Spectral Stability of the ${\overline{\partial }}-$Neumann Laplacian: Domain Perturbations

Abstract

We study spectral stability of the ${\bar{\partial }}$-Neumann Laplacian on a bounded domain in ${\mathbb {C}}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the ${\bar{\partial }}$-Neumann Laplacian on bounded pseudoconvex domains in ${\mathbb {C}}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property (P), and quantitative estimates on smooth bounded pseudoconvex domains of finite D’Angelo type in ${\mathbb {C}}^n$.

1 Introduction

The ${\overline{\partial }}$-Neumann Laplacian is a prototype of an elliptic operator with non-coercive boundary conditions ([27]). Since the fundamental work of Kohn [25] and Hörmander [24], it has been known that existence and regularity of the ${\overline{\partial }}$-Neumann Laplacian are closely connected to the boundary geometry of the underlying domains (see, e.g., [1, 8, 12, 16, 34] for expositions on the subject). Spectral behavior of the ${\overline{\partial }}$-Neumann Laplacian has also been shown to be sensitive to the geometry of the domains. Positivity of the ${\overline{\partial }}$-Neumann can be used to characterize pseudoconvexity (see [17, 22] and references therein). Spectral discreteness of the ${\overline{\partial }}$-Neumann Laplacian can be used to determine whether the boundary of a convex domain in ${\mathbb {C}}^n$ contains a complex variety ([19, 20]) and whether the boundary of a smooth bounded pseudoconvex Hartogs domain in ${\mathbb {C}}^2$ satisfies property (P), a potential theoretic property introduced by Catlin [5] (see [9, 21]). Asymptotic behavior of the eigenvalues can be used to establish whether a smooth bounded pseudoconvex domain in ${\mathbb {C}}^2$ is of finite type ([17]).

In physical sciences, exact values of quantities are oftentimes difficult—in some cases, impossible—to obtain and approximate values are observed and utilized instead. It is, thus, important to study how these quantities are affected when there are small perturbations of other parameters. Spectral stability of the classical Dirichlet and Neumann Laplacians on domains in ${\mathbb {R}}^n$ has been studied extensively in literatures (see, e.g., [3, 14, 23] and references therein). In this paper, we study spectral stability of the ${\overline{\partial }}$-Neumann Laplacian on a bounded domain $\Omega$ in ${\mathbb {C}}^n$ when the underlying domain is perturbed. There are several ways to measure spectral stability. Our focus here is on the variational eigenvalues. The kth-variational eigenvalues $\lambda ^q_k(\Omega )$ of the ${\overline{\partial }}$-Neumann Laplacian $\Box$ on (0, q)-forms ($1\le q\le n-1$) on $\Omega$ are defined through the min–max principle and they are bona fide eigenvalues when the spectrum is discrete (see Sect. 2 below). We first establish the following upper semi-continuity property of the variational eigenvalues of the ${\overline{\partial }}$-Neumann Laplacian on a pseudoconvex domain.

Theorem 1.1

Let $\Omega _1$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$ with $C^1$-smooth boundary. Let k be a positive integer. For any $\varepsilon >0$, there exists $\delta >0$ such that for any pseudoconvex domain $\Omega _2$,

$$\begin{aligned} \lambda ^q_k(\Omega _2)\le \lambda ^q_k(\Omega _1)+\varepsilon , \quad 1\le q\le n-1, \end{aligned}$$

(1.1)

provided $d_H(\Omega _1, \Omega _2)<\delta$, where $d_H$ denotes the Hausdorff distance between the domains.

Spectral theory of the ${\overline{\partial }}$-Neumann Laplacian differs substantially from that of the classical Laplacians because of the non-coercive nature of the ${\overline{\partial }}$-Neumann boundary conditions. Unlike the classical Dirichlet or Neumann Laplacian, spectral discreteness of the ${\overline{\partial }}$-Neumann Laplacian on a bounded domain $\Omega$ in ${\mathbb {C}}^n$ depends not only on the smoothness of the boundary, but more importantly on geometric and potential properties of the boundary. One difficulty in studying spectral stability of the ${\overline{\partial }}$-Neumann Laplacian is due to the fact that unlike the classical Neumann Laplacian, the restriction $f\vert _{{\widehat{\Omega }}}$ of a form $f\in {{\,\mathrm{Dom}\,}}(Q_\Omega )$, the domain of definition of the quadratic form associated with the ${\overline{\partial }}$-Neuman Laplacian on $\Omega$, need not belong to ${{\,\mathrm{Dom}\,}}(Q_{{\widehat{\Omega }}})$, where ${\widehat{\Omega }}$ is a subdomain of $\Omega$. Additionally, unlike the Dirichlet Laplacian, the extension of f to zero outside of $\Omega$ does not make it belong to ${{\,\mathrm{Dom}\,}}(Q_{{\widetilde{\Omega }}})$ for a larger domain ${\widetilde{\Omega }}$. To overcome these difficulties, we decompose a form in ${{\,\mathrm{Dom}\,}}(Q_\Omega )$ into tangential and normal components and treat them separately. Roughly speaking, the tangential component is treated as in the case of the Neumann Laplacian and the normal component is treated as in the case of the Dirichlet Laplacian.

To establish the lower semi-continuity property of the variational eigenvalues, we will have to assume that the targeted domain satisfies property (P). Property (P) is a potential theoretic property introduced by Catlin [5] to study compactness in the ${\overline{\partial }}$-Neumann problem. Kohn and Nirenberg [27] showed that compactness of the ${\overline{\partial }}$-Green operator, the inverse of ${\overline{\partial }}$-Neuman Laplacian, implies exact global regularity of the ${\overline{\partial }}$-Neumann Laplacian. (Compactness of the ${\overline{\partial }}$-Green operator is equivalent to spectral discreteness of the ${\overline{\partial }}$-Neumann Laplacian.) Catlin showed that for a bounded pseudoconvex domain with smooth boundary in ${\mathbb {C}}^n$, property (P) implies compactness of the ${\overline{\partial }}$-Green operator. Straube showed that Catlin’s theorem holds without the boundary smoothness assumption ([33]). It remains an open problem whether or not the converse to Catlin’s theorem is also true.

Theorem 1.2

Let $\Omega _1$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$ with $C^1$-smooth boundary that satisfies Property $(P_{q-1})$, $2\le q\le n-1$. Let k be a positive integer. For any $\varepsilon >0$, there exists $\delta >0$ such that for any pseudoconvex domain $\Omega _2$ whose ${\overline{\partial }}$-Neumann Laplacian has discrete spectrum on (0, q)-forms,

$$\begin{aligned} \lambda ^q_k(\Omega _2)\ge \lambda ^q_k(\Omega _1)-\varepsilon \end{aligned}$$

(1.2)

provided $d_H(\Omega _1, \Omega _2)<\delta$.

To establish the quantitative estimates, we further assume that the domains are of finite type. A notion of finite type was introduced by Kohn for smooth bounded pseudoconvex domains in ${\mathbb {C}}^2$ in connection with subellipticity of the ${\overline{\partial }}$-Neumann Laplacian [26]. For domains in higher dimensions, a new finite-type notion was introduced by D’Angelo [10]: A smooth bounded domain in ${\mathbb {C}}^n$ is of finite type in the sense of D’Angelo if the normalized order of contact of complex analytic varieties with the boundary is finite. Catlin showed that for a smooth bounded pseudoconvex domain in ${\mathbb {C}}^n$, subellipticity of the ${\overline{\partial }}$-Neumann Laplacian is equivalent to the finite D’Angelo type condition [4, 7]. Here we study spectral stability of the ${\overline{\partial }}$-Neumann Laplacian on such domains. Our main result in this regard is as follows:

Theorem 1.3

Let $\Omega$ be a smooth bounded pseudoconvex domain of finite $D_q$-type in ${\mathbb {C}}^n$. Let $\Omega _j$ be a family of smooth bounded pseudoconvex domains of uniform finite $D_q$-type in ${\mathbb {C}}^n$, $1\le q\le n-1$. Let k be a positive integer. Then there exist constants $\delta >0$ and $C_k>0$ such that

$$\begin{aligned} |\lambda ^q_k(\Omega _j)-\lambda ^q_k(\Omega )|\le C_k \delta _j, \end{aligned}$$

(1.3)

provided $\delta _j=d_H(\Omega , \Omega _j)<\delta$.

We refer the reader to Sect. 5 for precise definition of uniform finite type. Our analysis is based on Catlin’s construction of bounded plurisubharmonic functions with large complex Hessians. We will also use a version of sharp Hardy inequality due to Brezis and Marcus [2] and an idea from Davies [14].

This paper is organized as follows. In Sect. 2, we recall the spectral theoretic setup of the ${\overline{\partial }}$-Neumann Laplacian and relevant facts regarding the variational eigenvalues. In Sect. 3, we establish upper semi-continuity property for the variational eigenvalues of the ${\overline{\partial }}$-Neumann Laplacian on bounded pseudoconvex domains in ${\mathbb {C}}^n$ and prove Theorem 1.1. In Sect. 4, we study lower semi-continuity of the variational eigenvalues on bounded pseudoconvex domains satisfying property (P) and establish Theorem 1.2. In Sect. 5, we obtain quantitative estimates, including Theorem 1.3, for stability of the variational eigenvalues on pseudoconvex domains of finite type. Section 6 contains further results on convergence of the ${\overline{\partial }}$-Neumann Laplacian in resolvent sense.

2 Preliminary

We first review relevant elements in general spectral theory. Let Q be a non-negative, densely defined, and closed sesquilinear form on a complex Hilbert space ${\mathbb {H}}$ with domain ${{\,\mathrm{Dom}\,}}(Q)$. Then Q uniquely determines a non-negative self-adjoint operator S such that ${{\,\mathrm{Dom}\,}}(S^{1/2})={{\,\mathrm{Dom}\,}}(Q)$ and

$$\begin{aligned} Q(u, v)=\langle S^{1/2}u, \; S^{1/2}v\rangle \end{aligned}$$

for all $u, v\in {{\,\mathrm{Dom}\,}}(Q)$. Furthermore,

$$\begin{aligned} {{\,\mathrm{Dom}\,}}(S)=\{ u\in {{\,\mathrm{Dom}\,}}(Q) \mid \exists f\in {\mathbb {H}}, Q(u, v)=\langle f, v\rangle , \forall v\in {{\,\mathrm{Dom}\,}}(Q) \}. \end{aligned}$$

(See, e.g., Theorem 4.4.2 in [13].) For any subspace $L\subset {{\,\mathrm{Dom}\,}}(Q)$, let

$$\begin{aligned} \lambda _Q(L)=\sup \{Q(u, u) \mid u\in L, \Vert u\Vert =1\}. \end{aligned}$$

For any positive integer k, let

$$\begin{aligned} \lambda _{k} (S)=\inf \{\lambda _Q(L) \mid L\subset {{\,\mathrm{Dom}\,}}(Q), \dim (L)=k\} \end{aligned}$$

(2.1)

be the kth variational eigenvalues of S. The resolvent set $\rho (S)$ of the operator S consists of all $\lambda \in {\mathbb {C}}$ such that $S-\lambda I:{{\,\mathrm{Dom}\,}}(S)\rightarrow {\mathbb {H}}$ is both one-to-one and onto. It follows from the closed graph theorem that this operator has a bounded inverse, the resolvent operator $R_\lambda (S)=(S-\lambda I)^{-1}:{\mathbb {H}}\rightarrow {{\,\mathrm{Dom}\,}}(S)$. The spectrum $\sigma (S)$ is the complement of $\rho (S)$ in ${\mathbb {C}}$. It is a non-empty closed subset of $[0, \ \infty )$. The lowest point in the spectrum is $\lambda _1(S)$. The essential spectrum $\sigma _e(S)$ is the closed subset of $\sigma (S)$ that consists of isolated eigenvalues of infinite multiplicity and accumulation points of the spectrum. The bottom of the essential spectrum, $\inf \sigma _e(S)$, is the limit of $\lambda _k(S)$ as $k\rightarrow \infty$. The essential spectrum $\sigma _e(S)$ is empty if and only if $\lambda _{k}(S)\rightarrow \infty$ as $k\rightarrow \infty$. In this case, the variational eigenvalue $\lambda _{k}(S)$ is a bona fide eigenvalue of S. Indeed, it is the kth eigenvalue when the eigenvalues are arranged in increasing order and repeated according to multiplicity. One approach to measuring spectral stability of a self-adjoint operator is through study of how the variational eigenvalues vary as the operator is perturbed. The following simple lemma is well known (compare [3, Theorem 3.2]):

Lemma 2.1

Let $S_i$, $i=1, 2$, be non-negative self-adjoint operators on Hilbert spaces ${\mathbb {H}}_i$ with associated quadratic forms $Q_i$. Let $T:{{\,\mathrm{Dom}\,}}(Q_1)\rightarrow {{\,\mathrm{Dom}\,}}(Q_2)$ be a linear transformation from the domain of $Q_1$ to that of $Q_2$. Let k be a positive integer. Suppose there exist $0<\alpha _k<1/(2k)$ and $\beta _k>0$ such that for any orthonormal set $\{f_1, f_2, \ldots , f_k\}\subset {{\,\mathrm{Dom}\,}}(Q_1)$,

$$\begin{aligned} |\langle Tf_h, Tf_l\rangle _{2}{-}\delta _{hl}|\le \alpha _k \quad \text {and}\quad |Q_{2}(T f_h, T f_l){-}Q_{1}(f_h,f_l)|{\le } \beta _k, \quad 1\le h, l\le k.\nonumber \\ \end{aligned}$$

(2.2)

Then

$$\begin{aligned} \lambda _k(S_2)\le \lambda _k(S_1)+2 k(\alpha _k\lambda _k(S_1)+\beta _k). \end{aligned}$$

(2.3)

Proof

Let $L_k$ be any k-dimensional linear subspace of ${{\,\mathrm{Dom}\,}}(Q_1)$. Let $\{f_1, f_2, \ldots , f_k\}$ be an orthonormal basis for $L_k$. For $f=\sum _{j=1}^k a_j f_j\in L_k$, it follows from the Cauchy-Schwarz inequality that

$$\begin{aligned} |Q_2(Tf, Tf)-Q_1(f, f)|^2&=\left| \sum _{h,l=1}^{k}\left( Q_2(T f_{h},T f_{l}) -Q_1(f_{h},f_{l})\right) a_h\overline{a_l}\right| ^2\\&\le \sum _{h,l=1}^{k}\left| Q_{2}(T f_{h},Tf_{l})-Q_{1}(f_{h},f_{l})\right| ^2\sum _{h,l=1}^{k}|a_h\overline{a_l}|^2\\&\le k^2\beta _k^2 \Vert f\Vert _{1}^4. \end{aligned}$$

Thus

$$\begin{aligned} Q_2(Tf,Tf)\le Q_1(f, f)+k\beta _k \Vert f\Vert _{1}^2. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert Tf\Vert ^2_2\ge \Vert f\Vert ^2_1-k\alpha _k \Vert f\Vert _1^2>\frac{1}{2} \Vert f\Vert _1^2. \end{aligned}$$

Therefore, T is one-to-one on $L_k$ and $T(L_k)$ is a k-dimensional linear subspace of ${{\,\mathrm{Dom}\,}}(Q_2)$. It follows from (2.1) that

$$\begin{aligned} \begin{aligned} \lambda _k(S_2)&\le \sup \bigg \{\frac{Q_2(Tf, Tf)}{\Vert T f\Vert ^2_{2}} \mid f\in L_{k}\bigg \}\le \sup \bigg \{\frac{Q_1(f, f)+k\beta _k \Vert f\Vert _1^2}{(1-k\alpha _k)\Vert f\Vert _1^2} \mid f\in L_k\bigg \}\\&\le \dfrac{1}{1-k\alpha _k}\lambda _{Q_1}(L_k)+ \dfrac{k\beta _k}{1-k\alpha _k}=\lambda _{Q_1}(L_k)+\frac{k(\alpha _k\lambda _{Q_1}(L_k)+\beta _k)}{1-k\alpha _k} \\&\le \lambda _{Q_1}(L_k)+2k(\alpha _k\lambda _{Q_1}(L_k)+\beta _k). \end{aligned} \end{aligned}$$

(2.4)

Taking the infimum over all k-dimensional subspace $L_k$ in ${{\,\mathrm{Dom}\,}}(Q_1)$, we then obtain the desired inequality (2.3). $\square$

Remark 2.2

Condition (2.2) in Lemma 2.1 can be replaced by the following: For any k-dimensional subspace $L_k$ of ${\text {Dom}}\,(Q_1)$ and $f\in L_k$,

$$\begin{aligned} \Vert Tf\Vert ^2_2\ge (1-k\alpha _k) \Vert f\Vert _1^2 \quad {\text{ a }nd} \quad Q_2(Tf,Tf)\le Q_1(f, f)+k\beta _k \Vert f\Vert _1^2. \end{aligned}$$

(2.5)

This is easily seen from the proof above.

