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 \nabla 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-\frac{\sigma_1(M)+\sigma_2(M)}{2}) \nabla u_2) }[/math] so that

[math]\displaystyle{ \|\pi_{\nabla V_2^\perp}((M-\frac{\sigma_1(M)+\sigma_2(M)}{2}) \nabla u_2)\|_2 \leq \frac{\sigma_2(M)-\sigma_1(M)}{2} \|\nabla u_2\|_2 = \frac{\sigma_2(M)-\sigma_1(M)}{2} \lambda_2^{1/2} }[/math]

and so one has

[math]\displaystyle{ \|\nabla v \|_2 \leq \frac{(\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{(\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 \lt (\lambda_3/\lambda_2)^{1/2} }[/math], and is of the order of [math]\displaystyle{ O(\lambda_2^{1/2} (\kappa-1)) }[/math] when [math]\displaystyle{ \kappa }[/math] is close to 1. Using (8), we conclude in particular that

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

For these calculations performed on a third reference triangle, http://www.math.sfu.ca/~nigam/polymath-figures/Perturbation.pdf

An alternate approach to the L^2 and H^1 theory

Let us keep the reference triangle [math]\displaystyle{ \Omega }[/math] fixed, and view the matrix M=M(t) as being smoothly time dependent, so that the eigenvalues [math]\displaystyle{ \lambda_k = \lambda_k(t) }[/math] and L^2-normalised eigenfunctions [math]\displaystyle{ u_k = u_k(t) }[/math] are also time dependent. Assume for the sake of argument that eigenvalues stay simple and all functions depend smoothly on t. We have the eigenfunction equation

[math]\displaystyle{ -\nabla \cdot M \nabla u_k = \lambda_k u_k }[/math]

and the Neumann boundary condition

[math]\displaystyle{ -n \cdot M \nabla u_k = 0 }[/math]

and the L^2 normalisation

[math]\displaystyle{ \langle u_k, u_k \rangle = 1 }[/math].

Differentiating, we conclude that

[math]\displaystyle{ (-\nabla \cdot M \nabla - \lambda_k) \dot u_k = \nabla \cdot \dot M \nabla u_k + \dot \lambda_k u_k }[/math] (10)

and

[math]\displaystyle{ -n \cdot M \nabla \dot u_k = n \cdot \dot M \nabla u_k }[/math] (11)

and

[math]\displaystyle{ \langle \dot u_k, u_k \rangle = 0 }[/math]. (12)

Taking the inner product of (10) with u_k and using (12) yields

[math]\displaystyle{ \langle -\nabla \cdot M \nabla \dot u_k, u_k \rangle = \langle \nabla \cdot \dot M \nabla u_k, u_k \rangle + \dot \lambda_k }[/math].

Integrating by parts using the Neumann condition and (11) yields

[math]\displaystyle{ \langle -\nabla \cdot M \nabla \dot u_k, u_k \rangle = \langle \dot u_k, -\nabla \cdot M \nabla u_k \rangle + \int_{\partial \Omega} (n \cdot \dot M \nabla u_k) u_k }[/math].

By the eigenfunction equation and (12), the first inner product on the RHS vanishes. By Stokes theorem one has

[math]\displaystyle{ \int_{\partial \Omega} (n \cdot \dot M \nabla u_k) u_k = \langle \nabla \cdot \dot M \nabla u_k, u_k \rangle + \langle \dot M \nabla u_k, \nabla u_k \rangle }[/math]

and thus we have the variation formula

[math]\displaystyle{ \dot \lambda_k = \langle \dot M \nabla u_k, \nabla u_k \rangle }[/math]. (14)

Next, if we take the inner product of (10) with u_l for some l distinct from k, one has

[math]\displaystyle{ \langle (-\nabla \cdot M \nabla - \lambda_k) \dot u_k, u_l \rangle = \langle \nabla \cdot \dot M \nabla u_k, u_l \rangle }[/math].

Integrating by parts as before, we have

[math]\displaystyle{ \langle -\nabla \cdot M \nabla \dot u_k, u_l \rangle = \langle \dot u_k, -\nabla \cdot M \nabla u_l \rangle + \int_{\partial \Omega} (n \cdot \dot M \nabla u_k) u_l }[/math].

