Geometry and interior nodal sets of Steklov eigenfunctions

Abstract

We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein’s inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for the measure of interior nodal sets.

Appendix

In this section, we provide the proof of Lemma 1 and some arguments stated in the proof Proposition 2. Recall that

$$\begin{aligned} \Vert {\mathcal {L}}_\beta (v)\Vert ^2_\phi \ge \frac{1}{2}{\mathcal {A}}-{\mathcal {B}}, \end{aligned}$$

where

$$\begin{aligned} \mathcal {L}_\beta (v)= & {} \partial ^2_t v+\big (2\beta \phi '+e^{2t} {\bar{b}}_t+(n-2)+\partial _t \ln \sqrt{\gamma }\big )\partial _t v+ e^{2t} \bar{b}_i\partial _i v \nonumber \\&+\big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+\beta \phi ''+(n-2)\beta \phi '+\beta \partial _t \ln \sqrt{\gamma }\phi '\big ) v+\triangle _{\omega } v+ e^{2t} {{\bar{q}}} v \end{aligned}$$

(4.1)

and

$$\begin{aligned} {\mathcal {A}}=\Vert \partial ^2_t v+ & {} \triangle _{\omega } v +\big (2\beta \phi '+e^{2t} {\bar{b}}_t\big )\partial _t v+ e^{2t} \bar{b}_i\partial _i v \nonumber \\+ & {} \big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v\Vert ^2_{\phi } \end{aligned}$$

(4.2)

and

$$\begin{aligned} {\mathcal {B}}=\Vert \beta \phi '' v+\beta \partial _t \ln \sqrt{\gamma } \phi ' v+(n-2)\partial _t v+\partial _t \ln \sqrt{\gamma }\partial _t v\Vert ^2_{\phi }. \end{aligned}$$

(4.3)

Modifying the arguments in [2] and [35], we can obtain the following lemma, which verifies the proof of (2.18) and (2.19) in Proposition 2.

Lemma 3

There holds that

$$\begin{aligned} \Vert {\mathcal {L}}_\beta (v)\Vert ^2_\phi&\ge \frac{1}{4}{\mathcal {A}} \nonumber \\&\ge C\beta ^3\int |\phi ''||v|^2 \phi '^{-3}\sqrt{\gamma }\,dt d \omega +C\beta \int |\phi ''||D_\omega v|^2 \phi '^{-3}\sqrt{\gamma }\,dt d \omega \nonumber \\&+C\beta \int |\partial _t v|^2 \phi '^{-3}\sqrt{\gamma }\,dt d \omega . \end{aligned}$$

(4.4)

Proof

We decompose ${\mathcal {A}}$ as

$$\begin{aligned} {\mathcal {A}}={\mathcal {A}}'_1+{\mathcal {A}}'_2+{\mathcal {A}}'_3, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {A}}'_1=\Vert \partial ^2_t v +\big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v+ \triangle _{\omega } v\Vert ^2_{\phi } \end{aligned}$$

and

$$\begin{aligned} {\mathcal {A}}'_2=\Vert \big (2\beta \phi '+e^{2t} {\bar{b}}_t\big )\partial _t v+e^{2t}{\bar{b}}_i\partial _i v\Vert ^2_{\phi } \end{aligned}$$

and

$$\begin{aligned} {\mathcal {A}}'_3=2< \partial ^2_t v+ \big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v+\triangle _\omega v,\\ \big (2\beta \phi '+e^{2t} {{{\bar{b}}}}_t\big )\partial _t v+e^{2t}{\bar{b}}_i\partial _i v>_\phi . \end{aligned}$$

We first compute ${\mathcal {A}}'_1$. Let ${\hat{\alpha }}$ be some small positive constant. Recall that $\phi (t)=t-e^{\epsilon t}$. Since $|\phi ^{''}|\le 1$ and $\beta$ is large enough, it is true that

$$\begin{aligned} {\mathcal {A}}'_1\ge \frac{{\hat{\alpha }}}{\beta } {\mathcal {A}}^{''}_1, \end{aligned}$$