We now recall a spectral theoretic setup for the ${\bar{\partial }}$-Neumann Laplacian. (We refer the readers to [8, 16, 34] for an in-depth treatment on regularity theory of the ${\overline{\partial }}$-Neumann Laplacian.) Let $L^2_{(0, q)}(\Omega )$ be the space of (0, q)-forms with $L^2$-coefficients on $\Omega$ with respect to the standard Euclidean metric. Let ${\overline{\partial }}_q:L^2_{(0, q)}(\Omega )\rightarrow L^2_{(0, q+1)}(\Omega )$ be the maximally defined Cauchy-Riemann operator. Thus, ${{\,\mathrm{Dom}\,}}({\overline{\partial }}_q)$, the domain of ${\overline{\partial }}_q$, consists of those $u\in L^2_{(0, q)}(\Omega )$ such that ${\overline{\partial }}_q u$, defined in the sense of distribution, is in $L^2_{(0, q+1)}(\Omega )$. That is, there exists $v\in L^2_{(0, q+1)}(\Omega )$ such that

$$\begin{aligned} \langle u, \vartheta \varphi \rangle =\langle v, \varphi \rangle \end{aligned}$$

for all $\varphi \in {\mathcal {D}}_{(0, q+1)}(\Omega )$, where $\vartheta$ is the formal adjoint of ${\overline{\partial }}_q$ and ${\mathcal {D}}_{(0, q+1)}(\Omega )$ is the space of smooth $(0, q+1)$-forms with compact support in $\Omega$. Let ${\overline{\partial }}^*_q:L^2_{(0, q+1)}(\Omega )\rightarrow L^2_{(0, q)}(\Omega )$ be the adjoint of ${\overline{\partial }}_q$. Thus its domain is given by

$$\begin{aligned} {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_q)=\big \{u\in L^2_{(0, q+1)}(\Omega ) \mid \exists C>0, |\langle u, {\overline{\partial }}_q v\rangle |\le C\Vert v\Vert ,\ \forall v\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}_q)\big \}.\nonumber \\ \end{aligned}$$

(2.6)

The maximally defined ${\overline{\partial }}_q$-operator can be regarded as the adjoint of the formal adjoint $\vartheta _q:L^2_{(0, q+1)}(\Omega )\rightarrow L^2_{(0, q)}(\Omega )$ whose domain ${{\,\mathrm{Dom}\,}}(\vartheta _q)={\mathcal {D}}_{(0, q+1)}(\Omega )$. The ${\overline{\partial }}^*_q$-operator is then the closure of $\vartheta _q$ and it is sometimes referred to as the minimal extension of $\vartheta _q$. Let $\Omega =\{z\in {\mathbb {C}}^n \mid \rho (z)<0\}$ be a bounded domain with a $C^1$-smooth defining function $\rho$ such that $|\nabla \rho |=1$ on $\partial \Omega$ and let

Then $u\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{q-1})$ if and only if

on $\partial \Omega$, where

$$\begin{aligned} ({\overline{\partial }}\rho )^*=\sum _{j=1}^n \frac{\partial \rho }{\partial z_j}\frac{\partial }{\partial {\bar{z}}_j} \end{aligned}$$

is the dual (0, 1)-vector field of ${\overline{\partial }}\rho$ and $\lrcorner$ denotes the contraction operator.

For $1\le q\le n-1$, let

$$\begin{aligned} Q_q(u, v)=\langle {\overline{\partial }}_q u, {\overline{\partial }}_q v\rangle _\Omega +\langle {\overline{\partial }}^*_{q-1} u, {\overline{\partial }}^*_{q-1} v\rangle _\Omega \end{aligned}$$

be the sesquilinear form on $L^2_{(0, q)}(\Omega )$ with domain ${{\,\mathrm{Dom}\,}}(Q_{q})={{\,\mathrm{Dom}\,}}({\overline{\partial }}_q)\cap {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{q-1})$. The self-adjoint operator $\square _{q}$ associated with $Q_{q}$ is called the ${\overline{\partial }}$-Neumann Laplacian on $L^2_{(0, q)}(\Omega )$. Consequently, $\square _q$ is given by

$$\begin{aligned} \square _q={\overline{\partial }}_{q-1}{\overline{\partial }}^*_{q-1}+{\overline{\partial }}^*_q{\overline{\partial }}_q \end{aligned}$$

with

$$\begin{aligned} {{\,\mathrm{Dom}\,}}(\square _q)=\{u\in L^2_{(0, q)}(\Omega ) \mid u\in {{\,\mathrm{Dom}\,}}(Q_q), {\overline{\partial }}_q u\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{q}), {\overline{\partial }}_{q-1}^* u\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}_{q-1})\}. \end{aligned}$$

It is an elliptic operator with non-coercive boundary conditions [27]. We will use $\lambda ^q_k(\Omega )$ to denote the kth-variational eigenvalue of the ${\overline{\partial }}$-Neumann Laplacian $\Box$ on (0, q)-forms on $\Omega$, defined as above by

$$\begin{aligned} \lambda ^q_k(\Omega )=\inf _{\begin{array}{c} L\subset {{\,\mathrm{Dom}\,}}(Q_q)\\ \dim L=k \end{array}}\sup \limits _{u\in L\setminus \{0\}}\,Q_q(u,u)/\Vert u\Vert ^2, \end{aligned}$$

(2.7)

where the infimum takes over all linear subspace of ${{\,\mathrm{Dom}\,}}(Q_q)$ of dimension k. We will study spectral stability of the ${\overline{\partial }}$-Neumann Laplacian as the underlying domain $\Omega$ is perturbed. There are several ways to study spectral stability of the ${\overline{\partial }}$-Neumann Laplacian. In this paper, we will focus on stability of the variational eigenvalues and the convergence in resolvent sense. Let $T_j$ and T be self-adjoint operators on a Hilbert space ${{\mathbb {H}}}$. We say $T_j$ converges to T in norm (respectively strong) resolvent sense if for all $\lambda \in {{\mathbb {C}}}\setminus {{\mathbb {R}}}$, the resolvent operator $R_\lambda (T_j)=(T_j-\lambda I)^{-1}$ converges to $R_\lambda (T)= (T-\lambda I)^{-1}$ in norm (strongly). It is well known that if $T_j$ converges to T in norm resolvent sense, then for any $\lambda \not \in \sigma (T)$, $\lambda \not \in \sigma (T_j)$ for sufficiently large j, and if $T_j$ converges to T in strong resolvent sense, then for any $\lambda \in \sigma (T)$, there exist $\lambda _j\in \sigma (T_j)$ so that $\lambda _j\rightarrow \lambda$. We refer the reader to [31, §VIII.7] for relevant material.

Perturbation of the domains will be measured by the Hausdorff distance. Recall that for two sets A and B in a metric space (X, d), the Hausdorff distance between A and B is given by

$$\begin{aligned} {\tilde{d}}_H(A,B)=\max \left\{ \sup \limits _{x\in A}\inf \limits _{y\in B}d(x,y), \ \sup \limits _{y\in B}\inf \limits _{x\in A}d(x,y)\right\} . \end{aligned}$$

In this paper, we will measure the closeness between two domains $\Omega _1$ and $\Omega _2$ in ${\mathbb {C}}^n$ by the Hausdorff distance between them and their complements using the Euclidean metric. We set

$$\begin{aligned} d_H(\Omega _1, \Omega _2)=\max \{{\tilde{d}}_H(\Omega _1, \Omega _2), \ {\tilde{d}}_H(\Omega ^c_1, \Omega ^c_2)\}. \end{aligned}$$

For $\delta >0$, let

$$\begin{aligned} \Omega ^-_\delta =\{z\in \Omega \mid {{\,\mathrm{dist}\,}}(z, \Omega ^c)>\delta \} \quad \text {and}\quad \Omega ^+_\delta =\{z\in {\mathbb {C}}^n \mid {{\,\mathrm{dist}\,}}(z, \Omega )<\delta \}. \end{aligned}$$

It is easy to see that $d_H(\Omega _1, \Omega _2)<\delta$ if and only if

$$\begin{aligned} \overline{(\Omega _2)^-_\delta }\subset \Omega _1\subset (\Omega _2)^+_\delta \quad \text {and}\quad \overline{(\Omega _1)^-_\delta }\subset \Omega _2\subset (\Omega _1)^+_\delta . \end{aligned}$$

3 Upper Semi-continuity

In this section, we establish several upper semi-continuity properties for the variational eigenvalues of the ${\overline{\partial }}$-Neumann Laplacian when the underlying domain is perturbed. We first study spectral stability of the ${\overline{\partial }}$-Neumann Laplacian when the underlying domain is exhausted by subdomains from inside.

We will use $Q_{q, \Omega }$ to denote the quadratic form associated with the ${\bar{\partial }}$-Neumann Laplacian $\Box _{q, \Omega }$ acting on (0, q)-forms on $\Omega$. Let $\Omega _2\subset \Omega _1$ be bounded pseudoconvex domains in ${\mathbb {C}}^n$. Unlike the classical Neumann Laplacian, for a (0, q)-form $f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega _1})$, its restriction to $\Omega _2$ is no longer in ${{\,\mathrm{Dom}\,}}(Q_{q, \Omega _2})$. The following regularization procedure was introduced by Straube [33] (compare also [30]) to overcome this difficulty: For $f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega _1})$, we define

$$\begin{aligned} T f={\overline{\partial }}^*_{q, \Omega _2} N_{q+1, \Omega _2}({\overline{\partial }}_{q, \Omega _1} f)\big \vert _{\Omega _2}+{\overline{\partial }}_{q-1, \Omega _2} N_{q-1, \Omega _2}({\overline{\partial }}^*_{q-1, \Omega _1} f)\big \vert _{\Omega _2}. \end{aligned}$$

(3.1)

When $q=1$, $N_{0, \Omega _2}$ is the inverse of the restriction of $\Box _{0, \Omega _2}$ to the orthogonal complement $\ker ({\overline{\partial }}_{0, \Omega _2})^\perp ={{\mathcal {R}}}\,({\overline{\partial }}^*_{0, \Omega _2})$ of the kernel of ${\overline{\partial }}_{0, \Omega _2}$ such that $\Box _{0, \Omega _2}N_{0, \Omega _2}=I-P_{0, \Omega _2}$, where $P_{0, \Omega _2}$ is the Bergman projection on $\Omega _2$ (see [8, Theorem 4.4.3]). Hereafter, for economy of notations, we will suppress the subscripts involving q when doing this causes no confusion and instead use subscript 1 and 2 to indicate that the operators act on $\Omega _1$ and $\Omega _2$, respectively. Evidently, T is a linear transformation from ${{\,\mathrm{Dom}\,}}(Q_1)$ into ${{\,\mathrm{Dom}\,}}(Q_2)$. In light of Lemma 2.1, in order to estimate the difference between variational eigenvalues on $\Omega _1$ and $\Omega _2$, we need to compare f and Tf.

Lemma 3.1

Let $\Omega _1$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$ and let $f\in {{\,\mathrm{Dom}\,}}(Q_1)$. For any $\varepsilon >0$, there exists $\delta >0$ such that for any pseudoconvex domain $\Omega _2\subset \Omega _1$ with $d_H(\Omega _1, \Omega _2)<\delta$,

$$\begin{aligned} \Vert f-Tf\Vert _{\Omega _2}+\Vert {\overline{\partial }}(f-Tf)\Vert _{\Omega _2}+\Vert \vartheta (f-Tf)\Vert _{\Omega _2}<\varepsilon , \end{aligned}$$

(3.2)

where $\vartheta$ is the formal adjoint of ${\overline{\partial }}$.

Proof

Since ${\mathcal {D}}_{(0,q)}(\Omega _1)$ is dense in ${{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_1)$ in the graph norm $\Vert f\Vert _{\Omega _1}+\Vert {\overline{\partial }}^*f\Vert _{\Omega _1}$, for any $0<\varepsilon <1$, there exists $\phi \in {\mathcal {D}}_{(0,q)}(\Omega _1)$ such that $\Vert f-\phi \Vert _{\Omega _1}+\Vert \vartheta (f-\phi )\Vert _{\Omega _1}<\varepsilon$. We choose $\delta$ sufficiently small such that ${{\,\mathrm{supp}\,}}\phi \subset \subset \Omega _2$. Thus $\phi \in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_2)$. It follows that

$$\begin{aligned} \begin{aligned} Tf&= {\overline{\partial }}^*_2{\overline{\partial }}_2N_2 f+{\overline{\partial }}_2N_2\vartheta f \\&=f-{\overline{\partial }}_2{\overline{\partial }}^*_2 N_2 f+{\overline{\partial }}_2N_2\vartheta f\\&=f-{\overline{\partial }}_2{\overline{\partial }}^*_2N_2 (f-\phi )+{\overline{\partial }}_2N_2\vartheta (f-\phi ).\\ \end{aligned} \end{aligned}$$

(3.3)

Here, we have used the orthogonal decomposition $u={\overline{\partial }}{\overline{\partial }}^*N u +{\overline{\partial }}^*{\overline{\partial }}N u$ and commutative properties $N_2{\overline{\partial }}_2 ={\overline{\partial }}_2 N_2$ on ${{\,\mathrm{Dom}\,}}({\overline{\partial }}_2)$ and $N_2{\overline{\partial }}^*_2={\overline{\partial }}^*_2 N_2$ on ${{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_2)$. Moreover, we have

$$\begin{aligned} {\overline{\partial }}_2 T f={\overline{\partial }}f \end{aligned}$$

(3.4)

and

$$\begin{aligned} \begin{aligned} {\overline{\partial }}^*_2 Tf&={\overline{\partial }}^*_2{\overline{\partial }}_2 N_2\vartheta f=\vartheta f -{\overline{\partial }}_2{\overline{\partial }}^*_2 N_2\vartheta f\\&=\vartheta f-{\overline{\partial }}_2{\overline{\partial }}^*_2 N_2 \vartheta (f-\phi ). \end{aligned} \end{aligned}$$

(3.5)

The desired inequality (3.2) then follows from Hörmander’s $L^2$-estimates for the ${\overline{\partial }}$-operator which imply that ${\overline{\partial }}_2 N_2$ is a bounded operator whose norm is bounded from above by a constant depending only on the diameter of $\Omega _2$ (see, e.g., [8, Theorem 4.4.1]). $\square$

We have the following upper semi-continuity property for the variational eigenvalues defined by (2.7).

Theorem 3.2

Let $\Omega _1$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$. Given $1\le q\le n-1$ and $k\in {\mathbb {N}}$, for any $\varepsilon >0$, there exists $\delta >0$ such that for any pseudoconvex domain $\Omega _2\subset \Omega _1$,

$$\begin{aligned} \lambda ^q_k(\Omega _2)\le \lambda ^q_k(\Omega _1)+\varepsilon \end{aligned}$$

(3.6)

provided $d_H(\Omega _1, \Omega _2)<\delta$.