By the eigenfunction equation, the first inner product on the RHS is [math]\displaystyle{ \lambda_l \langle \dot u_k, u_l \rangle }[/math]. By Stokes theorem we have

[math]\displaystyle{ \int_{\partial \Omega} (n \cdot \dot M \nabla u_k) u_l = \langle \nabla \cdot \dot M \nabla u_k, u_l \rangle + \langle \dot M \nabla u_k, \nabla u_l \rangle }[/math]

and thus

[math]\displaystyle{ (\lambda_l - \lambda_k) \langle \dot u_k, u_l \rangle = \langle \dot M \nabla u_k, \nabla u_l \rangle }[/math]

and thus by eigenfunction expansion (and (12))

[math]\displaystyle{ \dot u_k = \sum_{l \neq k} \frac{1}{\lambda_l-\lambda_k} \langle \dot M \nabla u_k, \nabla u_l \rangle u_l }[/math]

where the convergence is in an unconditional L^2 sense. (Note that [math]\displaystyle{ \nabla u_l / \lambda_l^{1/2} }[/math] is an orthonormal system and so from Bessel's inequality we know that [math]\displaystyle{ \sum_l \langle \dot M \nabla u_k, \nabla u_l \rangle^2 / \lambda_l \lt \infty }[/math], which is enough decay to justify the L^2 convergence of the RHS.) In fact we may differentiate and conclude that

[math]\displaystyle{ \dot \nabla u_k = \sum_{l \neq k} \frac{1}{\lambda_l-\lambda_k} \langle \dot M \nabla u_k, \nabla u_l \rangle \nabla u_l }[/math]

where the convergence is again in the unconditional L^2 sense (i.e. the previous convergence was in the unconditional H^1 sense). From the Bessel inequality we see in particular that

[math]\displaystyle{ \| \dot \nabla u_k \|_{L^2}^2 = \sum_{l \neq k} \frac{\lambda_l^2}{(\lambda_l-\lambda_k)^2} |\langle \dot M \nabla u_k, \nabla u_l \rangle|/\lambda_l }[/math]

[math]\displaystyle{ \leq \| \dot M \nabla u_k \|_{L^2}^2 \sup_{l \neq k} \frac{\lambda_l^2}{(\lambda_l - \lambda_k)^2} }[/math];

in particular, we have

[math]\displaystyle{ \| \dot \nabla u_2 \|_{L^2}^2 \leq \frac{\lambda_3}{\lambda_3 - \lambda_2} \| \dot M \nabla u_2 \|_{L^2} }[/math]

and thus

[math]\displaystyle{ \| \dot u_2 \|_{L^2}^2 \leq \frac{1}{\lambda_3 - \lambda_2} \| \dot M \nabla u_2 \|_{L^2} }[/math]

which can be viewed as infinitesimal variants of the estimates in the previous section.

A Sobolev inequality

Lemma Suppose that [math]\displaystyle{ u: B(0,R) \to {\mathbf R} }[/math] obeys the inhomogeneous eigenfunction equation [math]\displaystyle{ -\Delta u = \lambda u + F }[/math]. Suppose also that [math]\displaystyle{ \eta: {\mathbf R} \to {\mathbf R} }[/math] is a smooth function that equals 1 at 0 and vanishes on [math]\displaystyle{ [R,+\infty) }[/math]. Then

[math]\displaystyle{ |u(0)| \leq \frac{\sqrt{\pi}}{2\sqrt{2}} (\|F\|_{L^2(B(0,R)} (\int_0^\infty \eta(r)^2 Y_0(\sqrt{\lambda} r)^2\ r dr)^{1/2} + \lambda^{1/2} \|u\|_{L^2(B(0,R))} (\int_0^R \eta'(r)^2 Y'_0(\sqrt{\lambda}r)^2\ r dr)^{1/2} }[/math]

[math]\displaystyle{ + \|\nabla u\|_{L^2(B(0,R))} (\int_0^R \eta'(r)^2 Y_0(\sqrt{\lambda}r)^2\ r dr)^{1/2} ). }[/math]

Proof We introduce the averaged functions

[math]\displaystyle{ \bar u(r) := \frac{1}{2\pi} \int_0^{2\pi} u(r,\theta)\ d\theta }[/math]

[math]\displaystyle{ \bar F(r) := \frac{1}{2\pi} \int_0^{2\pi} F(r,\theta)\ d\theta }[/math]

and observe that [math]\displaystyle{ \bar u }[/math] obeys the inhomogeneous Bessel equation

[math]\displaystyle{ -\bar u''(r) - \frac{1}{r} \bar u'(r) = \lambda \bar u(r) + \bar F(r) }[/math]

with initial condition [math]\displaystyle{ \bar u(0) = u(0) }[/math].

