On Nonlinear Stability of Maxwellian states for discrete velocity models of the extended Boltzmann equation

M. Miklavčič^{∗} and G. Spiga^{†}

Abstract

By introducing appropriate weights in the energy method we propose a new technique for studying nonlinear stability of Maxwellian states for discrete velocity models (McKean and Broadwell) of the extended Boltzmann equation.

1 Introduction

Discrete velocity models of the Boltzmann equation have become quite popular in a recent past, both for their fluid dynamics applications [1] and for the mathematical problems they have raised [2]. Among the latter, a significant role is played by stability of equilibria (the so called Maxwellian states) and related relaxation problems [3] [4]. In some recent papers [5] [6] such a question has been addressed for the extended version of the simplest and most ancient discrete models (Carleman, McKean, and Broadwell). In such a context, test particles are allowed to interact with the field particles of a given background by absorption and generation events, in addition to the usual elastic scattering process. External sources may also be present. This new feature adds simply linear and constant terms to the usual quadratic terms in the collision part of the kinetic equations, however this changes considerably the structure of the equations themselves. In [5] [6] linear stability of Maxwellian states to space dependent perturbations was studied. In this paper we shall perform the nonlinear stability analysis on the real line in the presence of sources. This requires a generalization of the standard energy methods, which have always proved very powerful and effective in fluid dynamics [7], but fail for the considered kinetic problems. If the generalization may be guessed properly by clever inspection for the Carleman model, a specific careful treatment is needed for the other less elementary models.

∗Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA (milan@math.msu.edu)

†Department of Mathematics, University of Parma, Via M. D’Azeglio 85, 43100 Parma, Italy (spiga@prmat.math.unipr.it). The research was sponsored by MURST through its 40% and 60% grants, and by the National Group for Mathematical Physics of the Italian National Research Council.

In Section 2 the modified energy method is presented and discussed, with reference to a set of n semilinear hyperbolic partial differential equations with quadratic nonlinearities describing a general discrete velocity model in kinetic theory, even though more general nonlinear terms could be easily included. The established differential inequality determines a criterion for nonlinear stability, and guarantees exponential relaxation to equilibrium when the initial perturbation is small enough. The criterion can be checked analytically for n = 2, and may be applied then in order to prove stability for all two-velocity models. In higher dimension (n ≥ 3) a complete analytical solution of the related minimum problem is not practicable, in general, but significant information may be obtained from the analysis.

Section 3 is devoted to the extended McKean two-velocity model (n = 2). Multiple Maxwelian states may occur in this case, and stability of all of them for varying parameters is studied analytically. In the final Section the extended Broadwell model in three dimensions is illustrated as a prototype of the stability problems to be dealt with when n ≥ 3. Even for the values of parameters excluded by analytical results, numerical verification indicates actual stability in all of the many cases tested.

2 Modified Energy Method

Suppose that u(x, t) ∈ R^{n} satisfies

u_{t} + Du_{x} = Au + F (u),  (1)

where D is a diagonal n × n real matrix, A is n × n real matrix and F (u) ∈ R^{n} is given by

F (u)_{i} = ∑_{j,k} f_{ijk}u_{j}u_{k} i = 1, …, n  (2)

where f_{ijk} are some real numbers. We assume that u(·, t) ∈ L^{2}(R)^{n} and u_{x}(·, t) ∈ L^{2}(R)^{n} for t ≥ 0, that is, u(·, t) ∈ H^{1}(R)^{n} for t ≥ 0. This is the most general form of the extended discrete Boltzmann equation for a single gas in a participating background medium when all interactions are binary, after translating the considered Maxwellian state to the origin.

In [5] and [6] we studied the linearized version of (1-2) for some special important models (Carleman, Mc Kean, Broadwell) and concluded their stability. Here we would like to show that solutions of the corresponding nonlinear systems decay exponentially for all choices of parameters, provided that the initial values of u_{i} are small enough. We use a modified energy method with a careful selection of weights. Our approach is general enough to be applicable in principle to any problem of the kind (1-2), and thus to essentially any discrete velocity model of the extended Boltzmann equation. A special case was worked out directly in [6] for the extended Carleman model. The Mc Kean and Broadwell models will be specifically considered in Sections 3 and 4.