Proof

For any $0<{\tilde{\varepsilon }}<1$, there exists a k-dimensional subspace $L_k\subset {{\,\mathrm{Dom}\,}}(Q_1)$ such that $\lambda _{Q_1}(L_k)=\sup \{Q_1(f,f)|\,f\in L_k, \Vert f\Vert _{\Omega _1}=1\} \le \lambda _k(\Omega _1)+{\tilde{\varepsilon }}$. (As before, we will drop the superscript q for economy of notations when doing so causes no confusion.) Consider $Q_1(\cdot , \cdot )$ as a sesquilinear form on $L_k\times L_k$. Then there exists an orthonormal basis $\{f_1,\cdots ,f_k\}$ of $L_k$ such that $Q_1(f_h,f_l)=\gamma _l\delta _{hl}$, $1\le h,l\le k$, and $0\le \gamma _1\le \cdots \le \gamma _k=\lambda _{Q_1}(L_k)$. Note that

$$\begin{aligned} \Vert {\overline{\partial }}f_l\Vert _{\Omega _1}^2+\Vert {\overline{\partial }}^*f_l\Vert _{\Omega _1}^2\le \lambda _k(\Omega _1)+{\tilde{\varepsilon }}. \end{aligned}$$

(3.7)

Furthermore, by choosing $\delta$ sufficiently small, we can assume that

$$\begin{aligned} \Vert f_l\Vert _{\Omega _1\setminus \Omega _2}+\Vert {\overline{\partial }}f_l\Vert _{\Omega _1\setminus \Omega _2}+\Vert {\overline{\partial }}^*f_l\Vert _{\Omega _1\setminus \Omega _2}\le {\tilde{\varepsilon }} \end{aligned}$$

(3.8)

for all $1\le l\le k$. We have

$$\begin{aligned} \begin{aligned}&\left| \langle Tf_{h},Tf_{l}\rangle _{\Omega _2}-\delta _{hl}\right| =\left| \langle Tf_{h},Tf_{l}\rangle _{\Omega _2}-\langle f_h,f_l\rangle _{\Omega _1}\right| \\&\qquad \le \left| \langle Tf_{h}-f_h,Tf_{l}\rangle _{\Omega _2}\right| +\left| \langle f_h,Tf_{l}-f_l\rangle _{\Omega _2}\right| +\big |\langle f_h,f_l\rangle _{\Omega _1\setminus \Omega _2}\big |\\&\qquad \le \Vert Tf_{h}-f_h\Vert _{\Omega _2}\Vert Tf_{l}\Vert _{\Omega _2} +\Vert f_h\Vert _{\Omega _2}\Vert Tf_{l}-f_l\Vert _{\Omega _2}+\Vert f_h\Vert _{\Omega _1\setminus \Omega _2} \Vert f_l\Vert _{\Omega _1\setminus \Omega _2}\\&\qquad \le C{\tilde{\varepsilon }}. \end{aligned} \end{aligned}$$

(3.9)

Since

$$\begin{aligned} \begin{aligned} Q_{2}(Tf_{h},Tf_{l})-Q_1(f_h,f_l)&= \langle {\overline{\partial }}_2 Tf_{h},{\overline{\partial }}_2 Tf_{l} -{\overline{\partial }}f_l\rangle _{\Omega _2}+\langle {\overline{\partial }}_2 Tf_{h} -{\overline{\partial }}f_h,{\overline{\partial }}f_{l}\rangle _{\Omega _2} \\&\quad -\langle {\overline{\partial }}f_{h},{\overline{\partial }}f_{l}\rangle _{\Omega _1\backslash \Omega _2} +\langle {\overline{\partial }}^*_2 Tf_{h},{\overline{\partial }}^*_2 Tf_{l}-{\overline{\partial }}^*f_l\rangle _{\Omega _2}\\&\quad +\langle {\overline{\partial }}^*_2 Tf_{h}-{\overline{\partial }}^*f_h,{\overline{\partial }}^*f_{l}\rangle _{\Omega _2}-\langle {\overline{\partial }}^*f_{h},{\overline{\partial }}^*f_{l}\rangle _{\Omega _1\backslash \Omega _2}, \end{aligned}\nonumber \\ \end{aligned}$$

(3.10)

it follows from (3.2), (3.7), (3.8) and the Cauchy–Schwarz inequality that

$$\begin{aligned} \left| Q_{2}(Tf_{h}, Tf_{l})-Q_1(f_h, f_l)\right| \le C(\lambda ^{1/2}_k(\Omega _1)+{\tilde{\varepsilon }}^{1/2}){\tilde{\varepsilon }}. \end{aligned}$$

(3.11)

By Lemma 2.1, we have

$$\begin{aligned} \lambda _k(\Omega _2)\le \lambda _k(\Omega _1)+\varepsilon , \end{aligned}$$

provided ${\tilde{\varepsilon }}$ is sufficiently small. $\square$

As a direct consequence of Theorem 3.2, we have the following:

Corollary 3.3

Let $\Omega$, $\Omega _j$ be bounded pseudoconvex domains in ${\mathbb {C}}^n$ such that $\Omega _j\subset \Omega$ and $d_H(\Omega _j, \Omega )\rightarrow 0$ as $j\rightarrow \infty$. Let $1\le q\le n-1$ and $k\in {\mathbb {N}}$. Then

$$\begin{aligned} \limsup _{j\rightarrow \infty }\lambda ^q_k(\Omega _j)\le \lambda ^q_k(\Omega ). \end{aligned}$$

(3.12)

Remark 3.4

The above result for the first eigenvalues (i.e., the case when $k=1$), is implicit in the paper of Michel and Shaw (see the proof of Theorem 1.2 on p. 123 in [29]; see also the proof of Theorem 4.3.4 on p. 76 in [8]).

We now study stability of variational eigenvalues of the ${\overline{\partial }}$-Neumann Laplacian on a bounded pseudoconvex domain $\Omega$ as it is encroached—not necessarily from inside—by pseudoconvex domains. Let $\Omega$ be a pseudoconvex domain in ${\mathbb {C}}^n$ with $C^1$-smooth boundary. Let $\rho (z)$ be the signed distance function such that $\rho (z)=-{{\,\mathrm{dist}\,}}(z, \partial \Omega )$ on $\Omega$ and $\rho (z)={{\,\mathrm{dist}\,}}(z, \partial \Omega )$ on ${\mathbb {C}}^n\setminus \Omega$. Then $\rho$ is $C^1$ in a neighborhood U of $\partial \Omega$ and $|\nabla \rho (z)|=1$ on U (see [28]). Let $z'\in \partial \Omega$ and let $U'\subset U$ be a tubular neighborhood of $z'$ such that $|\nabla \rho (z)-\nabla \rho (z')|<1/2$ when $z\in U'$. Denote $\vec {n}(z)=\nabla \rho (z)$ and

$$\begin{aligned} \Omega ^\pm _\delta =\{z\in {\mathbb {C}}^n \mid \rho (z)<\pm \delta \}. \end{aligned}$$

Shrinking $U'$ if necessary, then for sufficiently small $\delta >0$, we have $z-2\delta \vec {n}(z')\in \Omega$ for any $z\in U'\cap \Omega ^+_\delta$ and $z+2\delta \vec {n}(z')\not \in \Omega$ for any $z\in U'\setminus \Omega ^-_\delta$. Furthermore,

$$\begin{aligned} {{\,\mathrm{dist}\,}}(z- 2\delta \vec {n}(z'),\partial \Omega )\ge {{\,\mathrm{dist}\,}}(z - 2\delta \vec {n}(z),\partial \Omega )-2\delta |\vec {n}(z)-\vec {n}(z')|>2\delta -\delta =\delta \end{aligned}$$

for all $z\in U'\cap \Omega ^+_\delta$. We choose a finite covering $\{U^l\}_{l=0}^{m}$ of $\overline{\Omega }$ such that $U^0$ is relatively compact in $\Omega$ and each $U^l$, $1\le l\le m,$ is a tubular neighborhood about some $z^l\in \partial \Omega$ constructed as above. Write $\vec {n}^l=\vec {n}(z^l)$. We then have

$$\begin{aligned} \bigcup _{l=1}^{m}\left\{ z-2\delta \vec {n}^l\,|\,z\in U^l\cap \Omega \right\} \bigcup U^0 \subset \Omega ^-_\delta \end{aligned}$$

and

$$\begin{aligned} \bigcup _{l=1}^{m}\left\{ z+2\delta \vec {n}^l\,|\,z\in U^l\cap \Omega \right\} \bigcup U^0 \supset \Omega ^+_\delta . \end{aligned}$$

Let $\{\psi ^l\}_{l=0}^{m}$ be a partition of unity subordinated the covering $\{U^l, \ 0\le l\le m\}$ such that ${{\,\mathrm{supp}\,}}\psi ^l\subset U^l$. Let $f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega })$. Let $\widetilde{f}$ be the form obtained by extending f to 0 outside of $\Omega$. Then $\vartheta {\widetilde{f}}=\widetilde{{\overline{\partial }}^*_\Omega f}\in L^2_{(0, q-1)}({\mathbb {C}}^n)$ (see [8, p. 31]). Let

$$\begin{aligned} {{\widehat{f}}}_\delta (z)=\psi ^0(z) f(z)+\sum _{l=1}^{m}\psi ^l(z)f(z-2\delta \vec {n}^l) \end{aligned}$$

(3.13)

for $z\in \Omega ^+_\delta$ and let

$$\begin{aligned} \check{f}_\delta (z)=\psi ^0(z)\widetilde{f}(z)+\sum _{l=1}^{m}\psi ^l(z)\widetilde{f}(z+2\delta \vec {n}^l) \end{aligned}$$

(3.14)

for $z\in \Omega$. Here we use $f(z\pm 2\delta \vec {n}^l)$ to denote the form obtained by replacing the coefficient $f_J(z)$ of the form f by $f_J(z\pm 2\delta \vec {n}^l)$:

Notice that $\check{f}_\delta (z)$ is supported on $\Omega ^-_\delta$. Roughly speaking, the form ${{\widehat{f}}}_\delta$ and $\check{f}_\delta$ are respectively the push-out and push-in of f along the normal direction by $\delta$ unit. These constructions are used to counter the fact that the restriction of f to a subdomain does not necessarily belong to ${{\,\mathrm{Dom}\,}}({\overline{\partial }}^*)$ on the subdomain and the extension of f to zero outside of $\Omega$ does not necessarily belong to ${{\,\mathrm{Dom}\,}}({\overline{\partial }})$ on a larger domain.

Lemma 3.5

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n$ with $C^1$-smooth boundary. Let $f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega })$. Then ${{\widehat{f}}}_\delta \in {{\,\mathrm{Dom}\,}}({\overline{\partial }}_{\Omega _\delta ^+})$, $\check{f}_\delta \in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{\Omega _\delta ^-})$, and

$$\begin{aligned} \Vert {{\widehat{f}}}_\delta \Vert _{\Omega ^+_\delta }\le C\Vert f\Vert _\Omega , \quad \Vert {\overline{\partial }}{{\widehat{f}}}_\delta \Vert _{\Omega ^+_\delta }\le C(\Vert f\Vert _\Omega +\Vert {\overline{\partial }}f\Vert _\Omega ) \end{aligned}$$

(3.15)

and

$$\begin{aligned} \Vert \check{f}_\delta \Vert _{\Omega }\le C\Vert f\Vert _\Omega , \quad \Vert {\overline{\partial }}^*\check{f}_\delta \Vert _{\Omega }\le C(\Vert f\Vert _\Omega +\Vert {\overline{\partial }}^*f\Vert _\Omega ) \end{aligned}$$

(3.16)

for some constant $C>0$ independent of $\delta$. Furthermore,

$$\begin{aligned} \Vert {{\widehat{f}}}_\delta -\widetilde{f}\Vert _{\Omega ^+_\delta }+\Vert {\overline{\partial }}{{\widehat{f}}}_\delta -\widetilde{{\overline{\partial }}f}\Vert _{\Omega ^+_\delta }\rightarrow 0 \end{aligned}$$

(3.17)

and

$$\begin{aligned} \Vert \check{f}_\delta -f\Vert _{\Omega }+\Vert \vartheta \check{f}_\delta -\vartheta f\Vert _{\Omega }\rightarrow 0 \end{aligned}$$

(3.18)

as $\delta \rightarrow 0$, where $\widetilde{{\overline{\partial }}f}$, as before, is the extension of ${\overline{\partial }}f$ to 0 outside of $\Omega$.

Proof

The first part of the lemma follows directly from the definitions of ${{\widehat{f}}}_\delta$ and $\check{f}_\delta$. Notice that ${{\,\mathrm{supp}\,}}\check{f}_\delta \subset \Omega ^-_\delta$ and $\vartheta \check{f}_\delta \in L^2_{(0, q)}({\mathbb {C}}^n)$. Hence $\check{f}_\delta \in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{\Omega ^-_\delta })$. Since

$$\begin{aligned}&\Vert {{\widehat{f}}}_\delta -\widetilde{f}\Vert _{\Omega ^+_\delta } +\Vert {\overline{\partial }}{{\widehat{f}}}_\delta -\widetilde{{\overline{\partial }}f}\Vert _{\Omega _\delta ^+}\nonumber \\&\lesssim \sum _{l=1}^{m}\left\| f(z-2\delta \vec {n}^l)-\widetilde{f}(z)\right\| _{\Omega _\delta ^+\cap U^l}+\sum _{l=1}^{m}\left\| {\overline{\partial }}f(z-2\delta \vec {n}^l)-\widetilde{{\overline{\partial }}f}(z)\right\| _{\Omega _\delta ^+\cap U^l}\nonumber \\&\le \sum _{l=1}^{m}\left\| {{\widetilde{f}}}(z-2\delta \vec {n}^l) -\widetilde{f}(z)\right\| _{{\mathbb {C}}^n}+\sum _{l=1}^{m}\left\| \widetilde{{\overline{\partial }}f}(z-2\delta \vec {n}^l)-\widetilde{{\overline{\partial }}f}(z)\right\| _{{\mathbb {C}}^n}, \end{aligned}$$

(3.19)

we then obtain (3.17) from the dominated convergence theorem. The proof of (3.18) is similar and is left to the reader. $\square$

Theorem 3.6

Let $\Omega _1$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$ with $C^1$-smooth boundary. Let $1\le q\le n-1$ and $k\in {\mathbb {N}}$. For any $\varepsilon >0$, there exists $\delta >0$ such that for any pseudoconvex domain $\Omega _2$, we have

$$\begin{aligned} \lambda ^q_k(\Omega _2)\le \lambda ^q_k(\Omega _1)+\varepsilon \end{aligned}$$

(3.20)

provided $d_H(\Omega _1, \Omega _2)<\delta$.

Proof

Since $d_H(\Omega _1, \Omega _2)<\delta$, we have $(\Omega _1)^-_\delta \subset \Omega _2\subset (\Omega _1)^+_\delta$. For $f\in {{\,\mathrm{Dom}\,}}(Q_1)$, let ${{\widetilde{f}}}$ be the form obtained by extending f to 0 outside $\Omega _1$ and let ${{\widehat{f}}}_\delta$ be the forms constructed by (3.13) as above (with $\Omega$ replaced by $\Omega _1$). Let

$$\begin{aligned} T_\delta f={\overline{\partial }}^*_2 N_2{\overline{\partial }}_2 {{\widehat{f}}}_\delta +{\overline{\partial }}_2 N_2\vartheta \widetilde{f}. \end{aligned}$$

(3.21)

Then $T_\delta f\in {{\,\mathrm{Dom}\,}}(Q_2)$. Furthermore, for any $\phi \in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_2)$, we have

$$\begin{aligned} T_\delta f&={\overline{\partial }}^*_2{\overline{\partial }}_2N_2 {\widehat{f}}_\delta +{\overline{\partial }}_2 N_2\vartheta (\widetilde{f}-\phi ) +{\overline{\partial }}_2 {\overline{\partial }}^*_2N_2\phi \nonumber \\&={\overline{\partial }}^*_2{\overline{\partial }}_2N_2 \widetilde{f}+{\overline{\partial }}^*_2{\overline{\partial }}_2N_2 ({\widehat{f}}_\delta -\widetilde{f}) +{\overline{\partial }}_2 N_2\vartheta (\widetilde{f}-\phi )+{\overline{\partial }}_2 {\overline{\partial }}^*_2N_2(\phi -\widetilde{f}) +{\overline{\partial }}_2 {\overline{\partial }}^*_2N_2\widetilde{f}\nonumber \\&=\widetilde{f}+{\overline{\partial }}^*_2{\overline{\partial }}_2N_2 ({\widehat{f}}_\delta -\widetilde{f}) +{\overline{\partial }}_2 N_2\vartheta (\widetilde{f}-\phi )+{\overline{\partial }}_2 {\overline{\partial }}^*_2N_2(\phi -\widetilde{f}). \end{aligned}$$

(3.22)

Moreover,

$$\begin{aligned} {\overline{\partial }}_2 T_\delta f={\overline{\partial }}_2{\overline{\partial }}^*_2N_2{\overline{\partial }}_2{\widehat{f}}_\delta ={\overline{\partial }}_2{\widehat{f}}_\delta \end{aligned}$$