(4.5)

where ${\mathcal {A}}^{''}_1$ is given by

$$\begin{aligned} {\mathcal {A}}^{''}_1=\left\| \sqrt{|\phi ^{''}|}[\partial ^2_t v +\big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v+ \triangle _{\omega } v ]\right\| _\phi ^2. \end{aligned}$$

We split ${\mathcal {A}}^{''}_1$ into three parts:

$$\begin{aligned} {\mathcal {A}}^{''}_1={\mathcal {K}}_1+{\mathcal {K}}_2+{\mathcal {K}}_3, \end{aligned}$$

(4.6)

where

$$\begin{aligned} {\mathcal {K}}_1=\left\| \sqrt{|\phi ^{''}|}\big (\partial ^2_t v+\triangle _{\omega } v\big )\right\| _\phi ^2 \end{aligned}$$

and

$$\begin{aligned} {\mathcal {K}}_2=\left\| \sqrt{|\phi ^{''}|}\big (\beta ^2 \phi '^2+ \beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v \right\| _\phi ^2 \end{aligned}$$

and

$$\begin{aligned} {\mathcal {K}}_3=2\bigg < |\phi ^{''}|(\partial ^2_t v+\triangle _{\omega } v), \ \big (\beta ^2 \phi '^2+ \beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v \bigg >_\phi . \end{aligned}$$

The expression ${\mathcal {K}}_1$ is considered to be a nonnegative term. We estimate ${\mathcal {K}}_2$. By the triangle inequality,

$$\begin{aligned} {\mathcal {K}}_2\ge \beta ^4 \left\| \sqrt{|\phi ^{''}|} \phi ' v\right\| _\phi ^2- \left\| \sqrt{|\phi ^{''}|}\big (\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v \right\| _\phi ^2. \end{aligned}$$

(4.7)

Using the fact that $\beta >C(1+\Vert {{\bar{b}}}\Vert _{W^{1, \infty }}+\Vert \bar{q}\Vert ^{1/2}_{W^{1, \infty }})$, we have

$$\begin{aligned} \left\| \sqrt{|\phi ^{''}|}\big (\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) v \right\| _\phi ^2&\le C\beta ^4 \left\| \sqrt{|\phi ^{''}|} e^t v \right\| _\phi ^2 \nonumber \\&+ C\beta ^2 \left\| \sqrt{|\phi ^{''}|} v \right\| _\phi ^2. \end{aligned}$$

(4.8)

Since t is close to negative infinity and then $\phi '$ is close to 1, from (4.7) and (4.8), we obtain that

$$\begin{aligned} {\mathcal {K}}_2\ge C\beta ^4 \left\| \sqrt{|\phi ^{''}|} v \right\| _\phi ^2, \end{aligned}$$

(4.9)

where we also used the fact that $\phi '$ is close to 1. We derive a lower bound for ${\mathcal {K}}_3$. Integration by parts shows that

$$\begin{aligned} {\mathcal {K}}_3&=-2\int |\phi ^{''}||\partial _t v|^2 \big (\beta ^2\phi '^2+ \beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) \phi '^{-3} \sqrt{\gamma } \ dtd\omega \nonumber \\&-2 \int \partial _t v v \partial _t\big [|\phi ^{''}|\big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) \phi '^{-3} \sqrt{\gamma } \big ]\ dtd\omega \nonumber \\&-2\int |\phi ^{''}| |D_\omega v|^2 \big (\beta ^2\phi '^2+\beta \phi '{{{\bar{b}}}}_t e^{2t}+(n-2)\beta \phi '+ e^{2t} {{\bar{q}}} \big ) \phi '^{-3} \sqrt{\gamma } dtd\omega \nonumber \\&-2 \int \beta |\phi ^{''}|\phi ' \gamma ^{ij}\partial _i v \partial _j{{{\bar{b}}}}_t e^{2t} \phi '^{-3} \sqrt{\gamma } dtd\omega \nonumber \\&-2 \int |\phi ^{''}| \gamma ^{ij}\partial _i v\partial _j {\bar{q}} e^{2t} v \phi '^{-3} \sqrt{\gamma } dtd\omega . \end{aligned}$$