Let W be a diagonal matrix with diagonal entries w_{i} ≡ W_{ii} > 0, i = 1, …, n. Taking a scalar product, denoted as usually by (·, ·), of (1) with W u gives

(W u, u)_{t} + (W Du, u)_{x} = 2(W Au, u) + 2(W F (u), u).  (3)

Let λ_{w} be the smallest of real numbers λ for which

(W Az, z) ≤ λ(W z, z) for all z ∈ R^{n}.  (4)

Thus

\frac{1}{2} \frac{d}{dt} ∫_{−∞}^{∞}(W u, u)dx ≤ λ_{w} ∫_{−∞}^{∞}(W u, u)dx + ∫_{−∞}^{∞}(W F (u), u)dx.  (5)

Using Holder’s inequality twice gives

(W F (u), u) = ∑_{ijk} w_{i}f_{ijk}u_{i}u_{j}u_{k} = ∑_{jk} (∑_{i} w_{i}f_{ijk}u_{i}) u_{j}u_{k}

(W F (u), u)^{2} ≤ (∑_{jk} w_{j}w_{k}u_{j}^{2}u_{k}^{2}) ∑_{jk} \frac{(∑_{i} w_{i}f_{ijk}u_{i})^{2}}{w_{j}w_{k}}

= (W u, u)^{2} ∑_{jk} \frac{(∑_{i} w_{i}f_{ijk}u_{i})^{2}}{w_{j}w_{k}}

≤ C_{1}^{2}(W u, u)^{3}, where C_{1} = (∑_{ijk} \frac{w_{i}f_{ijk}^{2}}{w_{j}w_{k}} )^{1/2} .  (6)

Using (6) in (5) gives

\frac{1}{2} \frac{d}{dt} ∫_{−∞}^{∞}(W u, u)dx ≤ λ_{w} ∫_{−∞}^{∞}(W u, u)dx + C_{1}(W u, u)_{∞}^{1/2} ∫_{−∞}^{∞}(W u, u)dx  (7)

where (W u, u)_{∞} ≡ sup_{x}(W u(x, t), u(x, t)).

Taking a derivative of (1) with respect to x and taking the scalar product with W u_{x} gives

(W u_{x}, u_{x})_{t} + (W Du_{x}, u_{x})_{x} = 2(W Au_{x}, u_{x}) + 2(W F (u)_{x}, u_{x})
≤ 2λ_{w}(W u_{x}, u_{x}) + 2(W F (u)_{x}, u_{x}).  (8)

Note that

(F (u)_{i})_{x} = ∑_{jk}(f_{ijk} + f_{ikj})u_{k}(u_{j})_{x}

hence we can proceed, as in (6), to obtain

(W F (u)_{x}, u_{x}) ≤ C_{2}(W u, u)^{1/2}(W u_{x}, u_{x}) where C_{2} = (∑_{ijk} \frac{w_{i}(f_{ijk} + f_{ikj})^{2}}{w_{j}w_{k}} )^{1/2} . (9)

Using (9) in (8) and integrating gives

\frac{1}{2} \frac{d}{dt} ∫_{−∞}^{∞}(W u_{x}, u_{x})dx ≤ λ_{w} ∫_{−∞}^{∞}(W u_{x}, u_{x})dx + C_{2}(W u, u)_{∞}^{1/2} ∫_{−∞}^{∞}(W u_{x}, u_{x})dx. (10)

Define E(t) ≥ 0 by

E(t)^{2} = ∫_{−∞}^{∞}((W u, u) + (W u_{x}, u_{x}))dx.  (11)

Adding (7) and (10) gives

EE′ ≤ λ_{w}E^{2} + C_{3}(W u, u)_{∞}^{1/2}E^{2}, C_{3} = max{C_{1}, C_{2}}.  (12)

Note that f(x)^{2} = 2 ∫_{−∞}^{x} f(t)f′(t)dt ≤ 2∥f∥_{2}∥f′∥_{2} and hence

∥f∥_{∞}^{2} ≤ ∥f∥_{2}^{2} + ∥f′∥_{2}^{2} for f ∈ H^{1}(R).  (13)

(13) implies that (W u, u)_{∞}^{1/2} ≤ E and therefore

E′ ≤ λ_{w}E + C_{3}E^{2}  (14)

which implies that if λ_{w} < 0 then E(t) → 0 exponentially as t → ∞ provided that E(0) < −λ_{w}/C_{3}. In other words, if λ_{w} < 0 then all solutions of (1), which are small enough initially, decay exponentially.

