Stability of eigenfunctions

In [CZ1994] some bounds for the Neumann heat kernels [math]\displaystyle{ P_t(x,y) }[/math] on a general domain are given; in particular, one can bound this kernel by a multiple of the Euclidean heat kernel for t small enough. Using this bound, [BP2008] showed uniform stability of the second eigenfunction (in the uniform topology) and eigenvalue with respect to uniform perturbations of the domain.

Formal theory

Suppose one has a one-parameter family [math]\displaystyle{ t \mapsto L(t) }[/math] of self-adjoint operators (on some Hilbert space, e.g. [math]\displaystyle{ L^2(\Omega) }[/math], though for this formal computation the domain will not be important), and one-parameter families [math]\displaystyle{ t \mapsto u(t) }[/math], [math]\displaystyle{ t \mapsto \lambda(t) }[/math] to the eigenfunction equation

[math]\displaystyle{ L(t) u(t) = \lambda(t) u(t) }[/math]. (1)

We normalise [math]\displaystyle{ u(t) }[/math] to have norm 1:

[math]\displaystyle{ \langle u(t), u(t) \rangle = 1 }[/math]. (2)

Formally, if we differentiate the norm equation (2) at time zero, we get

[math]\displaystyle{ \langle \dot u(0), u(0) \rangle = 0 }[/math] (3)

while if we differentiate (1) at time zero we obtain

[math]\displaystyle{ \dot L(0) u(0) + L(0) \dot u(0) = \dot \lambda(0) u(0) + \lambda(0) \dot u(0) }[/math]. (4)

Taking the inner product of (4) with u(0) and using (2), (3) we conclude the Hadamard first variation formula for eigenvalues:

[math]\displaystyle{ \dot \lambda(0) = \langle \dot L(0) u(0), u(0) \rangle }[/math]. (5)

If we instead take the orthogonal projection [math]\displaystyle{ \pi_{u(0)}^\perp }[/math] onto the orthogonal complement of [math]\displaystyle{ u(0) }[/math], we obtain the Hadamard first variation formula for eigenfunctions:

[math]\displaystyle{ (L(0)-\lambda) \dot u(0) = - \pi_{u(0)}^\perp( \dot L(0) u(0) ) }[/math]. (6)

Formally, if [math]\displaystyle{ \lambda }[/math] is a simple eigenvalue, then [math]\displaystyle{ L(0)-\lambda }[/math] is invertible on the orthogonal complement of [math]\displaystyle{ u(0) }[/math], and so (6) and (3) allow one to solve for [math]\displaystyle{ \dot u(0) }[/math].

L^2 and H^1 theory

Suppose that one has a domain [math]\displaystyle{ \Omega }[/math] with second Neumann eigenvalue [math]\displaystyle{ \lambda_2 }[/math], and third Neumann eigenvalue (not counting multiplicity) [math]\displaystyle{ \lambda_3 \gt \lambda_2 }[/math]. Thus, one has

[math]\displaystyle{ \int_\Omega |\nabla u|^2 \geq \lambda_2 \int_\Omega |u|^2 }[/math] (7)

for all mean zero u, with equality when u lies in the second eigenspace [math]\displaystyle{ V_2 }[/math], and one can improve this to

[math]\displaystyle{ \int_\Omega |\nabla u|^2 \geq \lambda_3 \int_\Omega |u|^2 }[/math] (8)

when u is orthogonal to [math]\displaystyle{ V_2 }[/math].

