On the Isoperimetric Inequality for the Magnetic Robin Laplacian with Negative Boundary Parameter

Abstract

We consider the magnetic Robin Laplacian with a negative boundary parameter on a bounded, planar $C^2$-smooth domain. The respective magnetic field is homogeneous. Among a certain class of domains, we prove that the disk maximises the ground-state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all convex centrally symmetric domains.

Appendices

Appendix A: Closedness and Semi-boundedness of the Quadratic Form $\mathfrak q_\Omega ^{\beta ,b}$

In this appendix, we show that the quadratic form $\mathfrak q_\Omega ^{\beta ,b}$ in (2.2) satisfies all the assumptions of the first representation theorem.

Lemma A.1

The symmetric densely defined quadratic form $\mathfrak q_\Omega ^{\beta ,b}$ in (2.2) is closed and semi-bounded.

Proof

Using that $\mathbf {A}\in L^\infty (\Omega ;{\mathbb R}^2)$ we find that for all $u\in H^1(\Omega )$ one has

$$\begin{aligned} \begin{aligned} \Vert (\nabla -{\mathsf {i}}b\mathbf {A})u\Vert _{L^2(\Omega ;{\mathbb C}^2)}^2&\le 2\Vert \nabla u \Vert ^2_{L^2(\Omega ;{\mathbb C}^2)} + 2b^2\Vert \mathbf {A}\Vert _{\infty }^2\Vert u\Vert ^2_{L^2(\Omega )},\\ \Vert (\nabla -{\mathsf {i}}b\mathbf {A})u\Vert _{L^2(\Omega ;{\mathbb C}^2)}^2&\ge \frac{1}{2}\Vert \nabla u \Vert ^2_{L^2(\Omega ;{\mathbb C}^2)} - b^2\Vert \mathbf {A}\Vert _{\infty }^2\Vert u\Vert ^2_{L^2(\Omega )}. \end{aligned} \end{aligned}$$

(A.1)

From the inequalities in (A.1), we conclude that the non-negative symmetric densely defined quadratic form $\mathfrak q_\Omega ^{0,b}$ corresponding to the magnetic Neumann Laplacian on $\Omega$ with the homogeneous magnetic field is closed, because the norm induced by the quadratic form $\mathfrak q_\Omega ^{0,b}$ is equivalent to the standard norm in the Sobolev space $H^1(\Omega )$.

Recall that according to the diamagnetic inequality [37, Thm. 7.21]

$$\begin{aligned} \Vert \nabla |u|\Vert _{L^2(\Omega ;{\mathbb C}^2)}^2\le \Vert (\nabla -{\mathsf {i}}b\mathbf {A})u\Vert _{L^2(\Omega ;{\mathbb C}^2)}^2 \end{aligned}$$

(A.2)

for all $u\in H^1(\Omega )$. Combining (A.2) with the inequality in [5,  Lem. 2.6], we obtain that for any $\varepsilon >0$ there exists a constant $C(\varepsilon ) > 0$ such that

$$\begin{aligned} \Vert u|_{\partial \Omega }\Vert ^2_{L^2(\partial \Omega )} \le \varepsilon \Vert (\nabla -{\mathsf {i}}b\mathbf {A})u\Vert ^2_{L^2(\Omega ;{\mathbb C}^2)} + C(\varepsilon )\Vert u\Vert ^2_{L^2(\Omega )},\quad \text {for all}\, u\in H^1(\Omega ).\nonumber \\ \end{aligned}$$

(A.3)

From the above inequality, we deduce that the quadratic form $H^1(\Omega )\ni u\mapsto \beta \Vert u|_{\partial \Omega }\Vert ^2_{L^2(\partial \Omega )}$, $\beta \in {\mathbb R}_-$, is form bounded with respect to the quadratic form $\mathfrak q^{0,b}_\Omega$ with the form bound $< 1$. Hence, by [32, Thm. VI.1.33] the quadratic form $\mathfrak q^{\beta ,b}_\Omega$ is closed and semi-bounded. $\square$

Appendix B: Continuity of the Ground-State Energy

In this appendix, we present a standard proof that the ground-state energy of the magnetic Robin Laplacian ${\mathsf H}_{\Omega }^{\beta ,b}$ depends continuously on the intensity of the magnetic field b and the Robin parameter $\beta$.

Recall that $\Omega \subset {\mathbb R}^2$ is a bounded simply connected $C^2$-smooth domain. For definiteness we use the convention $\Vert u\Vert ^2_{H^1(\Omega )} := \Vert \nabla u\Vert ^2_{L^2(\Omega ;{\mathbb C}^2)} + \Vert u\Vert ^2_{L^2(\Omega )}$ for the standard norm in the Sobolev space $H^1(\Omega )$. Recall also that by the trace theorem [39, Thm. 3.38] there exists a constant $c >0$ such that $\Vert u|_{\partial \Omega }\Vert ^2\le c\Vert u\Vert ^2_{H^1(\Omega )}$ for any $u\in H^1(\Omega )$.