2.1 About λ_{w}

Let A be a n × n real matrix and let W be a diagonal matrix with diagonal entries w_{i} ≡ W_{ii} > 0, i = 1, …, n. We defined λ_{w} to be the smallest of real numbers λ for which

(W Az, z) ≤ λ(W z, z) for all z ∈ R^{n}.  (15)

Note that λ_{w} is the largest eigenvalue of

A_{w} ≡ \frac{1}{2}(W^{1/2}AW^{−1/2} + W^{−1/2}A^{∗}W^{1/2}).  (16)

Consider for example

A = ( \begin{matrix}−1 & 4 \\ 0 & −1 \end{matrix} ) .  (17)

The matrix has eigenvalues -1. If we simply take w_{i} = 1, like in the usual energy approach, then λ_{w} = 1, which is useless in (14) - we need λ_{w} < 0. In general λ_{w} = −1 + 2\sqrt{w_{1}/w_{2}} in this example.

Thus, for given A we need to find weights w_{i} that give the smallest λ_{w}. If μ is any eigenvalue of A and Ax = μx then

Re μ (W x, x) = (A_{w}W^{1/2}x, W^{1/2}x) ≤ λ_{w}(W^{1/2}x, W^{1/2}x)  (18)

which implies

Re μ ≤ λ_{w}.  (19)

Thus there exists inf_{w_{i}>0} λ_{w} and we will denote it by Λ. Note that Λ may not be attained as the above example shows. In case of 2 × 2 matrices it is easy to see directly that

Λ = \frac{A_{11} + A_{22}}{2} + \sqrt{(\frac{A_{11} − A_{22}}{2} )^{2} + max\{0, A_{12}A_{21}\}}.  (20)

Λ is actually attained when A_{12}A_{21} ≠ 0, and the corresponding weights are given by w_{1}/w_{2} = |A_{21}/A_{12}|. Note that when A_{12}A_{21} ≥ 0 then Λ is equal to the largest eigenvalue of the 2 × 2 matrix A, whereas Λ is greater than the real part of any eigenvalue of A when A_{12}A_{21} < 0. Finding Λ for larger matrices is much more involved.

Note that λ_{w} is also the largest eigenvalue of

\frac{1}{2}(A + W^{−1}A^{∗}W).  (21)

Assume it to be simple and let z be the corresponding eigenvector. Hence

(W A + A^{∗}W)z = 2λ_{w}W z.  (22)

An infinitesimal change dW produces changes dz, dλ_{w} which satisfy

(dW A + A^{∗}dW)z + (W A + A^{∗}W)dz = 2dλ_{w} W z + 2λ_{w}dW z + 2λ_{w}W dz. (23)

Taking a scalar product with z gives

((dW A + A^{∗}dW)z, z) = 2dλ_{w}(W z, z) + 2λ_{w}(dW z, z)  (24)

which simplifies to

dλ_{w}(W z, z) = (Az − λ_{w}z, dW z)  (25)

and therefore

\frac{∂λ_{w}}{∂w_{i}} = \frac{z_{i}(Az − λ_{w}z)_{i}}{(W z, z)} , i = 1, …, n.  (26)

When the multiplicity of λ_{w} is bigger than 1 then (26) gives different directional derivatives that correspond to different eigenvectors and hence λ_{w} surface has a corner in such a case.

When the minimum λ_{w} is attained and is not in a corner then one can use (26) to find it. (26) suggests that we examine various cases separately. In the case when all z_{i} ≠ 0 we have to have

Az = λ_{w}z and (W A + A^{∗}W)z = 2λ_{w}W z  (27)

and hence

Az = λ_{w}z and A^{∗}W z = λ_{w}W z.  (28)

Thus, in this case, the smallest λ_{w} equals to the largest eigenvalue of A and it occurs when

w_{i} = y_{i}/z_{i} for i = 1, …, n  (29)

where y ≠ 0 is such that A^{∗}y = λ_{w}y. Thus (29) provides the optimal choice of the weights for all values of the matrix elements in A for which the previous assumptions hold.