Now consider a perturbation [math]\displaystyle{ B\Omega }[/math] of [math]\displaystyle{ \Omega }[/math], where B is an invertible linear transformation. Then a second eigenfunction of [math]\displaystyle{ B\Omega }[/math], after change of variables, becomes a function u on [math]\displaystyle{ \Omega }[/math] that minimizes the modified Rayleigh quotient

[math]\displaystyle{ \int_\Omega M \nabla u \cdot \nabla u / \int_\Omega |u|^2 }[/math]

where [math]\displaystyle{ M := (B^{-1}) (B^{-1})^T }[/math]. We may normalize this eigenfunction as [math]\displaystyle{ u = u_2 + v }[/math], where u_2 is a unit eigenfunction in V_2 and v is orthogonal to V_2, so that [math]\displaystyle{ \|u\|_2^2 = 1 + \|v\|_2^2 }[/math]. Then the modified Rayleigh quotient of u is less than or equal to that of u_2, and hence

[math]\displaystyle{ \int_\Omega M \nabla u \cdot \nabla u \leq (\int_\Omega M \nabla u_2 \cdot \nabla u_2) (1 + \|v\|_2^2 ). }[/math]

Expanding out u as u_2+v and rearranging, we end up at

[math]\displaystyle{ \int_\Omega M \nabla v \cdot \nabla v \leq \|v\|_2^2 \int_\Omega M \nabla u_2 \cdot \nabla u_2 - 2 \int_\Omega M \nabla u_2 \cdot \nabla v. }[/math]

Note that [math]\displaystyle{ \nabla v }[/math] is orthogonal to [math]\displaystyle{ \nabla V_2 }[/math] by integration by parts, and so we may replace [math]\displaystyle{ M \nabla u_2 }[/math] on the RHS by the orthogonal projection [math]\displaystyle{ \pi_{\nabla V_2}^\perp(M \nabla u_2) }[/math]. Letting [math]\displaystyle{ \sigma_1(M) = \sigma_2(B)^{-2} }[/math] be the least singular value of M, we now apply Cauchy-Schwarz and conclude that

[math]\displaystyle{ \sigma_1(M) \|\nabla v \|_2^2 \leq \|v\|_2^2 \int_\Omega M \nabla u_2 \cdot \nabla u_2 + 2 \| \pi_{\nabla V_2}^\perp(M \nabla u_2) \|_2 \|\nabla v\|_2 }[/math].

By (8) we may bound [math]\displaystyle{ \|v\|_2^2 \leq \frac{1}{\lambda_3} \|\nabla v\|_2^2 }[/math], and so we conclude that

[math]\displaystyle{ (\sigma_1(M) - \frac{\int_\Omega M \nabla u_2 \cdot u_2}{\lambda_3}) \|\nabla v \|_2 \leq 2 \| \pi_{\nabla V_2}^\perp(M \nabla u_2) \|_2 }[/math]. (9)

This gives an H^1 bound on the error v between the perturbed eigenfunction u and the original eigenfunction u_2. To understand this bound, note that we may upper bound

[math]\displaystyle{ \int_\Omega M \nabla u_2 \cdot u_2 \leq \sigma_2(M) \int_\Omega |\nabla u_2|^2 = \lambda_2 \sigma_2(M) }[/math]

and also [math]\displaystyle{ \pi_{\nabla V_2}^\perp(M \nabla u_2) = \pi_{\nabla V_2^\perp}((M-\sigma_1) \nabla u_2) }[/math] and so one has

[math]\displaystyle{ \|\nabla v \|_2 \leq \frac{2 (\sigma_2(M)-\sigma_1(M)) \lambda_2^{1/2}}{\sigma_1(M) - \sigma_2(M) \frac{\lambda_2}{\lambda_3}} }[/math]

provided that the denominator is positive. In terms of the condition number [math]\displaystyle{ \kappa := \sigma_2(B)/\sigma_1(B) }[/math] of B, this becomes

[math]\displaystyle{ \|\nabla v \|_2 \leq \frac{2 (\kappa^2-1) \lambda_2^{1/2}}{1 - \kappa^2 \frac{\lambda_2}{\lambda_3}} }[/math]

which is a non-trivial bound when [math]\displaystyle{ \kappa \leq (\lambda_3/\lambda_2)^{1/2} }[/math], and is of the order of [math]\displaystyle{ O(\kappa-1) }[/math] when [math]\displaystyle{ \kappa }[/math] is close to 1.