We can solve this equation by separation of variables, writing

[math]\displaystyle{ \bar{u}(r) = a(r) J_0(\sqrt{\lambda} r) + b(r) Y_0(\sqrt{\lambda} r) }[/math]

[math]\displaystyle{ \bar{u}'(r) = \sqrt{\lambda} a(r) J_0'(\sqrt{\lambda} r) + \sqrt{\lambda} b(r) Y_0'(\sqrt{\lambda} r) }[/math].

Using the standard Wronskian identity

[math]\displaystyle{ J_0(\sqrt{\lambda} r) Y'_0(\sqrt{\lambda} r) - Y_0(\sqrt{\lambda} r) J'_0(\sqrt{\lambda} r) = \frac{2}{\sqrt{\lambda} \pi r} }[/math]

one soon arrives at the equations

[math]\displaystyle{ a(r) = \frac{\pi r}{2} ( \sqrt{\lambda} \bar u(r) Y'_0(\sqrt{\lambda} r) - \bar u'(r) Y_0(\sqrt{\lambda} r) ) }[/math]

and

[math]\displaystyle{ a'(r) = -\frac{\pi r}{2} \bar F(r) Y_0(\sqrt{\lambda} r) }[/math].

Meanwhile, by integration by parts

[math]\displaystyle{ a(0) = -\int_0^R a'(r) \eta(r)\ dr - \int_0^R a(r) \eta'(r)\ dr }[/math]

and thus

[math]\displaystyle{ |a(0)| \leq \frac{\pi}{2} ( \int_0^R |\bar F(r)| |Y_0(\sqrt{\lambda} r)|\ r dr + \int_0^R \sqrt{\lambda} |\bar u(r)| |Y'_0(\sqrt{\lambda} r)|\ r dr + \int_0^R |\bar u'(r)| |Y_0(\sqrt{\lambda} r)|\ r dr). }[/math]

From Cauchy-Schwarz one has

[math]\displaystyle{ 2\pi \int_0^R |\bar u(r)|^2\ r dr \leq \| u \|_{L^2(B(0,R))}^2 }[/math]

and

[math]\displaystyle{ 2\pi \int_0^R |\bar u'(r)|^2\ r dr \leq \| \nabla u \|_{L^2(B(0,R))}^2 }[/math]

and similarly

[math]\displaystyle{ 2\pi \int_0^R |\bar F(r)|^2\ r dr \leq \| F \|_{L^2(B(0,R))}^2. }[/math]

The claim then follows from further applications of the Cauchy-Schwarz inequality. []

In principle, the above inequality, when combined with the previous L^2 and H^1 estimates, gives L^infty control on [math]\displaystyle{ \dot u }[/math] in the interior of the triangle. One should also be able to get control near the interior of an edge by reflection. One needs to modify the argument a bit though to handle what is going on at vertices.

Here is an alternate approach that avoids the use of Bessel functions of the first and second kinds. Suppose one has a solution to the inhomogeneous eigenfunction equation [math]\displaystyle{ -\Delta u = \lambda u + F }[/math] on a ball B(0,R). Let [math]\displaystyle{ \eta }[/math] be a smooth (or at least C^2) cutoff on this ball which equals 1 at the origin. Then [math]\displaystyle{ u\eta }[/math] is compactly supported with Laplacian

[math]\displaystyle{ -\Delta(\eta u) = \lambda \eta u + \eta F - 2 \nabla \eta \cdot \nabla u -\Delta \eta u }[/math]

and hence by the fundamental solution to the Laplacian

[math]\displaystyle{ u(0) = \eta u(0) = \frac{1}{2\pi} \int_{B(0,R)} \log |x| (\lambda \eta u + \eta F - 2 \nabla \eta \cdot \nabla u -\Delta \eta u)(x)\ dx }[/math]

and hence by Cauchy-Schwarz

[math]\displaystyle{ |u(0)| \leq \frac{1}{2\pi} (\|u\|_{L^2(B(0,R))} \| \log|x| (\lambda \eta - \Delta \eta) \|_{L^2(B(0,R)} + 2 \|\nabla u \|_{L^2(B(0,R)} \| \nabla \eta \|_{L^2(B(0,R))} + \|F\|_{L^2(B(0,R)} \| \log|x| \eta \|_{L^2(B(0,R))}. }[/math]

Because [math]\displaystyle{ \Delta(\eta u) }[/math] has mean zero, one can also replace [math]\displaystyle{ \log |x| }[/math] here by [math]\displaystyle{ \log |x|-C }[/math] for any constant C, for instance one can use [math]\displaystyle{ \log(|x|/R) }[/math].