(3.23)

and

$$\begin{aligned} {\overline{\partial }}^*_2 T_\delta f={\overline{\partial }}^*_2{\overline{\partial }}_2 N_2\vartheta \widetilde{f}=\vartheta \widetilde{f}-{\overline{\partial }}_2{\overline{\partial }}^*_2 N_2\vartheta \widetilde{f}=\vartheta \widetilde{f}-{\overline{\partial }}_2{\overline{\partial }}^*_2 N_2\vartheta (\widetilde{f}-\phi ). \end{aligned}$$

(3.24)

Let $L_k$ be a k-dimensional subspace of ${{\,\mathrm{Dom}\,}}(Q_1)$ with an orthonormal basis $\{f_1,\cdots ,f_k\}$. For any $0<\varepsilon <1$, by choosing $\delta$ sufficiently small, we have that

$$\begin{aligned} \sum _{l=1}^k \big ( \Vert f_l\Vert ^2_{\Omega _1\setminus \Omega _2}+\Vert {\overline{\partial }}f_l\Vert ^2_{\Omega _1\setminus \Omega _2}+\Vert {\overline{\partial }}^*f_l\Vert ^2_{\Omega _1\setminus \Omega _2}\big )< \varepsilon ^2. \end{aligned}$$

(3.25)

Since ${\mathcal {D}}_{(0,q)}(\Omega _1)$ is dense in ${{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_1)$ in the graph norm $\Vert f\Vert _{\Omega _1}+\Vert {\overline{\partial }}^*f\Vert _{\Omega _1}$, there exists a $\phi _l\in {\mathcal {D}}_{(0,q)}(\Omega _1)$ such that

$$\begin{aligned} \sum _{l=1}^k\big (\Vert f_l-\phi _l\Vert ^2_{\Omega _1}+\Vert {\overline{\partial }}^*f_l-{\overline{\partial }}^*\phi _l\Vert ^2_{\Omega _1}\big )<\varepsilon ^2. \end{aligned}$$

(3.26)

By choosing $\delta$ sufficiently small, we have ${{\,\mathrm{supp}\,}}\phi _l\subset \subset \Omega _2$. Thus $\phi _l\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_2)$. Let

$$\begin{aligned} f=\sum _{l=1}^k c_l f_l \quad \text {and} \quad \phi =\sum _{l=1}^k c_l \phi _l. \end{aligned}$$

It follows from the Cauchy-Schwarz inequality that

$$\begin{aligned} \Vert f\Vert ^2_{\Omega _1\setminus \Omega _2}+\Vert {\overline{\partial }}f\Vert ^2_{\Omega _1\setminus \Omega _2}+\Vert {\overline{\partial }}^*f\Vert ^2_{\Omega _1\setminus \Omega _2}\le k \varepsilon ^2 \Vert f\Vert ^2_{\Omega _1}. \end{aligned}$$

(3.27)

From (3.22), we have

$$\begin{aligned} \Vert T_\delta f -{\widetilde{f}}\Vert _{\Omega _2}\le \Vert {\widehat{f}}_\delta -{\widetilde{f}}\Vert _{\Omega _2}+\Vert {\widetilde{f}}-\phi \Vert _{\Omega _2} +C\Vert \vartheta ({\widetilde{f}}-\phi )\Vert _{\Omega _2}, \end{aligned}$$

where the constant C depending only on the diameter of $\Omega _2$, which can be assumed to be uniformly bounded from above. It follows from Lemma 3.5 and (3.26) that

$$\begin{aligned} \begin{aligned} \big |\Vert T_\delta f\Vert ^2_{\Omega _2} - \Vert f\Vert ^2_{\Omega _1} \big |&=\big |\Vert T_\delta f\Vert ^2_{\Omega _2}-\Vert {{\widetilde{f}}}\Vert ^2_{\Omega _2} -\Vert f\Vert ^2_{\Omega _1\setminus \Omega _2}\big | \\&\le \big (\Vert T_\delta f\Vert _{\Omega _2}+\Vert {{\widetilde{f}}}\Vert _{\Omega _2}\big ) \Vert T_\delta f-{{\widetilde{f}}}\Vert _{\Omega _2}+\Vert f\Vert ^2_{\Omega _1\setminus \Omega _2}\\&\le C\varepsilon \Vert f\Vert ^2_{\Omega _1}. \end{aligned} \end{aligned}$$

(3.28)

From (3.23), (3.24), (3.26) and Lemma 3.5, we have

$$\begin{aligned} \Vert {\overline{\partial }}_2 T_\delta f -\widetilde{{\overline{\partial }}f}\Vert _{\Omega _2}=\Vert {\overline{\partial }}{\widehat{f}}_\delta -\widetilde{{\overline{\partial }}f}\Vert _{\Omega _2}\le C\varepsilon \Vert f\Vert _{\Omega _1} \end{aligned}$$

and

$$\begin{aligned} \Vert {\overline{\partial }}^*_2 T_\delta f-\vartheta {{\widetilde{f}}}\Vert _{\Omega _2}\le \Vert \vartheta ({{\widetilde{f}}}-\phi )\Vert _{\Omega _2}\le C\varepsilon \Vert f\Vert _{\Omega _1}. \end{aligned}$$

Therefore, similar to (3.28), we have

$$\begin{aligned} \big | Q_2(T_\delta f, T_\delta f)-Q_1(f, f)\big |\le C\varepsilon \Vert f\Vert ^2_{\Omega _1}. \end{aligned}$$

By Lemma 2.1 and the subsequent remark, we then have

$$\begin{aligned} \lambda _k(\Omega _2)\le \lambda _k(\Omega _1)+C\varepsilon . \end{aligned}$$

$\square$

As a direct consequence of Theorem 3.6, we have:

Corollary 3.7

Let $\Omega$, $\Omega _j$ be bounded pseudoconvex domains in ${\mathbb {C}}^n$ such that $\partial \Omega$ is $C^1$-smooth and $d_H(\Omega _j, \Omega )\rightarrow 0$ as $j\rightarrow \infty$. Then for any $k\in {\mathbb {N}}$,

$$\begin{aligned} \limsup _{j\rightarrow \infty }\lambda ^q_k(\Omega _j)\le \lambda ^q_k(\Omega ), \quad 1\le q\le n-1. \end{aligned}$$

(3.29)

The upper semi-continuity property of the variational eigenvalues also holds without the pseudoconvexity assumption when restricted to level sets.

Theorem 3.8

Let $\Omega =\{z\in {\mathbb {C}}^n \mid \rho <0\}$ be a bounded domain in ${\mathbb {C}}^n$ with $C^2$-smooth boundary where $\rho \in C^2$ is a defining function of $\Omega$ with $|\nabla \rho |=1$ on $\partial \Omega$. For $\delta >0$, let $\Omega ^-_\delta =\{z\in \Omega \mid \rho ^-_\delta =\rho +\delta <0\}$ and $\Omega ^+_\delta =\{z\in {\mathbb {C}}^n \mid \rho ^+_\delta =\rho -\delta <0\}$. Then for any $k\in {\mathbb {N}}$,

$$\begin{aligned} \limsup _{\delta \rightarrow 0^+} \lambda _k(\Omega ^{\pm }_\delta )\le \lambda _k(\Omega ). \end{aligned}$$

Proof

Since $C^1_{(0, q)}(\overline{\Omega })\cap {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_\Omega )$ is dense in ${{\,\mathrm{Dom}\,}}(Q_\Omega )$ in the graph norm $(\Vert f\Vert ^2_\Omega +Q_\Omega (f, f))^{1/2}$ (see, e.g., [8, Lemma 4.3.2]), it is sufficient to work on forms in $C^1_{(0, q)}(\overline{\Omega })\cap {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_\Omega )$. Let $f\in C^1_{(0,q)}(\overline{\Omega })\cap {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_\Omega )$. For $z\in \Omega$ near the boundary, we write

$$\begin{aligned} f^\nu ={\overline{\partial }}\rho \wedge f_N/|\partial {\bar{\rho }}|^2,\quad f^\tau =f-f^\nu , \end{aligned}$$

where

Notice that $f_N=0$ on $\partial \Omega$ and $({\overline{\partial }}\rho )^*\lrcorner f^\tau =0$ on $\Omega$. We extend $f_N$ to be 0 outside of $\Omega$. Let $\{U^l\}_{l=0}^m$ be an open covering of $\overline{\Omega }$ and let $\{\psi ^l\}_{l=0}^m$ be a partition of unity subordinated to the covering as in the setup preceding Lemma 3.5. Set

$$\begin{aligned} f_{\delta }^{-\nu } (z)=\psi ^0(z)f_N(z)\wedge {\overline{\partial }}\rho +\sum _{l=1}^{m}\psi ^l(z)f_N(z+2\delta \vec {n}^l)\wedge {\overline{\partial }}\rho \end{aligned}$$

and

$$\begin{aligned} f_{\delta }^{-\tau }(z)=f^\tau (z). \end{aligned}$$

for $z\in \Omega ^-_\delta$. Define $T^-_\delta f=f_\delta ^{-\nu }+ f_\delta ^{-\tau }$. Since

$$\begin{aligned} ({\overline{\partial }}\rho ^-_\delta )^*\lrcorner T^-_\delta f=({\overline{\partial }}\rho )^*\lrcorner f^\tau +({\overline{\partial }}\rho )^*\lrcorner f_\delta ^{-\nu }=\psi ^0(z)f_N(z)+\sum _{l=1}^{m}\psi ^l(z)f_N(z+2\delta \vec {n}^l)=0 \end{aligned}$$

on $\partial \Omega ^-_\delta$, we have $T^-_{\delta } f\in {{\,\mathrm{Dom}\,}}(Q_{\Omega ^-_\delta })$. Furthermore,

$$\begin{aligned} \Vert T^-_{\delta } f-f\Vert _{\Omega ^-_\delta }+\Vert {\overline{\partial }}T^-_{\delta } f-{\overline{\partial }}f\Vert _{\Omega ^-_\delta }+\Vert \vartheta T^-_{\delta } f-\vartheta f\Vert _{\Omega ^-_\delta }\rightarrow 0 \end{aligned}$$

as $\delta \rightarrow 0^+$. The proof for the case $\Omega ^-_\delta$ then follows along the same lines as in the proof of Theorem 3.6. For $\Omega ^+_\delta$, we set

$$\begin{aligned} f_{\delta }^{+\nu }(z)=f^\nu (z) \end{aligned}$$

and

$$\begin{aligned} f_{\delta }^{+\tau } (z)=\psi ^0(z)f^\tau (z)+\sum _{l=1}^{m}\psi ^l(z)f^\tau (z-2\delta \vec {n}^l) \end{aligned}$$

and define $T^+_\delta f=f_{\delta }^{+\tau }+f_\delta ^{+\nu }$, and then proceed similarly. $\square$

4 Lower Semi-continuity and Property (P)

Property (P) was introduced by Catlin as a potential theoretic sufficient condition for compactness of the inverse of the ${\overline{\partial }}$-Neumann Laplacian on bounded pseudoconvex domains in ${\mathbb {C}}^n$. A compact set $K\subset {\mathbb {C}}^n$ is said to satisfy Property (P) if for any $M>0$, there exists a neighborhood U of K and a function $\varphi \in C^\infty (U)$ such that $0\le \varphi \le 1$ and any eigenvalue of the hermitian matrix $(\partial ^2\varphi /\partial z_j\partial \bar{z}_k)_{j, k=1}^n$ is greater than or equal to M on U. It is said to satisfy Property ($P_q$), $1\le q\le n$, if any sum of q eigenvalues of the hermitian metric is greater than or equal to M on U. We start with the following well-known lemma.

Lemma 4.1

Let $\Omega$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$ and $b\in C^2(\overline{\Omega })$ with $-1\le b\le 0$. Then

$$\begin{aligned} Q_\Omega (f,f)\ge \frac{1}{e}\int _\Omega H_q(b)(f) dV \end{aligned}$$

(4.1)

for all $f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega })$, where

is the complex Hessian of b, acting on (0, q)-forms .

Proof

When $\partial \Omega$ is smooth, the above lemma is essentially due to Catlin (see (2.3) in [5]; see also (2–10) in [1]). When no boundary smoothness is assumed, the lemma was proved in [33] (see also [34, Corollary 2.13]). It can also be proved by exhausting $\Omega$ from inside by pseudoconvex domains with smooth boundaries and applying Lemma 3.1. $\square$

Lemma 4.2

Let $\Omega$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$. Suppose that $\partial \Omega$ satisfies property $(P_q)$. Then for any $\varepsilon >0$, there exists a $\delta >0$, such that for any pseudoconvex domain $\Omega _j$ with $d_H(\Omega , \Omega _j)<\delta$, we have

$$\begin{aligned} \Vert f_j\Vert ^2_{A_{j\delta }}\le \varepsilon ^2 Q_{\Omega _j}(f_j,f_j) \end{aligned}$$

(4.2)

for all $f_j\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega _j})$, where $A_{j\sigma }=\{z\in \Omega _j|{{\,\mathrm{dist}\,}}(z,\partial \Omega _j)<\sigma \}$. Furthermore, if $f_j$ is an eigenform of $\Box _{\Omega _j}$ with associated eigenvalue $\lambda (\Omega _j)$, then

$$\begin{aligned} \Vert {\overline{\partial }}_j f_j\Vert ^2_{A_{j\delta }}\le \varepsilon ^2\lambda ^2(\Omega _j)\Vert f_j\Vert _{\Omega _j}^2. \end{aligned}$$

(4.3)

Moreover, if $\partial \Omega$ satisfies property $(P_{q-1})$, then

$$\begin{aligned} \Vert {\overline{\partial }}^*_j f_j\Vert ^2_{A_{j\delta }}\le \varepsilon ^2\lambda ^2(\Omega _j)\Vert f_j\Vert _{\Omega _j}^2. \end{aligned}$$

(4.4)

Proof

For any $\varepsilon >0$, since $\partial \Omega$ satisfies property $(P_q)$, there exists a neighborhood U of $\partial \Omega$ and $b\in C^\infty (U)$ with $-1<b\le 0$ such that

(4.5)

for any (0, q)-form f on U. Let $\delta =\frac{1}{2} {{\,\mathrm{dist}\,}}(\partial \Omega ,\partial U)$. If $d_H(\Omega , \Omega _j)<\delta$, then $\partial \Omega _j\subset U$ and $A_{j\delta }\subset U\cap \Omega _j$. Applying Lemma 4.1 to $\Omega _j$ and $f_j$, we have

$$\begin{aligned} \int _{A_{j\delta }}|f_j|^2dV\le \int _{\Omega _j\cap U}|f_j|^2dV\le \frac{\varepsilon ^2}{e}\int _{\Omega _j\cap U}H_q(b)(f_j)dV\le \varepsilon ^2\,Q_{\Omega _j}(f_j,f_j).\nonumber \\ \end{aligned}$$

(4.6)

This concludes the proof of (4.2).

