Sharp bounds for domain perturbations of Dirichlet Laplacians defined on smooth domains

Abstract

We obtain sharp quantitative bounds which describe how an eigenspace of the Dirichlet Laplacian defined on a smooth domain changes when the smooth domain is replaced by a slightly smaller domain.

Appendix

In [11], $H \geqslant 0$ is a second order uniformly elliptic operator in divergence form defined on a domain $\Omega \subseteq \mathbb {R}^N, N \geqslant 3,$ with Dirichlet boundary conditions. Let $E_1 > 0$ be the ground state eigenvalue of $H$, $\phi _1 > 0$ be the ground state eigenfunction of $H$, normalized so that $\left\| \phi _1\right\| _{L^2(\Omega )} = 1$, and let $K(t, x, y)$ be the heat kernel of $e^{-Ht}$. It is assumed in [11] that (i) there exist $0 < \lambda \leqslant \Lambda < \infty$ such that

$$\begin{aligned} -\lambda \Delta _{\Omega } \leqslant H \leqslant -\Lambda \Delta _{\Omega }, \end{aligned}$$

where $-\Delta _{\Omega } \geqslant 0$ denotes the Dirichlet Laplacian defined on $\Omega$, that (ii) there exists $c_1 = c_1(\Omega ) \geqslant 1$ such that $\Omega$ satisfies a strong Hardy’s inequality of the form

$$\begin{aligned} \int _{\Omega } \frac{\left| f (x) \right| ^2}{{d_{\Omega }(x)}^2} \,\, dx \le c_1 \int _{\Omega } \left| \nabla f\right| ^2 \,\,dx \qquad \qquad (f \in C^{\infty }_{c}(\Omega )), \end{aligned}$$

and that (iii) there exist $a \geqslant 1$ and $b > 0$ such that $\phi _1$ satisfies an inequality of the form

$$\begin{aligned} \phi _1(x) \geqslant b{d_{\Omega }(x)}^a \qquad \qquad (x \in \Omega ). \end{aligned}$$

Then [11, Theorem 16] asserts that there exists $c_{2}=c_{2}(\Omega ) \geqslant 1$ such that if we put

$$\begin{aligned} \Gamma =&\,\,\Gamma (\Omega , a, b, \lambda , \Lambda , E_1)\\ =&\,\, \max \left\{ c_{2} e^{c_{2} a\log a}\lambda ^{-(\frac{N}{2}+a)}b^{-2}e^{E_1}, (c_{2}e^{c_{2} a\log a} \lambda ^{-(\frac{N}{2}+a)}2^{\frac{N}{2}+a}b^{-2})^2\right. \\&\,\, \left. \times \exp \left\{ E_1 + \frac{diam(\Omega )^2}{8\Lambda }\right\} , 1\right\} \\ \geqslant&\,\,1, \end{aligned}$$

then we have

$$\begin{aligned} 0 \leqslant&\,\,K(t,x,y)\nonumber \\ \leqslant&\,\,\Gamma \max \{t^{-(\frac{N}{2}+a)}, 1\}e^{-E_1 t} \exp \left\{ {-\frac{\left| {x-y}\right| ^2}{8\Lambda t}}\right\} \phi _1(x)\phi _1(y) \end{aligned}$$

(145)

for all $t>0$ and all $x,y\in \Omega .$

In the main body of this paper if $\Omega$ is a bounded $C^{1,1}$ domain in $\mathbb {R}^N$, $N \geqslant 3$, and $H=-\Delta _{\Omega },$ then it is well known that the assumptions in [11] are satisfied and we can put $\lambda =\Lambda =a=1$ in (145). Noting in this case that $b$ and $E_1 = \mu _1[\Omega ]$ both depend only on $\Omega$, we get the upper heat kernel bound, in the notations of the main body of this paper,

$$\begin{aligned} K_{\Omega }(t, x, y) \leqslant c(\Omega ) \max \{t^{-(\frac{N}{2}+1)}, 1\}e^{-\mu _1[\Omega ] t} \exp \left\{ {-\frac{\left| {x-y}\right| ^2}{8 t}}\right\} \varphi _1[\Omega ](x)\varphi _1[\Omega ](y) \end{aligned}$$

(146)

for some $c(\Omega )\geqslant 1$. Clearly (146) implies that for all $0 < \delta < 1$ we have

$$\begin{aligned} K_{\Omega }(t, x, y)\leqslant c(\Omega , \delta ) t^{-(\frac{N}{2}+1)}e^{-\mu _1[\Omega ] (1-\delta )t} \exp \left\{ {-\frac{\left| {x-y}\right| ^2}{8 t}}\right\} \varphi _1[\Omega ](x)\varphi _1[\Omega ](y) \end{aligned}$$

(147)

for some $c(\Omega , \delta ) \geqslant 1$. If we fix $\delta = 1/3$, then we get the bound (2) in this paper.

