < 1} the open unit disc and by = ∂ = {x ∈ R2 : |x| = 1} its boundary. We study the lowest eigenvalue of the magnetic Robin Laplacian in L2 (), Pγb = −(∇ − ibA0 )2 , (1.1) with domain D(Pγb ) = {u ∈ H 2 () : ν · (∇u − ibA0 )u + γ u = 0
on ∂} .
Mikael P. Sundqvist
mikael.persson [email protected] Ayman Kachmar [email protected] 1
Department of Mathematics, Lebanese University, Nabatieh, Lebanon
2
Department of Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden
(1.2)
27
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Math Phys Anal Geom
(2020) 23:27
Here ν is the unit outward normal vector of , γ < 0 the Robin parameter and b > 0 is the intensity of the applied magnetic field. The vector field A0 generates the unit magnetic field and is defined as follows A0 (x1 , x2 ) =
1 (−x2 , x1 ) . 2
(1.3)
To be more precise, the operator Pγb is defined as the Friedrichs extension, starting from the quadratic form [8, Ch. 4], 2 H 1 () u → Qbγ (u) := (∇ − ibA0 )u(x) dx + γ |u(x)|2 ds(x) . (1.4)
1.2 Main Result The operator Pγb has a compact resolvent, and thus its spectrum consists of an increasing sequence of eigenvalues. We are interested in examining the asymptotics of the principal eigenvalue Qbγ (u) (1.5) λ1 (b, γ ) = inf u∈H 1 () u2 2 L () when b > 0 is fixed and the Robin parameter γ tends to −∞. Theorem 1.1 Let b > 0. Then, as γ → −∞, b 2 1 λ1 (b, γ ) = −γ 2 + γ + inf m − − + o(1). m∈Z 2 2 The first two terms in the asymptotic expansion given in Theorem 1.1 are well known after many contributions (see [17–19] for the case b = 0 and [13] for the case b > 0); however, the third correction term is new for the disc geometry for b > 0. The recent contribution [15, Thm. 1.5] shows that Theorem 1.1 continues to hold in the case b = 0. The proof of Theorem 1.1 can be easily modified to cover the situation of the annulus A = {x ∈ R2 : < |x| < 1} (that is when take = A and = {|x| = 1} ∩ {|x| = } in (1.4)). The aymptotics for the principal eigenvalue in this case is the same as the one in Theorem 1.1. The reason is that the corresponding eigenfunctions are concentrated near the circle {|x| = 1}. More details on this point are given in Remark 2.1 1.3 Lack of Strong Diamagnetism The celebrated diamagnetic inequality yields λ1 (b, γ ) λ1 (γ , 0) . By using Theorem 1.1, we can quantify the dia
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