To prove (4.3) and (4.4), we first note that if $f_j$ is an eigenform for $\Box _{\Omega _j}$ associated with eigenvalue $\lambda (\Omega _j)$, then

$$\begin{aligned} {\overline{\partial }}^*_j{\overline{\partial }}_j f_j&=\lambda (\Omega _j) f_j-{\overline{\partial }}_j{\overline{\partial }}^*_j f_j\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}_{j}) \end{aligned}$$

(4.7)

and

$$\begin{aligned} {\overline{\partial }}_j{\overline{\partial }}^*_j f_j&=\lambda (\Omega _j) f_j-{\overline{\partial }}^*_j{\overline{\partial }}_j f_j\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_j). \end{aligned}$$

(4.8)

Moreover,

$$\begin{aligned} \Box _{\Omega _j}{\overline{\partial }}_jf_j&={\overline{\partial }}_j\Box _{\Omega _j} f_j=\lambda (\Omega _j){\overline{\partial }}_j f_j \end{aligned}$$

(4.9)

and

$$\begin{aligned} \Box _{\Omega _j}{\overline{\partial }}^*_jf_j&={\overline{\partial }}^*_j\Box _{\Omega _j}f_j=\lambda (\Omega _j){\overline{\partial }}^*_j f_j. \end{aligned}$$

(4.10)

Since $\partial \Omega$ satisfies property ($P_q$), it also satisfies property ($P_{q+1}$). Therefore, applying (4.2) to ${\overline{\partial }}_j f_j$, we have

$$\begin{aligned} \Vert {\overline{\partial }}_j f_j\Vert ^2_{A_{j\delta }}&\le \varepsilon ^2Q_{\Omega _j}({\overline{\partial }}_j f_j,{\overline{\partial }}_j f_j) =\varepsilon ^2\lambda (\Omega _j)\,\Vert {\overline{\partial }}_j f_j\Vert ^2_{\Omega _j}\\&\le \varepsilon ^2\lambda (\Omega _j)\,Q_{\Omega _j}(f_j,f_j)=\varepsilon ^2\lambda ^2(\Omega _j)\Vert f_j\Vert _{\Omega _j}^2. \end{aligned}$$

Similarly, when $\partial \Omega$ satisfies property ($P_{q-1}$), we have

$$\begin{aligned} \Vert {\overline{\partial }}^*_jf_j\Vert ^2_{A_{j\delta }}\le \varepsilon ^2Q_{\Omega _j}({\overline{\partial }}^*_j f_j,{\overline{\partial }}^*_j f_j)\le \varepsilon ^2\lambda ^2(\Omega _j)\Vert f_j\Vert _{\Omega _j}^2. \end{aligned}$$

This concludes the proof of Lemma 4.2. $\square$

Lemma 4.3

$\Omega$ is a bounded pseudoconvex domain in ${\mathbb {C}}^n$ that satisfies property $(P_q)$. Let M be a positive constant. Let $\{\Omega _j\}$ be a family of pseudoconvex domains such that $d_H(\Omega ,\Omega _j)\rightarrow 0$ as $j\rightarrow \infty$. Suppose $f_j\in {{\,\mathrm{Dom}\,}}(Q_{\Omega _j})$ is a sequence of (0, q)-forms such that $\Vert f_j\Vert ^2_{\Omega _j}+Q_{\Omega _j}(f_j,f_j)\le M$. Let $\widetilde{f}_j$ be the extension of $f_j$ to 0 outside of $\Omega _j$. Then $\{\widetilde{f}_j\}$ is a pre-compact family in $L^2_{(0,q)}({\mathbb {C}}^n)$.

Proof

For any $\varepsilon >0$, it follows from (4.6) that there exist a neighborhood U of $\partial \Omega$ such that

$$\begin{aligned} \int _{U}|\widetilde{f}_j|^2dV=\int _{\Omega _j\cap U}|f_j|^2dV\le \varepsilon ^2\,Q_{\Omega _j}(f_j,f_j)\le \varepsilon ^2 M \end{aligned}$$

for sufficiently large j. Let $V\subset \subset U$ be a neighborhood of $\partial \Omega$. Choosing sufficiently large j such that $(\Omega \setminus \Omega _j)\cup (\Omega _j\setminus \Omega )\subset V$. Let $\eta \in C_0^{\infty }({\mathbb {C}}^n)$ with $0\le \eta \le 1$, $\eta \equiv 1$ on $\Omega \setminus U$ and ${{\,\mathrm{supp}\,}}\eta \subset \Omega \setminus V$. Then there exists a constant $M_1$, such that $Q_{\Omega _j}(\eta f_j,\eta f_j)\le M_1$ and hence $\Vert \eta \widetilde{f}_j\Vert _{W^1(\Omega )}\le M_1$. (Hereafter $\Vert f\Vert _{W^\alpha }$ denotes the norm of $L^2$-Sobolev space of order $\alpha$.) By Rellich’s compactness theorem, $\{\eta \widetilde{f}_j\}$ has a subsequence $\{\eta \widetilde{f}_{j_l}\}$ that conveges in $L^2_{(0,q)}(\Omega )$. Thus

$$\begin{aligned} \Vert \widetilde{f}_{j_h}-\widetilde{f}_{j_l}\Vert _{{\mathbb {C}}^n}&=\Vert \widetilde{f}_{j_h}-\widetilde{f}_{j_l}\Vert _{U} +\Vert \widetilde{f}_{j_h}-\widetilde{f}_{j_l}\Vert _{\Omega \setminus U}\\&\le 2M^{1/2}\varepsilon +\Vert \eta \widetilde{f}_{j_h}-\eta \widetilde{f}_{j_l}\Vert _{\Omega }, \end{aligned}$$

when h and l are sufficiently large. Thus $\{\widetilde{f}_{j_l}\}$ is a subsequence of $\{\widetilde{f}_j\}$ that converges in $L^2_{(0,q)}({\mathbb {C}}^n)$. $\square$

Remark 4.4

Let $f_j$ be an eigenform associated with $k^{\mathrm{th}}$ eigenvalue $\lambda _k(\Omega _j)$ of $\Box _{\Omega _j}$. From the proof of Lemma 4.2, we know that ${\overline{\partial }}_jf_j$ is also an eigenform of $\Box _{\Omega _j}$. Moreover,

$$\begin{aligned} \Vert {\overline{\partial }}_j f_j\Vert ^2_{\Omega _j}+Q_{\Omega _j}({\overline{\partial }}_jf_j,{\overline{\partial }}_jf_j)\le \lambda _k(\Omega _j)(1+\lambda _k(\Omega _j)) \Vert f_j\Vert ^2_{\Omega _j}, \end{aligned}$$

which, by Corollary 3.3, is bounded from above by a constant independent of j. Therefore, $\{\widetilde{{\overline{\partial }}_j f_j}\}$ is also a pre-compact family in $L^2_{(0,q+1)}({\mathbb {C}}^n)$. Similarly, when $\partial \Omega$ satisfies property $(P_{q-1})$, $\{\vartheta \widetilde{f}_j\}$ is also a pre-compact family in $L^2_{(0, q-1)}({\mathbb {C}}^n)$.

Theorem 4.5

Let $\Omega$ be a bounded pseudoconvex domains in ${\mathbb {C}}^n$ with $C^1$-smooth boundary that satisfies Property $(P_{q-1})$, $2\le q\le n-1$. Let $\Omega _j$ be a sequence of bounded pseudoconvex domains whose ${\overline{\partial }}$-Neumann Laplacian $\Box _{\Omega _j}$ has purely discrete spectrum on (0, q)-forms. If $d_H(\Omega _j, \Omega )\rightarrow 0$ as $j\rightarrow \infty$, then for any $k\in {\mathbb {N}}$,

$$\begin{aligned} \liminf _{j\rightarrow \infty }\lambda ^q_k(\Omega _j)\ge \lambda ^q_k(\Omega ). \end{aligned}$$

(4.11)

Proof

The proof is similar in some respects to Theorem 3.6. The difference here is to use Lemma 4.3 and the Kolmogorov-Riesz theorem to establish estimates that are uniform with regard to j.

We first construct the transition operator $T_{j\delta }$ from ${{\,\mathrm{Dom}\,}}(Q_{\Omega _j})$ into ${{\,\mathrm{Dom}\,}}(Q_\Omega )$. Let $\{U^l\}_{l=0}^m$ be an open covering of $\overline{\Omega }$ and let $\{\psi ^l\}_{l=0}^m$ be a partition of unity subordinated to this covering, constructed as in the setup preceding Lemma 3.5. Let $U=\cup _{l=1}^m U^l$ and let $V\subset \subset U$ be a tubular neighborhood of $\partial \Omega$ such that ${{\,\mathrm{dist}\,}}(\partial V, \partial \Omega )<{{\,\mathrm{dist}\,}}(\partial U, \partial \Omega )$. We assume that j is sufficiently large so that $(\Omega \setminus \Omega _j)\cup (\Omega _j\setminus \Omega )\subset \subset V$. Let $f_j\in {{\,\mathrm{Dom}\,}}(Q_{\Omega _j})$. For any $\delta <{{\,\mathrm{dist}\,}}(\partial V, \partial \Omega )$ and any sufficiently large j such that $\delta _j=d_H(\Omega _j, \Omega )<\delta$, we define

$$\begin{aligned} {\widehat{f}}_{j\delta }(z)=\psi ^0(z)f_j(z)+\sum _{l=1}^{m}\psi ^l(z)f_j(z-2\delta \vec {n}^l) \end{aligned}$$

(4.12)

and

$$\begin{aligned} \check{f}_{j\delta }(z)=\psi ^0(z)\widetilde{f}_j(z)+\sum _{l=1}^{m}\psi ^l(z)\widetilde{f}_j(z+2\delta \vec {n}^l). \end{aligned}$$

(4.13)

(Throughout this proof, we will use ${{\widetilde{f}}}_j$ to denote the form obtained by extending $f_j$ to 0 outside of $\Omega _j$.) Notice that $z-2\delta \vec {n}^l\in \Omega _j$ and $z+2\delta \vec {n}^l\not \in \Omega _j$ for $z\in \Omega \cap U^l$ (see the proof of Lemma 3.5). It follows that ${\widehat{f}}_{j\delta }\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}_\Omega )$ and $\check{f}_{j\delta }\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_\Omega )$. Define $T_{j\delta }:{{\,\mathrm{Dom}\,}}(Q_{\Omega _j})\rightarrow {{\,\mathrm{Dom}\,}}(Q_\Omega )$ by

$$\begin{aligned} T_{j\delta } f_j={\overline{\partial }}^*N{\overline{\partial }}{{\widehat{f}}}_{j\delta }+{\overline{\partial }}N{\overline{\partial }}^*\check{f}_{j\delta }. \end{aligned}$$

(4.14)

Then

$$\begin{aligned} T_{j\delta }f_j={\overline{\partial }}^*{\overline{\partial }}N {\widehat{f}}_{j\delta }+{\overline{\partial }}{\overline{\partial }}^*N\check{f}_{j\delta } \end{aligned}$$

(4.15)

and

$$\begin{aligned} {\overline{\partial }}T_{j\delta } f_j={\overline{\partial }}{\overline{\partial }}^*{\overline{\partial }}N{\widehat{f}}_{j\delta }={\overline{\partial }}{\widehat{f}}_{j\delta } \quad \text {and}\quad {\overline{\partial }}^*T_{j\delta } f_j={\overline{\partial }}^*{\overline{\partial }}{\overline{\partial }}^*N \check{f}_{j\delta }={\overline{\partial }}^*\check{f}_{j\delta }. \end{aligned}$$

(4.16)

We first fix $1\le l\le k$ and let $f_{j}$ be the normalized eigenform of $\Box _{\Omega _j}$ associated eigenvalue $\lambda _l(\Omega _j)$. Since

$$\begin{aligned} T_{j\delta } f_j-\widetilde{f}_j={\overline{\partial }}^*{\overline{\partial }}N ({\widehat{f}}_{j\delta }-\widetilde{f}_{j})+{\overline{\partial }}{\overline{\partial }}^*N(\check{f}_{j\delta }-\widetilde{f}_j), \end{aligned}$$

we have

$$\begin{aligned} \begin{aligned} \Vert T_{j\delta } f_j -\widetilde{f}_j\Vert _\Omega&\le \Vert {\widehat{f}}_{j\delta }-\widetilde{f}_{j}\Vert _\Omega +\Vert \check{f}_{j\delta }-\widetilde{f}_j\Vert _\Omega \\&\le \sum _{l=1}^m \left( \left\| \widetilde{f}_j(z-2\delta \vec {n}^l)-\widetilde{f}_j(z)\right\| _{\Omega \cap U^l} + \left\| \widetilde{f}_j(z+2\delta \vec {n}^l)-\widetilde{f}_j(z)\right\| _{\Omega \cap U^l}\right) \\&\le \sum _{l=1}^m \left( \left\| \widetilde{f}_j(z-2\delta \vec {n}^l)-\widetilde{f}_j(z)\right\| _{{\mathbb {C}}^n} + \left\| \widetilde{f}_j(z+2\delta \vec {n}^l)-\widetilde{f}_j(z)\right\| _{{\mathbb {C}}^n}\right) . \end{aligned} \end{aligned}$$

By Lemma 4.3 and the subsequent remark, $\{\widetilde{f}_j\}$ is a pre-compact family in $L^2_{(0, q)}({\mathbb {C}}^n)$. For any $0<\varepsilon <1$, it the follows from the Kolmogorov-Riesz theorm that

$$\begin{aligned} \Vert T_{j\delta } f_j -\widetilde{f}_j\Vert _\Omega <\varepsilon \end{aligned}$$

for all sufficiently small $\delta$ and sufficiently large j.

Furthermore, we have

$$\begin{aligned} \begin{aligned}&\Vert {\overline{\partial }}T_{j\delta } f_j -\widetilde{{\overline{\partial }}f_j}\Vert _\Omega =\Vert {\overline{\partial }}{\widehat{f}}_{j\delta } -\widetilde{{\overline{\partial }}f_j}\Vert _\Omega \\&\quad \le C\sum _{l=1}^m \big (\big \Vert \widetilde{f}_j(z-2\delta \vec {n}^l)-\widetilde{f}_j(z)\big \Vert _{\Omega \cap U^l} + \big \Vert {\overline{\partial }}f_j(z-2\delta \vec {n}^l)-\widetilde{{\overline{\partial }}f_j}(z) \big \Vert _{\Omega \cap U^l}\big ) \\&\quad \le C\sum _{l=1}^m \big (\big \Vert \widetilde{f}_j(z-2\delta \vec {n}^l)-\widetilde{f}_j(z)\big \Vert _{{\mathbb {C}}^n} + \big \Vert \widetilde{{\overline{\partial }}f_j}(z-2\delta \vec {n}^l)-\widetilde{{\overline{\partial }}f_j}(z)\big \Vert _{{\mathbb {C}}^n}\big ), \end{aligned} \end{aligned}$$

where the constant C depends only on the partition of unity and is independent of $\delta$ or j. Note that $\widetilde{{\overline{\partial }}^*f_j}={\overline{\partial }}^*\widetilde{f_j}$. Using the pre-compactness of the families $\{\widetilde{f}_j\}$ and $\{\widetilde{{\overline{\partial }}f_j}\}$ in $L^2$-spaces, we then have

$$\begin{aligned} \Vert {\overline{\partial }}T_{j\delta } f_j -\widetilde{{\overline{\partial }}f_j}\Vert _\Omega <\varepsilon \end{aligned}$$

for all sufficiently small $\delta$ and all sufficiently large j. Similarly,

$$\begin{aligned} \Vert {\overline{\partial }}^*T_{j\delta } f_j -\widetilde{{\overline{\partial }}^*f_j}\Vert _\Omega =\Vert {\overline{\partial }}^*\check{f}_{j\delta } -\widetilde{{\overline{\partial }}^*f_j}\Vert _\Omega <\varepsilon . \end{aligned}$$

The rest of the proof follows the same lines of arguments as in the proof of Theorem 3.6. We sketch the proof below. Let $f_{jl}$ be the normalized eigenform of $\Box _{\Omega _j}$ associated with eigenvalue $\lambda _l(\Omega _j)$. Let $L_{jk}$ be the k-dimensional linear subspace of ${{\,\mathrm{Dom}\,}}(Q_{\Omega _j})$ spanned by $\{f_{jl}\}_{l=1}^k$. Let $f_j=\sum _{l=1}^k c_l f_{jl}$ be a (0, q)-form in $L_{jk}$. Then

$$\begin{aligned} \begin{aligned} \big |\Vert T_{j\delta } f_j\Vert ^2_{\Omega } - \Vert f_j\Vert ^2_{\Omega _j} \big |&=\big |\Vert T_{j\delta } f_j\Vert ^2_{\Omega }-\Vert \widetilde{f_j}\Vert ^2_{\Omega }-\Vert f_j\Vert ^2_{\Omega _j\setminus \Omega } \big | \\&\le \big (\Vert T_{j\delta } f_j\Vert _{\Omega }+\Vert \widetilde{f_j}\Vert _{\Omega }\big )\Vert T_{j\delta } f_j-\widetilde{f_j}\Vert _{\Omega }+\Vert f_j\Vert ^2_{\Omega _j\setminus \Omega }\\&\le C\varepsilon \Vert f_j\Vert ^2_{\Omega _j} \end{aligned}\nonumber \\ \end{aligned}$$

(4.17)

for all sufficiently small $\delta$ and sufficiently large j. Note that in the last inequality, we have used Lemma 4.2. Similarly, we have

$$\begin{aligned} \big | Q_{\Omega }(T_{j\delta } f_j, T_{j\delta } f_j)-Q_{\Omega _j}(f_j, f_j)\big |\le C\varepsilon \Vert f_j\Vert ^2_{\Omega _j}. \end{aligned}$$

The desired inequality (4.11) then follows from Lemma 2.1 and the subsequent remark. $\square$

Theorem 1.2 is a direct consequence of Theorem 4.5 by reductio ad absurdum. Combining Theorems 3.2 and  1.2, we then have:

Corollary 4.6

Let $\Omega , \Omega _j$ be bounded pseudoconvex domains in ${\mathbb {C}}^n$. Suppose $\partial \Omega$ is $C^1$-smooth and satisfies Property $(P_{q-1})$, $2\le q\le n-1$ and $\partial \Omega _j$ satisfies property $(P_q)$. If $d_H(\Omega _j, \Omega )\rightarrow 0$ as $j\rightarrow \infty$, then for any $k\in {\mathbb {N}}$,

$$\begin{aligned} \lim _{j\rightarrow \infty }\lambda ^q_k(\Omega _j)= \lambda ^q_k(\Omega ). \end{aligned}$$

(4.18)

Remark 4.7

Unlike Theorem 3.8, lower semicontinuity property does not hold on level sets of a smooth bounded domain without additional assumption. For example, let $\Omega$ be the Diederich-Fornaess worm domain with winding greater than $\pi$. Since $\Omega$ does not have Stein neighborhood basis ([15]), we have $\liminf _{\delta \rightarrow 0^+} \lambda ^1_k(\Omega ^+_\delta )=0$ but $\lambda ^1_k(\Omega )>0$. This follows from the fact that pseudoconvexity of a smooth bounded domain in ${\mathbb {C}}^2$ is characterized by positivity of one of the variational eigenvalues $\lambda ^1_k(\Omega )$ (see [18]).