3 Extended McKean Model

The most typical two velocity model of the extended Boltzmann equation, together with the Carleman model considered already in [6], is the extended McKean model, given by

\frac{∂N_{1}}{∂t} + \frac{∂N_{1}}{∂x} = −N_{1}N_{2} + N_{2}^{2} − εN_{1} + χηρ + S_{1}  (30)
\frac{∂N_{2}}{∂t} − \frac{∂N_{2}}{∂x} = N_{1}N_{2} − N_{2}^{2} − εN_{2} + (1 − χ)ηρ + S_{2}  (31)

where ρ = N_{1} + N_{2}. ε > 0, η > 0 and 0 ≤ χ ≤ 1 are constants. S_{i} ≥ 0 are sources and S = S_{1} + S_{2} > 0. ε represents an absorption coefficient, and η takes generation processes (like fission for neutrons, or ionization for electrons) into account. The factor χ describes the angular distribution of generated particles. The standard Mc Kean model used in classical kinetic theory corresponds to the limiting case η = ϵ = S = 0.

Nontrivial, nonnegative, constant solutions of (30-31) are called Maxwellian states. To obtain them, add (30) to (31), which gives again ρ(ε − η) = S and ε > η. Solving (31) when (1 − χ)ηρ + S_{2} > 0 gives

N_{2} = \frac{ρ − ε}{4} + \sqrt{\frac{(ρ − ε)^{2}}{16} + \frac{(1 − χ)ηρ + S_{2}}{2}} , N_{1} = ρ − N_{2}.  (32)

When (1 − χ)ηρ + S_{2} = 0, i.e. when χ = 1 and S_{2} = 0, then we have

N_{2} = 0, N_{1} = ρ  (33)

and when ρ > ε, i.e. S > ε(ε − η), we also have another solution

N_{2} = \frac{ρ − ε}{2}, N_{1} = \frac{ρ + ε}{2}.  (34)

Perturbations u_{i} of a Maxwellian state N_{1}, N_{2} satisfy

\frac{∂u_{1}}{∂t} + \frac{∂u_{1}}{∂x} = (−N_{2} − ε + χη)u_{1} + (2N_{2} − N_{1} + χη)u_{2} + u_{2}^{2} − u_{1}u_{2}  (35)
\frac{∂u_{2}}{∂t} − \frac{∂u_{2}}{∂x} = (N_{2} + (1 − χ)η)u_{1} + (N_{1} − 2N_{2} − ε + (1 − χ)η)u_{2} − u_{2}^{2} + u_{1}u_{2}. (36)

We will show now that perturbations decay exponentially, if they are initially small enough, for all values of parameters, except when the Maxwellian state is given by (33) with ρ ≥ ε. So, the Maxwellian state given by (33) is nonlinearly stable for ρ < ε and it looses its stability to the coexisting Maxwellian state (34) when ρ > ε. The Maxwellian state given by (32) is always nonlinearly stable.

Let us examine first the case when the Maxwellian state is given by (33). In this case

A = ( \begin{matrix}η − ε & η − ρ \\ 0 & ρ − ε \end{matrix} ) .  (37)

(20) implies that Λ = max{η − ε, ρ − ε}. Thus Λ < 0 iff ρ < ε which shows nonlinear stability of the Maxwellian state (33) for ρ < ε.

Let us examine next Maxwellian states (32) and (34). Note that N_{2} > 0 and

A = ( \begin{matrix}−N_{2} − ε + χη & 2N_{2} − N_{1} + χη \\ N_{2} + (1 − χ)η & N_{1} − 2N_{2} − ε + (1 − χ)η \end{matrix} )  (38)

and A_{21} > 0.

Let us consider first the case A_{12} ≥ 0. Note that Λ, given by (20), is in this case the largest eigenvalue of A. The characteristic polynomial of A can be written as

λ^{2} + (3N_{2} − N_{1} + 2ε − η)λ + (ε − η)(3N_{2} − N_{1} + ε) = 0  (39)

with roots η − ε < 0 and N_{1} − 3N_{2} − ε. Since N_{2} > 0, (31) implies that N_{1} − 3N_{2} − ε < 0 and hence Λ < 0.