(4.10)

By the Cauchy-Schwartz inequality and the condition that $\beta >C(1+\Vert {{\bar{b}}}\Vert _{W^{1, \infty }}+\Vert \bar{q}\Vert ^{1/2}_{W^{1, \infty }})$, we arrive at

$$\begin{aligned} {\mathcal {K}}_3\ge -C\beta ^2 \int |\phi ^{''}|( |\partial _t v|^2+|D_\omega v|^2+v^2) \phi '^{-3} \sqrt{\gamma } dtd\omega . \end{aligned}$$

(4.11)

Since ${\mathcal {K}}_1$ is nonnegative, the combination of (4.6), (4.9) and (4.11) yields that

$$\begin{aligned} {\mathcal {A}}^{''}_1&\ge C\beta ^4 \left\| \sqrt{|\phi ^{''}|} v \right\| _\phi ^2- C\beta ^2 \left\| \sqrt{|\phi ^{''}|} \partial _t v \right\| _\phi ^2 \nonumber \\&-C\beta ^2 \left\| \sqrt{|\phi ^{''}|} |D_\omega v| \right\| _\phi ^2. \end{aligned}$$

(4.12)

From (4.5), it follows that

$$\begin{aligned} {\mathcal {A}}^{'}_1&\ge C{\hat{\alpha }}\beta ^3 \left\| \sqrt{|\phi ^{''}|} v \right\| _\phi ^2- C{\hat{\alpha }}\beta \left\| \sqrt{|\phi ^{''}|} \partial _t v \right\| _\phi ^2 \nonumber \\&-C{\hat{\alpha }} \beta \left\| \sqrt{|\phi ^{''}|} |D_\omega v| \right\| _\phi ^2. \end{aligned}$$

(4.13)

Recall that

$$\begin{aligned} {\mathcal {A}}'_2=\Vert \big (2\beta \phi '+e^{2t} {\bar{b}}_t\big )\partial _t v+e^{2t}{\bar{b}}_i\partial _i v\Vert ^2_{\phi }. \end{aligned}$$

By the triangle inequality, one has

$$\begin{aligned} {\mathcal {A}}'_2\ge 2\beta ^2 \Vert \phi '\partial _t v\Vert ^2_{\phi }- \Vert e^{2t} {\bar{b}}_t \partial _t v+e^{2t}{\bar{b}}_i\partial _i v\Vert ^2_{\phi }. \end{aligned}$$

It is obvious that

$$\begin{aligned} {\mathcal {A}}'_2\ge \frac{1}{\beta }{\mathcal {A}}'_2. \end{aligned}$$

From the assumption that $\beta >C(1+\Vert {{\bar{b}}}\Vert _{W^{1, \infty }}+\Vert \bar{q}\Vert ^{1/2}_{W^{1, \infty }})$, we obtain that

$$\begin{aligned} {\mathcal {A}}'_2&\ge C\beta \Vert \phi '\partial _t v\Vert ^2_{\phi } - C\beta \Vert e^{t}\partial _t v\Vert ^2_{\phi } - C\beta \Vert e^{t}|D_\omega v|\Vert ^2_{\phi }\nonumber \\&\ge C\beta \Vert \phi '\partial _t v\Vert ^2_{\phi }-C\beta \Vert e^{t}|D_\omega v|\Vert ^2_{\phi }. \end{aligned}$$

(4.14)