5 Quantitative Estimates on Finite Type Domains

We continue our study of spectral stability on smooth bounded pseudoconvex domains of finite type. Our aim is to establish quantitative estimates for the stability on such domains. Notions of finite type were introduced by Kohn [26], D’Angelo [10, 11], and Catlin [4, 6, 7] in connection with subelliptic theory of the ${\overline{\partial }}$-Neumann Laplacian. A smooth bounded domain $\Omega$ in ${\mathbb {C}}^n$ is said to be of finite $D_q$-type if the order of contact of $\partial \Omega$ with any q-dimensional complex analytic variety is finite. (We refer the reader to [10, 11] for precise definitions.)

A fundamental theorem of Catlin states that a smooth bounded pseudoconvex domain $\Omega$ in ${\mathbb {C}}^n$ is of finite $D_q$-type if and only if the ${\overline{\partial }}$-Neumann Laplacian satisfies the following subelliptic estimate

$$\begin{aligned} \Vert f\Vert ^2_{W^\alpha }\le C Q_\Omega (f, f), \quad \forall f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega }) \end{aligned}$$

(5.1)

for some constants $0<\alpha \le 1/2$ and $C>0$. The constant $\alpha$ is referred to as the order of subellipticity. A key step in Catlin’s theory is the construction of plurisubharmonic functions with large complex Hessians. More precisely, if $\Omega$ is a smooth bounded pseudoconvex domain in ${\mathbb {C}}^n$ of finite type, then there exist constants $\alpha >0$, $\delta _0>0$, and $C>0$ such that for any $0<\delta <\delta _0$, there exists a smooth plurisubharmonic function $\lambda _\delta$ on $\overline{\Omega }$ with $|\lambda _\delta |\le 1$ and

$$\begin{aligned} H_q(\lambda _\delta )(f)\ge C |f|^2/\delta ^{2\alpha } \end{aligned}$$

(5.2)

on $A_\delta =\{z\in \Omega \mid d(z)={{\,\mathrm{dist}\,}}(z, \partial \Omega )<\delta \}$ ( [7, Theorem 9.2]). Subelliptic estimate (5.1) is then a consequence of the existence of such plurisubharmonic functions. Straube [33] showed that this last step also holds on bounded pseudoconvex domains with Lipschitz boundaries: Let $\Omega$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$ with Lipschitz boundary. Suppose there exist a continuous plurisubharmonic function $\lambda$ on $\Omega$ and constants $\alpha >0$, $C>0$ such that

$$\begin{aligned} H_q(\lambda )(f)\ge C |f|^2/(d(z))^{2\alpha } \end{aligned}$$

(5.3)

on $\Omega$ as currents, then subelliptic estimate (5.1) holds. For abbreviation, a bounded pseudoconvex domain $\Omega$ is said to satisfy property ($P_q^\alpha$) if condition (5.2) is satisfied. We have the following simple analogs of Lemma 4.2.

Lemma 5.1

Let $\Omega$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$. Suppose $\Omega$ satisfies property $(P_q^\alpha )$. Then there exists a constant C such that for all sufficiently small $\delta >0$,

$$\begin{aligned} \Vert f\Vert ^2_{A_\delta }\le C\delta ^{2\alpha } Q_\Omega (f,f),\qquad \forall f\in {{\,\mathrm{Dom}\,}}(Q_{q, \Omega }). \end{aligned}$$

(5.4)

Furthermore, if f is an eigenform for $\Box _\Omega$ associated with eigenvalue $\lambda (\Omega )$, then

$$\begin{aligned} \Vert f\Vert ^2_{A_\delta }\le C\delta ^{2\alpha }\lambda (\Omega )\Vert f\Vert ^2_\Omega \quad \text {and}\quad \Vert {\overline{\partial }}f\Vert ^2_{A_{\delta }}\le C\delta ^{2\alpha }\lambda ^2(\Omega )\Vert f\Vert _{\Omega }^2. \end{aligned}$$

(5.5)

Moreover, if $\Omega$ satisfies property $(P_{q-1}^\alpha )$, then

$$\begin{aligned} \Vert {\overline{\partial }}^*f\Vert ^2_{A_{\delta }}\le C\delta ^{2\alpha }\lambda ^2(\Omega )\Vert f\Vert _{\Omega }^2. \end{aligned}$$

(5.6)

Lemma 5.2

Let $\Omega$ be a bounded pseudoconvex domain in ${\mathbb {C}}^n$. Suppose $\Omega$ satisfies property $(P_q^\alpha )$. Then there exists a constant $\delta >0$ such that for any pseudoconvex domain $\Omega _j$ with $\delta _j=d_H(\Omega , \Omega _j)\le \delta$, we have

$$\begin{aligned} \Vert f_j\Vert ^2_{A_{j\delta _j}}\le C\delta _j^{2\alpha } Q_{\Omega _j}(f_j,f_j),\qquad \forall f_j\in {{\,\mathrm{Dom}\,}}(Q_{\Omega _j}), \end{aligned}$$

(5.7)

where $A_{j\sigma }{:}{=}\{z\in \Omega _j|{{\,\mathrm{dist}\,}}(z,\partial \Omega _j)<\sigma \}$. Furthermore, if $f_j\in {{\,\mathrm{Dom}\,}}(\Box _{\Omega _j})$ is an eigenform satisfies $\Box _{\Omega _j} f_j=\lambda (\Omega _j) f_j$, then

$$\begin{aligned} \Vert {\overline{\partial }}_j f_j\Vert ^2_{A_{j\delta _j}}\le C\delta _j^{2\alpha }\lambda ^2(\Omega _j)\Vert f_j\Vert _{\Omega _j}^2. \end{aligned}$$

(5.8)

Moreover, if $\Omega$ satisfies property $(P_{q-1}^\alpha )$, then

$$\begin{aligned} \Vert {\overline{\partial }}^*_j f_j\Vert ^2_{A_{j\delta _j}}\le C\delta _j^{2\alpha }\lambda ^2(\Omega _j)\Vert f_j\Vert _{\Omega _j}^2. \end{aligned}$$

(5.9)

These two lemmas are simple consequence of Lemma 4.1, following the same line of arguments as in Lemma 4.2. We omit the proofs. The following lemma is a direct consequence of the interior ellipticity of the ${\overline{\partial }}\oplus {\overline{\partial }}^*$.

Lemma 5.3

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n$. Let $\Omega _{\delta }=\{z\in \Omega \,|\,{{\,\mathrm{dist}\,}}(z,\partial \Omega )>\delta \}$. Then there exists a constant $C>0$ such that

$$\begin{aligned} \Vert f\Vert ^2_{W^1(\Omega _\delta )}\le C\big (Q_\Omega (f, f)+ \frac{1}{\delta ^2}\Vert f\Vert ^2_\Omega \big ) \end{aligned}$$

(5.10)

for all $f\in {{\,\mathrm{Dom}\,}}(Q_\Omega )$.

Proof

The lemma is also well know. We include a proof for the reader’s convenience. Let $\chi (t)=0$ for $t<1/2$, $\chi (t)=2(t-1/2)$ for $t\in [1/2, 1]$, and $\chi (t)=1$ for $t>1$. Let $d(z)={{\,\mathrm{dist}\,}}(z,\partial \Omega )$ and $\eta (z)=\chi (d(z)/\delta )$. Note that since the distance function is uniformly Lipschitz with Lipschitz constant 1, we have $|\nabla \eta (z)|\le 2/\delta$ almost everywhere on $\Omega$. Therefore

(See [34, Corollary 2.13] for a proof of the first inequality.) The desired inequality then follows from integration by part on the left-hand side. $\square$

We remark that the constant in (5.10) can be chosen to be independent of $\Omega$. We will use this fact in the proof of the next theorem.

Theorem 5.4

Let $\Omega$ be a bounded pseudoconvex domain with $C^1$-smooth boundary. Assume that $\Omega$ satisfies property $(P_{q-1}^\alpha )$. Let $\Omega _j$ be a bounded pseudoconvex domain whose ${\overline{\partial }}$-Neumann Laplacian has discrete spectrum on (0, q)-forms. Let $k\in {\mathbb {N}}$. Then there exist constants $\delta >0$ and $C>0$ such that

$$\begin{aligned} \left| \lambda _k(\Omega _j)-\lambda _k(\Omega )\right| \le Ck\delta _j^{\alpha /(\alpha +1)} (\lambda _k(\Omega )+1)^2, \end{aligned}$$

(5.11)

provided $\delta _j=d_H(\Omega ,\Omega _j)<\delta$.

Proof

The proof follows the same line of arguments as those for Theorem 4.5. The difference here is that we use Lemmas  5.1 and  5.2 to estimate terms near the boundary and use Lemma 5.3 to estimate terms inside the domain.

We provide the proof of the inequality

$$\begin{aligned} \lambda _k(\Omega )-\lambda _k(\Omega _j)\le Ck\delta _j^{\alpha /(\alpha +1)}. \end{aligned}$$

Following the same setup as in the proof of Theorem 4.5, for $f_j\in {{\,\mathrm{Dom}\,}}(Q_{\Omega _j})$, we set

$$\begin{aligned} {\widehat{f}}_{j}(z)=\psi ^0(z)f_j(z)+\sum _{l=1}^{m}\psi ^l(z)f_j(z-2\delta _j\vec {n}^l) \end{aligned}$$

(5.12)

and

$$\begin{aligned} \check{f}_{j}(z)=\psi ^0(z)\widetilde{f}_j(z)+\sum _{l=1}^{m}\psi ^l(z)\widetilde{f}_j(z+2\delta _j\vec {n}^l). \end{aligned}$$

(5.13)

Define $T_{j}:{{\,\mathrm{Dom}\,}}(Q_{\Omega _j})\rightarrow {{\,\mathrm{Dom}\,}}(Q_\Omega )$ by

$$\begin{aligned} T_{j} f_j={\overline{\partial }}^*N{\overline{\partial }}{{\widehat{f}}}_{j}+{\overline{\partial }}N{\overline{\partial }}^*\check{f}_{j}. \end{aligned}$$

(5.14)

We now assume that $f_j(z)$ is the normalized eigenform of $\Box _{\Omega _j}$ associated with the eigenvalue $\lambda (\Omega _j)$. As in the proofs of Theorems 3.6 and 4.5, it suffices to estimate the terms

$$\begin{aligned} \Vert f_j\Vert _{\Omega _j\setminus \Omega }, \quad \Vert {\overline{\partial }}f_j\Vert _{\Omega _j\setminus \Omega }, \quad \Vert {\overline{\partial }}^*f_j\Vert _{\Omega _j\setminus \Omega }, \quad \Vert {\widehat{f}}_j-\widetilde{f}_j\Vert _{\Omega },\quad \Vert \check{f}_j-\widetilde{f}_j\Vert _{\Omega }, \end{aligned}$$

(5.15)

and

$$\begin{aligned} \Vert {\overline{\partial }}{\widehat{f}}_j-\widetilde{{\overline{\partial }}f_j}\Vert _{\Omega }, \quad \Vert {\overline{\partial }}^*\check{f}_j-\widetilde{{\overline{\partial }}^*f_j}\Vert _{\Omega }. \end{aligned}$$

(5.16)

From (5.7) in Lemma 5.2, we have

$$\begin{aligned} \Vert f_j\Vert _{\Omega _j\setminus \Omega }\le \Vert f_j\Vert _{A_{j\delta _j}}\le C\delta _j^{\alpha }(\lambda (\Omega _j))^{1/2}. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert {\overline{\partial }}f_j\Vert _{\Omega _j\setminus \Omega }+\Vert {\overline{\partial }}^*f_j\Vert _{\Omega _j\setminus \Omega }\le \Vert {\overline{\partial }}f_j\Vert _{A_{j\delta _j}}+\Vert {\overline{\partial }}^*f_j\Vert _{A_{j\delta _j}}\le C\delta _j^{\alpha }\lambda (\Omega _j). \end{aligned}$$

Noticing that from Lemma 5.3, we have

$$\begin{aligned} \Vert \nabla f_j \Vert _{\Omega _{j, \delta _j^\beta }}\le C\delta _j^{-\beta }(\lambda (\Omega _j)+1)^{1/2}, \end{aligned}$$

where $\Omega _{j, \delta _j^\beta }=\{z\in \Omega _j \mid {{\,\mathrm{dist}\,}}(z, \partial \Omega _j)>\delta _j^\beta \}$. Taking $\beta =1/(1+\alpha )$, we then have

$$\begin{aligned} \Vert {\widehat{f}}_j -\widetilde{f}_j\Vert _{\Omega }&\le C\sum _{l=1}^{m} \left\| f_j(z-2\delta _j\vec {n}^l) -\widetilde{f}_j(z)\right\| _{\Omega \cap U^l}\nonumber \\&\le C\sum _{l=1}^m\left\| f_j(z-2\delta _j\vec {n}^l)-f_j(z)\right\| _{(\Omega \cap U^l) \setminus A_{j, \delta _j^{\beta }}}+C\Vert f_j(z)\Vert _{A_{j, 2\delta _j^{\beta }}}\nonumber \\&\le C\delta _j\left\| \nabla f_j\right\| _{\Omega _{j, \frac{1}{2}\delta _j^{\beta }}} +C\delta _j^{\alpha /(\alpha +1)}(\lambda (\Omega _j))^{1/2} \nonumber \\&\le C\delta _j^{\alpha /(\alpha +1)}(\lambda (\Omega _j)+1)^{1/2}. \end{aligned}$$

(5.17)

Noticing that in obtaining the last inequality, we have used the facts that

$$\begin{aligned} \Omega \setminus A_{j, \delta ^\beta _j}\subset \Omega _{j, \frac{1}{2}\delta ^\beta _j} \end{aligned}$$

and $\lambda (\Omega _j)$ is controlled from above by the corresponding eigenvalue $\lambda (\Omega )$ of $\Box _\Omega$ as shown in Corollary 3.7. Similar estimates also hold for the other three terms in (5.15) and (5.16). Notice that plugging ${\overline{\partial }}f_j$ and ${\overline{\partial }}^*f_j$ into (5.10), we have

$$\begin{aligned} \Vert \nabla \, {\overline{\partial }}f_j\Vert _{\Omega _{j, \delta }} \le C \delta ^{-1}(\lambda (\Omega _j)+1) \quad \text {and}\quad \Vert \nabla \, {\overline{\partial }}^*f_j\Vert _{\Omega _{j, \delta }} \le C \delta ^{-1}(\lambda (\Omega _j)+1) \end{aligned}$$

and the constants in the above estimates are independent of j. Using Lemma 2.1, we then obtain inequality (5.12). The proof of the other inequality in Theorem 5.4 is similar and is left to the interested reader. $\square$

Quantitative estimate (5.11) can be sharpened when more restriction is placed on the boundaries of $\Omega$ and $\Omega _j$. A family of smoothly bounded pseudoconvex domains $\Omega _j$ in ${\mathbb {C}}^n$ with defining functions $\rho _j$ is said to be of uniform finite $D_q$-type if there exist positive constants $\alpha$ and C such that inequality (5.2) holds for all $\Omega _j$ and the $C^\infty$-norm of $\rho _j$ is uniformly bounded. The following lemma is a direct consequence of Catlin’s subelliptic estimates ( [7]).