Suppose next that $\Omega$ is a $C^{1, 1}$ bounded domain in $\mathbb {R}^2$, then, as mentioned in [6], we consider $\Omega \times \Omega \subseteq \mathbb {R}^4$. Recall that a domain $D \subseteq \mathbb {R}^N$, $N \geqslant 2,$ is said to satisfy a uniform external ball condition with parameters $\alpha > 0$ and $\beta > 0$ if for any $y \in \partial \Omega$ and $0 < s < \beta$ there exists a ball $B_{p, r}$ with center $p$ satisfying $\left| p - y\right| \leqslant s$, and radius $r$ satisfying $r \geqslant 2\alpha s$, which does not meet $D$. This condition holds if $\partial D$ satisfies a uniform Lipschitz condition. It is easy to check that if $D \subseteq \mathbb {R}^N, N \geqslant 2,$ satisfies a uniform exterior ball condition with parameters $\alpha$ and $\beta$, then $D \times D \subseteq \mathbb {R}^{2N}$ satisfies a uniform exterior ball condition with parameter $\alpha /\sqrt{2}$ and $\beta$. By [5, Theorem 1.5.5.], a domain $D\subseteq \mathbb {R}^N, N \geqslant 2,$ that satisfies a uniform external ball condition and has finite inradius, i.e.,

$$\begin{aligned} Inr(D) = sup\{d_{\Omega }(x): x \in D\} < \infty , \end{aligned}$$

automatically satisfies a strong Hardy’s inequality. Applying these to $\Omega$ we see that $\Omega \times \Omega \subseteq \mathbb {R}^4$ satisfies a strong Hardy’s inequality. Moreover, since

$$\begin{aligned}\varphi _1[\Omega \times \Omega ](x_1, x_2) = \varphi _1[\Omega ](x_1)\varphi [\Omega ](x_2) \qquad \qquad (x_1, x_2 \in \Omega ),\end{aligned}$$

we have, by (1),

$$\begin{aligned}\varphi _1[\Omega \times \Omega ](x_1, x_2) \geqslant c_1^{-2}d_{\Omega }(x_1)d_{\Omega }(x_2) \geqslant c_1^{-2}d_{\Omega \times \Omega }(x_1, x_2)^{2}\qquad \qquad (x_1, x_2 \in \Omega ).\end{aligned}$$

for some $c_1=c_1(\Omega ) \geqslant 1.$ So $-\Delta _{\Omega \times \Omega }$ satisfies all the assumptions in [11] and we can apply the bound (145) to it with $\lambda = \Lambda = 1, a=2$, $b=b(\Omega ) = c_1(\Omega )^{-2}$ and $E_1 = E_1(\Omega ) = 2\mu _1[\Omega ] .$ So we get

$$\begin{aligned}&K_{\Omega \times \Omega }(t, (x_1, x_2), (y_1, y_2))\nonumber \\&\quad \leqslant c(\Omega )\max \{t^{-4}, 1\}e^{-2\mu _1[\Omega ] t} \exp \left\{ {-\frac{\left| {x_1-y_1}\right| ^2+\left| {x_2 - y_2}\right| ^2}{8 t}}\right\} \nonumber \\&\quad \times \varphi _1[\Omega \times \Omega ](x_1, x_2)\varphi _1[\Omega \times \Omega ](y_1, y_2)\qquad \qquad (x_1, x_2, y_1, y_2 \in \Omega ) \end{aligned}$$

(148)

for some $c(\Omega ) \geqslant 1$. Since for all $x_1, x_2, y_1, y_2 \in \Omega$ we have

$$\begin{aligned} K_{\Omega \times \Omega }(t, (x_1, x_2), (y_1, y_2)) = K_{\Omega }(t, x_1, y_1)\,K_{\Omega }(t, x_2, y_2), \end{aligned}$$

(149)

(148) implies that (146), and hence (147), hold also if $N = 2$. Hence (2) holds also if $\Omega \subseteq \mathbb {R}^2$.