For the inner product ${\mathcal {A}}'_3$, using the arguments of integration by parts, since $e^t \ll 1$ as $t<T_0$ and $|T_0|$ is large enough, we can show a lower bound of ${\mathcal {A}}'_3$,

$$\begin{aligned} {\mathcal {A}}'_3&\ge C\beta \left\| \sqrt{|\phi ^{''}|} |D_\omega v| \right\| _\phi ^2- C\beta ^3 \left\| e^t v\right\| _\phi ^2 -C\beta \left\| \sqrt{|\phi ^{''}|}\partial _t v \right\| _\phi ^2 \nonumber \\&-C\beta ^2\left\| \sqrt{|\phi ^{''}|} v \right\| _\phi ^2. \end{aligned}$$

(4.15)

Recall that ${\mathcal {A}}={\mathcal {A}}'_1+{\mathcal {A}}'_2 +{\mathcal {A}}'_3$. From (4.13), (4.14) and (4.15), it follows that

$$\begin{aligned} {\mathcal {A}}&\ge C{\hat{\alpha }} \beta ^3 \int |\phi ^{''}|v^2 \phi '^{-3} \sqrt{\gamma } dtd\omega +C\beta \int |\partial _t v|^2 \phi '^{-3} \sqrt{\gamma } \nonumber \\&+ C\beta \int |\phi ^{''}| |D_\omega v|^2 \phi '^{-3} \sqrt{\gamma } dtd\omega - C\beta ^2 \int |\phi ^{''}| v^2 \phi '^{-3} \sqrt{\gamma } dtd\omega \nonumber \\&-C\beta ^3 \int e^{2t} v^2 \phi '^{-3} \sqrt{\gamma } dtd\omega - C\beta \int |\phi ^{''}| |\partial _t v|^2 \phi '^{-3} \sqrt{\gamma } dtd\omega \nonumber \\&-C{\hat{\alpha }}\beta \int |\phi ^{''}| |D_\omega v|^2 \phi '^{-3} \sqrt{\gamma } dtd\omega - C\beta \int e^{2t} |D_\omega v|^2 \phi '^{-3} \sqrt{\gamma } dtd\omega . \end{aligned}$$

(4.16)

If we choose ${\hat{\alpha }}$ to be appropriately small and take the fact $|\phi ^{''}|>e^t$ into account, we obtain that

$$\begin{aligned} C{\mathcal {A}}\ge & {} \beta ^3\int |\phi ''||v|^2 \phi '^{-3}\sqrt{\gamma }\,dt d \omega +\beta \int |\phi ''||D_\omega v|^2 \phi '^{-3}\sqrt{\gamma }\,dt d \omega \nonumber \\&+\beta \int |\partial _t v|^2 \phi '^{-3}\sqrt{\gamma }\,dt d \omega . \end{aligned}$$

(4.17)

Now we show ${\mathcal {B}}$ can be absorbed into ${\mathcal {A}}$ for large $|T_0|$ and large $\beta$. Since

$$\begin{aligned} |\partial _t \ln \sqrt{\gamma }|\le C e^t\le |\phi ^{''}|, \end{aligned}$$

then

$$\begin{aligned} {\mathcal {B}}&=\Vert \beta \phi ''v+\beta \partial _t \ln \sqrt{\gamma } \phi ' v+(n-2)\partial _t v+\partial _t \ln \sqrt{\gamma }\partial _t v\Vert ^2_{\phi } \nonumber \\&\le \beta ^2 \int |\phi ''| v^2 \phi '^{-3} \sqrt{\gamma } dt d \omega + C\int |\partial _t v|^2 e^{2t} \phi '^{-3} \sqrt{\gamma } dt d \omega . \end{aligned}$$

(4.18)

Thus, the right hand side of (4.18) can be incorporated by the right hand side of (4.17). Hence the proof of the lemma is arrived. $\square$

Proof of Lemma 1

If we recall that $u=e^{-\beta \varphi (x)} v$, the proof of Lemma 3 just implies Lemma 1 stated in Sect. 2. $\square$