Lemma 5.5

Let $\Omega$ be a smooth bounded pseudoconvex domain of finite type in ${\mathbb {C}}^n$. Let l be a non-negative integer. Let f be an eigenform of the ${\overline{\partial }}$-Neumann Laplacian $\Box _{q, \Omega }$ with associated eigenvalue $\lambda (\Omega )$. Then there exist positive constants $\alpha$ and $C_l$ such that

$$\begin{aligned} \Vert f\Vert _{C^l(\overline{\Omega })}\le C_l (\lambda (\Omega ))^{[\frac{n+l}{2\alpha }]+1} \Vert f\Vert , \end{aligned}$$

(5.18)

where $[(n+l)/2\alpha ]$ denotes the integer part of $(n+l)/2\alpha$.

Proof

It follows from above-mentioned work of Catlin that $\Omega$ satisfies property $(P_q^\alpha )$ for some $\alpha \in (0, \ 1/2]$ and there exists a constant $C_s>0$ such that

$$\begin{aligned} \Vert N_\Omega f\Vert _{W^{s+2\alpha }(\Omega )}\le C_s\Vert f\Vert _{W^s(\Omega )}. \end{aligned}$$

(5.19)

Starting with $s=0$ and repeatedly applying (5.19) to $\Box _\Omega f=\lambda (\Omega ) f$, we then have

$$\begin{aligned} \Vert f\Vert _{W^{2m\alpha }(\Omega )}\le C(\lambda (\Omega ))^{m}\Vert f\Vert ,\quad m\in {\mathbb {N}}. \end{aligned}$$

(5.20)

The desired estimates (5.18) is then an immediate consequence of Sobolev embedding theorem. $\square$

We remark that the constant in (5.18) depends only on the constant in (5.2) and the $C^\infty$-norm of the defining function of $\Omega$. We will use this fact in proving the following theorem:

Theorem 5.6

Let $\Omega$ be a smooth bounded pseudoconvex domain of finite $D_q$-type in ${\mathbb {C}}^n$. Let $\Omega _j$ be a family of bounded pseudoconvex domains. Let $1\le q\le n-1$ and let $k\in {\mathbb {N}}$. Then there exist constants $C_k>0$ and $\delta >0$ such that

$$\begin{aligned} \lambda ^q_k(\Omega _j)-\lambda ^q_k(\Omega )\le C_k\delta ^{1/2}_j, \end{aligned}$$

(5.21)

provided $\delta _j=d_H(\Omega ,\Omega _j)<\delta$. Furthermore, if $\Omega _j$ is a family of smooth bounded pseudoconvex domains of uniform finite $D_q$-type, then

$$\begin{aligned} -C_k\delta _j\le \lambda ^q_k(\Omega _j)-\lambda ^q_k(\Omega )\le C_k \delta _j. \end{aligned}$$

(5.22)

We will prove this theorem using the following sharp Hardy’s inequality due to Brezis and Marcus (for functions) [2] and an idea from Davies [14].

Lemma 5.7

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n$ with $C^2$-smooth boundary. Then there exists a constant $A>0$ such that

$$\begin{aligned} \int _{\Omega }\frac{|f_N|^2}{(d(z))^2}dV\le 16\left( Q_{\Omega }(f_N,f_N)+A\Vert f_N\Vert ^2\right) \end{aligned}$$

(5.23)

for any $f\in {{\,\mathrm{Dom}\,}}(Q_{\Omega })$, where $f_N=({\overline{\partial }}d(z))^*\lrcorner f$ is the normal component of f and $d(z)=d(z, \partial \Omega )$ is the Euclidean distance from z to the boundary $\partial \Omega$. Furthermore, if $\Omega$ is pseudoconvex, then for any $\varepsilon >0$, there exists a constant $C_\varepsilon >0$ such that

$$\begin{aligned} Q_{\Omega }(f_N, f_N)\le (4q+ \varepsilon )Q_{\Omega }(f, f)+C_\varepsilon \Vert f\Vert ^2. \end{aligned}$$

(5.24)

Proof

For functions in $W^1_0(\Omega )$, inequality (5.23) was proved in [2]. We first provide a proof for functions, using only the divergence theorem. We can assume that $f\in C^\infty _0(\Omega )$. Replacing d(z) by a function that is identical to d(z) in a neighborhood of $\partial \Omega$ and $C^2$ inside $\Omega$, we may assume that $d\in C^2(\overline{\Omega })$. Then

$$\begin{aligned} \begin{aligned}&\int _{\Omega }|\nabla f|^2 dV=\int _{\Omega }\left| d^{1/2}\nabla (d^{-1/2}f) +\frac{\nabla d}{2d}f\right| ^2 dV\\&=\int _{\Omega } d|\nabla (d^{-1/2}f)|^2 dV+\frac{1}{2}\langle \nabla |d^{-1/2}f|^2, \nabla d\rangle _\Omega +\frac{1}{4}\int _{\Omega }\frac{|\nabla d|^2}{d^2}|f|^2 dV.\\ \end{aligned} \end{aligned}$$

(5.25)

Let $g=d^{-1/2}f$. Note that $g\in C^\infty _0(\Omega )$. By the divergence theorem, we have

$$\begin{aligned} \begin{aligned} 0&=\int _\Omega \nabla \cdot (|g|^2 d\nabla d)\, dV\\&=\int _\Omega \big (\nabla |g|^2\cdot (d\nabla d)+ |g|^2 |\nabla d|^2+ |g|^2 d\nabla ^2 d \big )\, dV. \end{aligned} \end{aligned}$$

(5.26)

Thus,

$$\begin{aligned} \int _\Omega |g|^2 |\nabla d|^2\, dV&=-2{{\,\mathrm{Re}\,}}\int _\Omega {\bar{g}}\nabla g\cdot d(\nabla d)\,dV-\int _\Omega |g|^2 d(\nabla ^2d)\, dV\nonumber \\&\le \varepsilon \int _\Omega d|\nabla g|^2dV\nonumber \\&\quad +\dfrac{1}{\varepsilon }\int _\Omega d|\nabla d|^2|g|^2dV +\int _\Omega d|\nabla ^2d||g|^2dV\nonumber \\&\le \varepsilon \int _\Omega d|\nabla g|^2dV+C_\varepsilon \int _\Omega d|g|^2dV. \end{aligned}$$

(5.27)

Note that $|\nabla d|=1$ near $\partial \Omega$. The middle term in the last expression of (5.25) is under controlled as above. Thus by choosing $A>0$ sufficiently large, we obtain the following version of Hardy’s inequality:

$$\begin{aligned} \begin{aligned} \int _{\Omega }\frac{|f|^2}{d^2} dV\le 4\int _{\Omega }|\nabla f|^2 dV+A\Vert f\Vert ^2. \end{aligned} \end{aligned}$$

(5.28)

The above inequality holds for all $f\in W^1_0(\Omega )$ as $C^\infty _0(\Omega )$ is dense in $W^1_0(\Omega )$.

Since $C^1_{(0, q)}(\overline{\Omega })\cap {{\,\mathrm{Dom}\,}}(Q_\Omega )$ is dense in ${{\,\mathrm{Dom}\,}}(Q_\Omega )$ in graph norm, it suffices to prove (5.23) for $f\in C^1_{(0,q)}({\bar{\Omega }})\cap {{\,\mathrm{Dom}\,}}(Q_\Omega )$. Notice that

Since $f\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_\Omega )$, it follows that $f_N^K=0$ on $\partial \Omega$ and hence $f_N^K\in W^1_0(\Omega )$ for any strictly increasing $(q-1)$-tuple K. Moreover,

(5.29)

We then obtain (5.23) by combining (5.28) and (5.29).

To establish (5.24), we note that

(5.30)

The desirable inequality then follows from a simple use of the Cauchy-Schwarz inequality and the facts that $\sum _{l=1}^n |\partial d (z)/\partial z_l|^2=1/4$ near $\partial \Omega$ and

$\square$

The following lemma is a direct consequence of Lemma 5.7 and a theorem of Davies [14, Theorem4]. We sketch the proof for the reader’s convenience.

Lemma 5.8

Let $\Omega$ be a smooth bounded pseudoconvex domain of finite type in ${\mathbb {C}}^n$. If f is an eigenform for the ${\overline{\partial }}$-Neumann Laplacian with associated eigenvalue $\lambda (\Omega )$, then there exist constants $0<\alpha <1$ and $C>0$ such that

$$\begin{aligned} \Vert f_N\Vert _{A_\delta }\le C\delta ^{3/2}(1+\lambda (\Omega ))^{\frac{1}{2}[\frac{1}{2\alpha }]+\frac{3}{4}}\Vert f\Vert , \end{aligned}$$

(5.31)

where $A_\delta =\{z\in \Omega | {{\,\mathrm{dist}\,}}(z,\partial \Omega )<\delta \}$.

Proof

It follows from [14, Theorem 4] that

$$\begin{aligned} \Vert f_N\Vert _{A_\delta }^2\le C\delta ^{3}\Vert (\Delta ^D+A)f_N\Vert _\Omega \Vert (\Delta ^D+A)^{1/2}f_N\Vert _\Omega , \end{aligned}$$

(5.32)

where $\Delta ^D$ is the Dirichlet Laplacian, acting componentwise on $f_N$. Note that

$$\begin{aligned} \Vert (\Delta ^D+A)^{1/2}f_N\Vert ^2=\Vert \nabla f_N\Vert ^2+A\Vert f_N\Vert ^2=4Q_\Omega (f_N, f_N)+A\Vert f_N\Vert ^2. \end{aligned}$$

Thus by Lemma 5.7, this term is dominated by a constant multiple of

$$\begin{aligned} Q_\Omega (f, f)+\Vert f\Vert ^2=(1+\lambda (\Omega ))\Vert f\Vert ^2. \end{aligned}$$

To estimate the term $\Vert (\Delta ^D+A)f_N\Vert _\Omega$, we observe that

$$\begin{aligned} \begin{aligned} \Vert \Delta ^D f_N\Vert ^2_\Omega&= \Vert \nabla ^2 f_N\Vert ^2_\Omega =\Bigg \Vert \nabla ^2 \mathop {{\sum }'}\limits _{|K|=q-1}\left( \frac{1}{q}\sum _{j=1}^{n} \frac{\partial d(z)}{\partial z_j}f_{jK}\right) d\bar{z}_K\Bigg \Vert ^2_\Omega \\&\le C\left( \Vert f\Vert ^2+\Vert \nabla f\Vert ^2+\Vert \lambda (\Omega )f_N\Vert ^2\right) \\&\le C (1+\lambda (\Omega ))^{2\big [\frac{1}{2\alpha }\big ]+2}\Vert f\Vert ^2, \end{aligned} \end{aligned}$$

(5.33)

for some constant C depending on the $C^3$-norm of the defining function. Here in the last inequality above we have used (5.20). Combining the above estimates, we then obtain the desired estimate (5.31). $\square$

We are now in a position to prove Theorem 5.6.

Proof of Theorem 5.6

Let f be a normalized (0, q)-eigenform of $\Box _\Omega$ associated with eigenvalue $\lambda (\Omega )$. Since $\Omega$ is of finite type, $f\in C^\infty _{(0, q)}(\overline{\Omega })$. Let $\text {d}(z)={{\,\mathrm{dist}\,}}(z,\partial \Omega )$ and let $\eta _{\delta _j}(z)=\chi \left( \text {d}(z)/\delta _j\right)$ where $\chi$ is a smooth function such that $\chi (t)=0$ if $t<1$, $\chi (t)=1$ if $t>2$, and $0\le \chi '(t)\le 1$. Then $|\nabla \eta _{\delta _j}|\le 1/\delta _j$, and ${{\,\mathrm{supp}\,}}\eta _{\delta _j}\subset \Omega$ provided $\delta _j$ is sufficiently small. Set $f_{\delta _j}(z)=\eta _{\delta _j}(z){\widetilde{f}}(z)$. Then ${{\,\mathrm{supp}\,}}f_{\delta _j}\subset \subset \Omega _j$ and hence $f_{\delta _j}\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{\Omega _j})$.

Let ${\mathcal {E}}:W^s(\Omega )\rightarrow W^s({\mathbb {C}}^n)$ be a continuous extension operator. Recall that the norm of this operator depends only on n, s, and the Lipschitz constant of $\Omega$ ( [32, Ch VI.3, Theorem 5]). We have

$$\begin{aligned} \Vert f_{\delta _j} -{\mathcal {E}}f\Vert ^2_{\Omega _j}\le \int _{A_{j, 3\delta _j}} |{\mathcal {E}}f|^2\, dV\le C\delta _j, \end{aligned}$$

(5.34)

where, as before, $A_{j, 3\delta _j}=\{z\in \Omega _j \mid {{\,\mathrm{dist}\,}}(z, \partial \Omega _j)\le 3\delta _j\}$. We have

$$\begin{aligned} \vartheta f_{\delta _j}=\eta _{\delta _j}\vartheta f+\delta _j^{-1}\chi '(d(z)/\delta _j) f_N, \end{aligned}$$

where

Note that since $f\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_{\Omega })$, $f^K_N(z)=0$ on $\partial \Omega$. It follows from Lemmas 5.5 and  5.8 that

$$\begin{aligned} \Vert \vartheta f_{\delta _j}\Vert _{A_{2\delta _j}}\le \Vert \vartheta f\Vert _{A_{2\delta _j}}+\delta _j^{-1}\Vert f_N\Vert _{A_{2\delta _j}} \le C\delta ^{1/2}_j \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&\Vert \vartheta f_{\delta _j}-\vartheta f\Vert _{\Omega _j}\le \Vert (\eta _{\delta _j}-1) \vartheta f\Vert _{\Omega _j}+\delta _j^{-1}\Vert \chi '(d/\delta _j)f_N\Vert _{\Omega _j}\\&\qquad \le \Vert \vartheta f\Vert _{A_{2\delta _j}}+\delta _j^{-1}\Vert f_N\Vert _{A_{2\delta _j}}\le C\delta _j^{1/2}. \end{aligned} \end{aligned}$$

(5.35)

Define $T_j:{{\,\mathrm{Dom}\,}}(\Box _\Omega )\rightarrow {{\,\mathrm{Dom}\,}}(Q_j)$ by

$$\begin{aligned} T_j f={\overline{\partial }}^*_j N_j{\overline{\partial }}_j {\mathcal {E}}f+{\overline{\partial }}_j N_j{\overline{\partial }}^*_j f_{\delta _j}={\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f+{\overline{\partial }}_j {\overline{\partial }}^*_jN_j f_{\delta _j}. \end{aligned}$$

(5.36)

Then

$$\begin{aligned} \begin{aligned} \Vert T_j f\Vert ^2_{\Omega _j}&=\Vert {\overline{\partial }}^*_j {\overline{\partial }}_j N_j {\mathcal {E}}f\Vert ^2_{\Omega _j}+ \Vert {\overline{\partial }}_j {\overline{\partial }}^*_j N_j f_{\delta _j}\Vert ^2_{\Omega _j} \\&=\Vert f_{\delta _j}\Vert ^2_{\Omega _j}+\Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\Vert ^2_{\Omega _j}-\Vert {\overline{\partial }}^*_j {\overline{\partial }}_j N_j f_{\delta _j}\Vert ^2_{\Omega _j}. \end{aligned} \end{aligned}$$

(5.37)

Notice that

$$\begin{aligned} \begin{aligned} \Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j f_{\delta _j}\Vert ^2_{\Omega _j}&=\Vert {\overline{\partial }}^*_j {\overline{\partial }}_jN_j (f_{\delta _j}-{\mathcal {E}}f)+{\overline{\partial }}^*_j{\overline{\partial }}_j N_j{\mathcal {E}}f\Vert ^2_{\Omega _j} \\&=\Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j (f_{\delta _j}-{\mathcal {E}}f)\Vert ^2_{\Omega _j} +\Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\Vert ^2_{\Omega _j}\\&\qquad +2{{\,\mathrm{Re}\,}}\langle {\overline{\partial }}^*_j {\overline{\partial }}_j N_j (f_{\delta _j}-{\mathcal {E}}f), \, {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\rangle _{\Omega _j}.\\ \end{aligned} \end{aligned}$$