When A_{12} < 0 then (20) implies that Λ = max{A_{11}, A_{22}}. Since obviously A_{11} < 0 we only need to show that A_{22} < 0 in order to conclude that Λ < 0. This is obvious for the Maxwellian state (34) since A_{22} = (ε − ρ)/2 < 0 (recall that χ = 1 in this case). The rest of this section is devoted to showing

A_{22} = N_{1} − 2N_{2} − ε + (1 − χ)η < 0  (40)

for the Maxwellian state (32). (40) is equivalent to

−\frac{ρ − ε}{4} + 3\sqrt{\frac{(ρ − ε)^{2}}{16} + \frac{(1 − χ)ηρ + S_{2}}{2}} − (1 − χ)η > 0  (41)

which is implied by

(ρ − ε)^{2} + (8ρ + ε)η(1 − χ) − 2η^{2}(1 − χ)^{2} + 9S_{2} > 0.  (42)

When χ = 1 then S_{2} > 0 for the Maxwellian state (32) and hence (42) holds. Assume from now on that χ < 1. (42) is implied by

F (η) ≡ (ρ − ε)^{2} + (8ρ + ε)η(1 − χ) − 2η^{2}(1 − χ)^{2} > 0.  (43)

Since F (0) ≥ 0, the concavity of F implies that it is enough to show that F (ε) ≥ 0 in order to conclude that F (η) > 0 for η ∈ (0, ε). To show that F (ε) ≥ 0 note that

F (ε) = ρ^{2} + (6 − 8χ)ρε + ε^{2}χ(3 − 2χ)  (44)

and also

F (ε) = (ρ + (3 − 4χ)ε)^{2} + 9ε^{2}(1 − χ)(2χ − 1).  (45)

So, when χ ≥ 1/2, then F (ε) ≥ 0 by (45). When χ < 3/4 the minimum, with respect to ρ, of (44) occurs at a negative value, however, (44) is nonnegative when ρ = 0.

4 Broadwell Model

Another typical discrete velocity model of the extended Boltzmann equation is the following six velocity Broadwell model

\frac{∂N_{1}}{∂t} + \frac{∂N_{1}}{∂x} = N_{3}^{2} − N_{1}N_{2} − εN_{1} + χ_{1}ηρ + S_{1}  (46)
\frac{∂N_{2}}{∂t} − \frac{∂N_{2}}{∂x} = N_{3}^{2} − N_{1}N_{2} − εN_{2} + χ_{2}ηρ + S_{2}  (47)
\frac{∂N_{3}}{∂t} = −\frac{1}{2}(N_{3}^{2} − N_{1}N_{2}) − εN_{3} + χ_{3}ηρ + S_{3}  (48)

on the line (−∞ < x < ∞), where ρ = N_{1} + N_{2} + 4N_{3}. Parameters ε > 0, η > 0, χ_{1} ≥ 0, χ_{2} ≥ 0, χ_{3} ≥ 0 are constants such that χ_{1} + χ_{2} + 4χ_{3} = 1. S_{i} ≥ 0 are sources and S = S_{1} + S_{2} + 4S_{3} > 0. The physical meaning of all of them is the same as before, with the χ_{i} describing the angular distribution of generated particles.

Maxwellian states are constants N_{1}, N_{2}, N_{3} that satisfy (46-48) and which are physically relevant, i.e. N_{i} ≥ 0 and ρ > 0. Adding (46-48) gives, in this case, ρ(ε − η) = S. Therefore

η < ε and ρ = S/(ε − η).  (49)

This enables us to define effective χ̃_{i} by

χ̃_{i}ερ = χ_{i}ηρ + S_{i} for i = 1, 2, 3.  (50)

Note that χ̃_{1} + χ̃_{2} + 4χ̃_{3} = 1. (46-48) imply, in this case, that

N_{1} = (q + χ̃_{1})ρ, N_{2} = (q + χ̃_{2})ρ, N_{3} = (−q/2 + χ̃_{3})ρ  (51)

where q = (N_{3}^{2} − N_{1}N_{2})/(ερ) and hence q has to satisfy

3q^{2}/4 + q(χ̃_{1} + χ̃_{2} + χ̃_{3} + ε/ρ^{∗}) + χ̃_{1}χ̃_{2} − χ̃_{3}^{2} = 0  (52)

which implies that

q = \frac{2(χ̃_{3}^{2} − χ̃_{1}χ̃_{2})}{χ̃_{1} + χ̃_{2} + χ̃_{3} + ε/ρ + \sqrt{(χ̃_{1} + χ̃_{2} + χ̃_{3} + ε/ρ)^{2} + 3(χ̃_{3}^{2} − χ̃_{1}χ̃_{2})}}.  (53)