Let $\beta _1\le 0$ and $b_1\ge 0$ be fixed and $\beta _2\le 0$ and $b_2\ge 0$ be such that $|\beta _1-\beta _2|,|b_1-b_2|\le 1$. It follows from the second inequality in (A.1) combined with [5, Lem. 2.6] that there exists $\gamma \in {\mathbb R}$ such that ${\mathsf H}^{\beta _2,b_2}_\Omega \ge \gamma$ for any $\beta _2\le 0$ and $b_2\ge 0$ satisfying $|\beta _1-\beta _2|,|b_1-b_2|\le 1$

For any $u \in H^1(\Omega )$, we get the following estimate

$$\begin{aligned} \begin{aligned}&\big |\mathfrak q^{\beta _1,b_1}_\Omega [u] - \mathfrak q^{\beta _2,b_2}_\Omega [u]\big |\\&\quad \!=\!\bigg | 2(b_1-b_2)\mathrm{Im}\,(\nabla u,\mathbf{A} u)_{L^2(\Omega ;{\mathbb C}^2)} + (b_1^2-b_2^2)(|\mathbf{A}|^2 u,u)_{L^2(\Omega )} + (\beta _1-\beta _2)\Vert u|_{\partial \Omega }\Vert ^2_{L^2(\partial \Omega )} \bigg |\\&\quad \!\le \! 2|b_1-b_2|\Vert \nabla u\Vert _{L^2(\Omega ;{\mathbb C}^2)} \Vert \mathbf{A} u\Vert _{L^2(\Omega ;{\mathbb C}^2)} \!+\! |b_1^2-b_2^2|\Vert \mathbf{A}\Vert _\infty ^2\Vert u\Vert ^2_{L^2(\Omega )}\!+\! |\beta _1-\beta _2|\Vert u|_{\partial \Omega }\Vert ^2_{L^2(\partial \Omega )}\\&\quad \!\le \! |b_1\!-\!b_2|\big [\Vert \nabla u\Vert _{L^2(\Omega ;{\mathbb C}^2)}^2 \!+\!\Vert \mathbf{A}\Vert ^2_\infty \Vert u\Vert ^2_{L^2(\Omega )}\big ] \!+\!|b_1^2\!-\!b_2^2|\Vert \mathbf{A}\Vert _\infty ^2\Vert u\Vert ^2_{L^2(\Omega )}\! +\! c|\beta _1\!-\!\beta _2|\Vert u\Vert ^2_{H^1(\Omega )}\\&\quad \!\le \! \max \{|b_1-b_2|,|b_1-b_2| \Vert \mathbf{A}\Vert _\infty ^2, |b_1^2-b_2^2|\Vert \mathbf{A}\Vert _\infty ^2,c|\beta _1-\beta _2|\}\Vert u\Vert ^2_{H^1(\Omega )}, \end{aligned} \end{aligned}$$

where we used the trace theorem in the penultimate step. Since the standard $H^1$-norm is equivalent to the norm $u \mapsto \mathfrak q_{\Omega }^{\beta _1,b_1}[u] +(-\gamma +1)\Vert u\Vert ^2_{L^2(\Omega )}$ induced by the quadratic form $\mathfrak q_\Omega ^{\beta _1,b_1}$ we conclude from the above estimate with the aid of [28, Thm. VI.3.6] that the operator ${\mathsf H}^{\beta _2,b_2}_\Omega$ converges in the norm resolvent sense to the operator ${\mathsf H}^{\beta _1,b_1}_\Omega$ as $(\beta _2,b_2)\rightarrow (\beta _1,b_1)$. Note also that the family of operators ${\mathsf H}^{\beta _2,b_2}_\Omega$ is uniformly lower semi-bounded. Hence, it follows from the spectral convergence result [47, Satz 9.24 (ii)] that $\lambda ^{\beta _2,b_2}_1(\Omega )\rightarrow \lambda ^{\beta _1,b_1}_1(\Omega )$ as $(\beta _2,b_2)\rightarrow (\beta _1,b_1)$ and thus the lowest eigenvalue $\lambda _1^{\beta ,b}(\Omega )$ of ${\mathsf H}^{\beta ,b}_\Omega$ is a continuous function of the parameters $\beta ,b\in (-\infty ,0]\times [0,\infty )$.

Appendix C: The Neumann Magnetic Ground-State Energy

Consider the Neumann eigenvalue $\lambda _1^{0,b}(\Omega )$ introduced in (2.5) with an associated normalised eigenfunction $u_b:\Omega \rightarrow {\mathbb C}$. Let us assume that $\lambda _1^{0,b}(\Omega )=0$. The diamagnetic inequality,

$$\begin{aligned} \int _\Omega \big |\nabla |u_b|\big |^2{\mathsf {d}}x\le \int _\Omega |(\nabla -{\mathsf {i}}b\mathbf {A})u_b|^2 {\mathsf {d}}x=0,\end{aligned}$$

yields that $|u_b|= |\Omega |^{-1/2}$, since the domain $\Omega$ is connected. Furthermore, $u_b$ satisfies

$$\begin{aligned} (\nabla -{\mathsf {i}}b \mathbf {A})u_b=0\,. \end{aligned}$$

(C.1)

Taking the inner product with $\overline{u}_b$, we infer from (C.1),

$$\begin{aligned} \overline{u}_b\nabla u_b=\frac{{\mathsf {i}}b}{|\Omega |}\mathbf {A}\,.\end{aligned}$$

Taking the $\mathrm{curl}$ in (C.1), we get,

$$\begin{aligned} {\mathsf {i}}b\,\mathrm{curl}(\mathbf {A}u_b)=\mathrm{curl}(\nabla u_b)=0 \,.\end{aligned}$$

(C.2)

Finally, we notice that (see (2.3))

$$\begin{aligned}\overline{u}_b\,\mathrm{curl}(\mathbf {A}u_b)=|u_b|^2\,\mathrm{curl}\mathbf {A}+\mathbf {A}^\bot \cdot (\overline{u}_b\nabla u_b)=\frac{1}{|\Omega |}+\frac{{\mathsf {i}}b}{|\Omega |}\mathbf {A}^\bot \cdot \mathbf {A}=\frac{1}{|\Omega |}\end{aligned}$$

where $\mathbf {A}^\bot =\frac{1}{2}(x_1,x_2)$. Consequently, we get from (C.2) that $b=0$.