(5.38)

The first term on the right hand side above is estimated by

$$\begin{aligned} \Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j (f_{\delta _j}-{\mathcal {E}}f)\Vert ^2_{\Omega _j}\le \Vert f_{\delta _j}-{\mathcal {E}}f\Vert ^2_{\Omega _j}\le C\delta _j. \end{aligned}$$

(5.39)

We also have

$$\begin{aligned} \big |\langle {\overline{\partial }}^*_j {\overline{\partial }}_j N_j (f_{\delta _j}-{\mathcal {E}}f), \, {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\rangle _{\Omega _j}\big |=\big |\langle f_{\delta _j}-{\mathcal {E}}f, \, {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\rangle _{\Omega _j}\big |, \end{aligned}$$

which is estimated from above by

$$\begin{aligned} \big |\langle f_{\delta _j}-{\mathcal {E}}f, \, {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\rangle _{\Omega _j}\big |\le C\Vert f_{\delta _j}-{\mathcal {E}}f\Vert _{A_{j, 3\delta _j}}\Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\Vert _{A_{j, 3\delta _j}}. \end{aligned}$$

(5.40)

When there is no finite type assumption on $\Omega _j$, since

$$\begin{aligned} \Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\Vert _{A_{j, 3\delta _j}}\le \Vert {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\Vert _{\Omega _j}\le \Vert {\mathcal {E}}f\Vert _{\Omega _j}\le C, \end{aligned}$$

it follows from (5.34) and (5.40) that

$$\begin{aligned} \big |\langle f_{\delta _j}-{\mathcal {E}}f, \, {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\rangle _{\Omega _j}\big | \le C\delta ^{1/2}_j. \end{aligned}$$

(5.41)

It then follows from (5.37), (5.38), (5.39), and (5.41) that in this case, we have

$$\begin{aligned} \left| \Vert T_jf\Vert ^2_{\Omega _j} -\Vert f\Vert ^2_{\Omega }\right| \le C\delta ^{1/2}_j. \end{aligned}$$

(5.42)

Under the uniform finite type assumption on $\Omega _j$, by Catlin’s subelliptic estimate, $N_j {\mathcal {E}}f$ is smooth and its $C^2$-norm is bounded from above by a constant independent of j (see Lemma 5.5 and the subsequent remark above). It follows that

$$\begin{aligned} \big |\langle f_{\delta _j}-{\mathcal {E}}f, \, {\overline{\partial }}^*_j{\overline{\partial }}_j N_j {\mathcal {E}}f\rangle _{\Omega _j}\big | \le C\delta _j. \end{aligned}$$

(5.43)

Thus, in this case, we have

$$\begin{aligned} \left| \Vert T_jf\Vert ^2_{\Omega _j} -\Vert f\Vert ^2_{\Omega }\right| \le C\delta _j. \end{aligned}$$

(5.44)

On the one hand, since

$$\begin{aligned} \Vert {\overline{\partial }}^*_j T f\Vert _{\Omega _j}^2=\Vert {\overline{\partial }}^*_j {\overline{\partial }}_j N_j {\overline{\partial }}^*_j f_{\delta _j}\Vert _{\Omega _j}^2=\Vert {\overline{\partial }}^*_j f_{\delta _j}\Vert _{\Omega _j}^2, \end{aligned}$$

we have

$$\begin{aligned}&\left| \Vert {\overline{\partial }}^*_j T_j f\Vert _{\Omega _j}^2-\Vert {\overline{\partial }}^*f\Vert _{\Omega }^2\right| \\&\quad \le \int _\Omega (1-\eta _{\delta _j}^2) |\vartheta f|^2+2|\eta _{\delta _j}\vartheta f|\cdot |\delta ^{-1}_j \chi '(\text {d}(z)/\delta _j) f_N|+ \delta ^{-2}_j |f_N|^2\, \text {d}V. \end{aligned}$$

By Lemma 5.8 and the fact that the $C^1$-norm of f is bounded from above (see Lemma 5.5), we have

$$\begin{aligned} \left| \Vert {\overline{\partial }}^*_j T_j f\Vert _{\Omega _j}^2-\Vert {\overline{\partial }}^*f\Vert _{\Omega }^2\right| \le C\delta _j. \end{aligned}$$

(5.45)

On the other hand, since ${\overline{\partial }}_j T_j f={\overline{\partial }}{\mathcal {E}}f$, we have

$$\begin{aligned} |\Vert {\overline{\partial }}_j T_jf\Vert _{\Omega _j}^2-\Vert {\overline{\partial }}f\Vert ^2_\Omega |\le \int _{\Omega _j\setminus \Omega } |{\overline{\partial }}{\mathcal {E}}f|^2\, \text {d}V+\int _{\Omega \setminus \Omega _j} |{\overline{\partial }}f|^2\, \text {d}V \le C\delta _j. \end{aligned}$$

(5.46)

Thus

$$\begin{aligned} |Q_{\Omega _j}(T_j f, T_j f)-Q_\Omega (f, f)|\le C\delta _j. \end{aligned}$$

(5.47)

When (5.47) is coupled with (5.42), we then obtain from Lemma 2.1 the inequality (5.21). When it is coupled with (5.44), we obtain the second inequality in (5.22). The first inequality of (5.22) is proved similarly and is left to the interested reader. $\square$

Remark 5.9

The power of $\delta _j$ in (5.22) is sharp. For example, let ${\mathbb {B}}$ be the unit ball in ${\mathbb {C}}^n$. Then $\lambda _k^q(r_j{\mathbb {B}})=\lambda _k^q({\mathbb {B}})/r^2_j$. Thus

$$\begin{aligned} \big |\lambda _k^q(r_j{\mathbb {B}})-\lambda _k^q ({\mathbb {B}})\big |=(1-r_j)(1+r_j) \lambda _k^q({\mathbb {B}})/r^2_j\approx \delta _j \end{aligned}$$

as $\delta _j=1-r_j\rightarrow 0$.

6 Resolvent Convergence

Let $\Omega$ be a bounded domain in ${\mathbb {C}}^n$. We consider $L^2_{(0, q)}(\Omega )$ be a subspace of $L^2_{(0, q)}({\mathbb {C}}^n)$ consisting of forms vanishing outside $\Omega$. For $\lambda \in {\mathbb {C}}\setminus {\mathbb {R}}$, we extend the resolvent operator $R_\lambda (\square _\Omega )=(\lambda I-\square _\Omega )^{-1}$ to act on $L^2_{(0,q)}({\mathbb {C}}^n)$ by setting $R_\lambda (\square _\Omega )=0$ on $L^2_{(0, q)}({\mathbb {C}}^n)\ominus L^2_{(0, q)}(\Omega )$. Let $\Omega _j$ and $\Omega$ be bounded domains in ${\mathbb {C}}^n$, we say that $\square _{\Omega _j}$ converges to $\square _\Omega$ in strong (respectively in norm) resolvent sense if for all $\lambda \in {\mathbb {C}}\setminus {\mathbb {R}}$, $R_\lambda (\square _{\Omega _j})$ converges strongly (respectively in norm) to $R_\lambda (\square _\Omega )$ as operators acting on $L^2_{(0, q)}({\mathbb {C}}^n)$. When $\Omega$ and $\Omega _j$ are pseudoconvex, we will extend $N_\Omega =\square ^{-1}_\Omega$ and $N_{\Omega _j}=\square ^{-1}_{\Omega _j}$ to act on $L^2_{(0, q)}({\mathbb {C}}^n)$ in a likewise manner. Since the spectra of $\Box _{\Omega _j}$ and $\Box _{\Omega }$ are uniformly bounded away from 0, it is easy to see that $\Box _{\Omega _j}$ converges to $\Box _{\Omega }$ in strong resolvent sense if $N _j$ converges to N strongly on $L^2_{(0, q)}({\mathbb {C}}^n)$ (see, e.g., [31, Theorem VIII.9]). Here, as before, to economize the notation, we write $N_{\Omega _j}$ and $N_\Omega$ simply as $N_j$ and N respectively.

Theorem 6.1

$\Omega$ is a bounded pseudoconvex domain in ${\mathbb {C}}^n$ with $C^1$-smooth boundary. Let $\{\Omega _j\}_{j\in {\mathbb {N}}}$ be a sequence of bounded pseudoconvex domain such that $\delta _j=d_H(\Omega ,\Omega _j)\rightarrow 0$ as $j\rightarrow \infty$. Then $\Box _{\Omega _j}$ converges to $\Box _\Omega$ in strong resolvent sense.

Proof

The proof follows the same lines of arguments as in Theorem 3.6. Let $f\in L^2_{(0,q)}({\mathbb {C}}^n)$ and let $g=Nf$. Then $f_\Omega {:}{=}f|_\Omega =\Box _\Omega g={\overline{\partial }}{\overline{\partial }}^*g+{\overline{\partial }}^*{\overline{\partial }}g$. Since $g,{\overline{\partial }}g\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}^*_\Omega )$, it follows from the minimality of ${\overline{\partial }}^*$ that for any $0<\varepsilon <1$, there exist $\phi \in {\mathcal {D}}_{(0,q)}(\Omega )$ and $\varphi \in {\mathcal {D}}_{(0,q+1)}(\Omega )$ such that

$$\begin{aligned} \Vert \phi - g\Vert _\Omega +\Vert {\overline{\partial }}^*(\phi - g)\Vert _\Omega +\Vert \varphi -{\overline{\partial }}g\Vert _\Omega +\Vert {\overline{\partial }}^*(\varphi - {\overline{\partial }}g)\Vert _\Omega <\varepsilon . \end{aligned}$$

(6.1)

By choosing j sufficiently large, we have $\phi \in {\mathcal {D}}_{(0,q)}(\Omega _{j})$ and $\varphi \in {\mathcal {D}}_{(0,q+1)}(\Omega _{j})$.

As in the proof of Theorem 3.6, we will use ${\widetilde{F}}$ to denote the extension of F to ${\mathbb {C}}^n$ by letting $\widetilde{F}=0$ outside of $\Omega$ and ${\widehat{F}}$ to denote the form constructed by (3.13) with $\delta =\delta _j$. Thus $\widehat{{\overline{\partial }}^*g}$, ${{\widehat{g}}}\in {{\,\mathrm{Dom}\,}}({\overline{\partial }}_j)$. Since

$$\begin{aligned} \Vert N _jf-N f\Vert _{{\mathbb {C}}^n}&= \Vert N _jf-N f\Vert _{\Omega _{j}}+\Vert Nf\Vert _{\Omega \setminus \Omega _j}\\&\le \Vert N_j\widetilde{f_\Omega }-{\widetilde{g}}\Vert _{\Omega _{j}}+ \Vert N_j (f-\widetilde{f_\Omega })\Vert _{\Omega _j}+\Vert Nf\Vert _{\Omega \setminus \Omega _j}\\&\le \Vert N_j\widetilde{f_\Omega }-{\widetilde{g}}\Vert _{\Omega _{j}}+C\Vert f-\widetilde{f_\Omega }\Vert _{\Omega _j}+\Vert Nf\Vert _{\Omega \setminus \Omega _j} \end{aligned}$$

and the last two terms above goes to 0 as $j\rightarrow \infty$, it suffices to prove that $\Vert N_j\widetilde{f_\Omega }-{\widetilde{g}}\Vert _{\Omega _{j}}\rightarrow 0$. We have

$$\begin{aligned} \Vert N_j\widetilde{f_\Omega }-\widetilde{g}\Vert _{\Omega _j}\le \Vert N_{j}\widetilde{{\overline{\partial }}\,{\overline{\partial }}^*g}-{\overline{\partial }}_j{\overline{\partial }}^*_jN_j{\widetilde{g}}\Vert _{\Omega _j}+\Vert N_{j}{\overline{\partial }}^*\widetilde{{\overline{\partial }}g}-{\overline{\partial }}^*_j{\overline{\partial }}_jN_j{\widetilde{g}}\Vert {:}{=}I+II. \end{aligned}$$

Note that

$$\begin{aligned} I&= \Vert {\overline{\partial }}_jN _j\widehat{{\overline{\partial }}^*g}+N_j(\widetilde{{\overline{\partial }}\,{\overline{\partial }}^*g} -{\overline{\partial }}\,\widehat{{\overline{\partial }}^*g})-{\overline{\partial }}_j{\overline{\partial }}^*_jN _j{\widetilde{g}}\Vert _{\Omega _j}\\&=\Vert {\overline{\partial }}_jN _j{\overline{\partial }}^*{\widetilde{g}}+ {\overline{\partial }}_jN _j(\widehat{{\overline{\partial }}^*g} -\widetilde{{\overline{\partial }}^*g})+N_j(\widetilde{{\overline{\partial }}\,{\overline{\partial }}^*g}-{\overline{\partial }}\, \widehat{{\overline{\partial }}^*g})-{\overline{\partial }}_j{\overline{\partial }}^*_jN _j{\widetilde{g}}\Vert _{\Omega _j}\\&\le \Vert {\overline{\partial }}_j{\overline{\partial }}^*_jN _j(\phi -{\widetilde{g}})\Vert _{\Omega _j} + \Vert {\overline{\partial }}_jN _j{\overline{\partial }}^*({\widetilde{g}}-\phi )\Vert _{\Omega _j}\\&\qquad + \Vert {\overline{\partial }}_jN _j(\widehat{{\overline{\partial }}^*g}-\widetilde{{\overline{\partial }}^*g})\Vert _{\Omega _j} +\Vert N_j(\widetilde{{\overline{\partial }}\,{\overline{\partial }}^*g}-{\overline{\partial }}\,\widehat{{\overline{\partial }}^*g})\Vert _{\Omega _j}\\&\le \Vert \phi -g\Vert _{\Omega }+ C\Vert {\overline{\partial }}^*(g-\phi )\Vert _{\Omega }+ C\Vert \widehat{{\overline{\partial }}^*g} -\widetilde{{\overline{\partial }}^*g}\Vert _{\Omega _j}+C\Vert \widetilde{{\overline{\partial }}\,{\overline{\partial }}^*g} -{\overline{\partial }}\,\widehat{{\overline{\partial }}^*g}\Vert _{\Omega _j}. \end{aligned}$$

It follows from (6.1) and Lemma 3.5 that $I\rightarrow 0$ as $j\rightarrow \infty$. Similarly, we have

$$\begin{aligned} II&\le \Vert {\overline{\partial }}^*_j{\overline{\partial }}_jN_j({\widehat{g}}-{\widetilde{g}})\Vert _{\Omega _j} +\Vert {\overline{\partial }}^*_jN_j(\widetilde{{\overline{\partial }}g}-{\overline{\partial }}{\widehat{g}})\Vert _{\Omega _j}\\&\qquad +\Vert {\overline{\partial }}^*_jN_j(\varphi -\widetilde{{\overline{\partial }}g})\Vert _{\Omega _j} +\Vert N_{j}{\overline{\partial }}^*(\widetilde{{\overline{\partial }}g}-\varphi )\Vert _{\Omega _j}\\&\le \Vert {\widehat{g}}-{\widetilde{g}}\Vert _{\Omega _j} +C\Vert \widetilde{{\overline{\partial }}g} -{\overline{\partial }}{\widehat{g}}\Vert _{\Omega _j}+C\Vert \varphi -{\overline{\partial }}g\Vert _{\Omega } +C\Vert {\overline{\partial }}^*({\overline{\partial }}g-\varphi )\Vert _{\Omega }, \end{aligned}$$

which again goes to 0 as $j\rightarrow \infty$. Thus $\Vert N_j\widetilde{f_\Omega }-{\widetilde{g}}\Vert _{\Omega _{j}}\rightarrow 0$. $\square$

Remark 6.2

(1) As for Theorem 3.2, Theorem 6.1 holds without the $C^1$ assumption on $\partial \Omega$ if the $\Omega _j$’s are contained in $\Omega$. (2) One cannot expect that $\Box _{\Omega _j}$ converges to $\Box _\Omega$ in norm resolvent sense. For example, let $\Omega _j$ and $\Omega$ be bounded convex domains in ${\mathbb {C}}^n$ such that $\Omega$ is exhausted by $\Omega _j$ from inside. Suppose $\partial \Omega$ contains a complex analytic variety but $\partial \Omega _j$ does not. Then $N_j$ is compact but N is not (see [19]). Thus in this case, $N_j$ does not converge to N in norm.