Let N_{1}, N_{2}, N_{3} be a Maxwellian state and assume that perturbations N_{j} + u_{j}(x, t) also satisfy (46-48). This implies that

\frac{∂u_{1}}{∂t} + \frac{∂u_{1}}{∂x} = (χ_{1}η − ε − N_{2})u_{1} + (χ_{1}η − N_{1})u_{2} + (4χ_{1}η + 2N_{3})u_{3} + u_{3}^{2} − u_{1}u_{2} (54)
\frac{∂u_{2}}{∂t} − \frac{∂u_{2}}{∂x} = (χ_{2}η − N_{2})u_{1} + (χ_{2}η − ε − N_{1})u_{2} + (4χ_{2}η + 2N_{3})u_{3} + u_{3}^{2} − u_{1}u_{2} (55)
\frac{∂u_{3}}{∂t} = (χ_{3}η + \frac{N_{2}}{2} ) u_{1} + (χ_{3}η + \frac{N_{1}}{2} ) u_{2} + (4χ_{3}η − ε − N_{3})u_{3} − \frac{u_{3}^{2} − u_{1}u_{2}}{2} . (56)

This case belongs of course to the class defined by (1,2), but an analytical determination of Λ is impossible. However, under the conditions for which (29) is valid, the optimal weights are given by (29) itself, and a long calculation gives

w_{1} = \frac{1}{2N_{1}(χ_{1} − χ_{2}) + N_{3}(1 − χ_{2} + χ_{1}) + 2χ_{1}η}  (57)

w_{2} = \frac{1}{2N_{2}(χ_{2} − χ_{1}) + N_{3}(1 − χ_{1} + χ_{2}) + 2χ_{2}η}  (58)
w_{3} = \frac{4}{N_{1}χ_{2} + N_{2}χ_{1} + 2χ_{3}(N_{1} + N_{2} + η)}  (59)

and λ_{w} = η − ε < 0 - exactly what we need and expect physically. However, note that w_{1}, w_{2} can become negative when |χ_{1} − χ_{2}| is large - so other cases have to be considered. There are 6 other cases, corresponding to various z_{i} = 0, that have to be considered in case of 3 × 3 matrices, and in all of them proof of existence of a negative λ_{w} would be required for stability.

We conclude our investigation on the Broadwell model by presenting the results of extensive numerical computations aimed at checking analytical predictions and at getting more insight for the cases in which such predictions are still missing. More then 400000 random selections of parameters 0 < χ_{1} < 1, 0 < χ_{2} < 1 − χ_{1}, 0 < ε < 10, 0 < η < ε, 0 < S_{i} < 1 were made and in each case the smallest λ_{w} was found by checking all 7 cases as well as possibility of corners. In all cases the smallest λ_{w} was negative - which indicates stability of the Maxwellian states for each specific case.

In about 77% of cases it has turned out that the smallest λ_{w} is obtained by weights given in (57-59); an example

χ_{1} = 0.7, χ_{2} = 0.2, ε = 2, η = 1, S_{1} = S_{2} = S_{3} = 0.6, Λ = −1.  (60)

In about 8% of cases the smallest λ_{w} is obtained by setting (26) to 0 with z_{1} = 0; an example

χ_{1} = 0.1, χ_{2} = 0.8, ε = 2, η = 1, S_{1} = S_{2} = S_{3} = 0.6, Λ = −0.968360. (61)

In about 8% of cases the smallest λ_{w} is obtained by setting (26) to 0 with z_{2} = 0; an example

χ_{1} = 0.8, χ_{2} = 0.1, ε = 2, η = 1, S_{1} = S_{2} = S_{3} = 0.6, Λ = −0.968360. (62)

In about 7% of cases the smallest λ_{w} is obtained in a corner - when the multiplicity of λ_{w} is equal to 2; an example

χ_{1} = 0.7, χ_{2} = 0.1, ε = 2, η = 1, S_{1} = S_{2} = S_{3} = 0.6, Λ = −0.992814. (63)

These are the only cases that showed up. Choosing w_{i} = 1, like in the usual energy method, gives positive λ_{w} in each of the above examples (60), (61), (62) and (63).

References

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