5.1 The Laplace method

Upon linearization and localization, the IBVP (5.1View Equation, 5.2View Equation, 5.3View Equation) reduces to a linear, constant-coefficient problem of the following form,
∑n j ∂- ut = A ∂xju + F0(t,x), x ∈ Σ, t ≥ 0, (5.6 ) j=1 u (0, x) = f(x), x ∈ Σ, (5.7 ) bu = g(t,x), x ∈ ∂Σ, t ≥ 0, (5.8 )
where Aj = Aj (t0,x0,u (0)(t0,x0)), b = b(t0,x0, u(0)(t0,x0)) denote the matrix coefficients corresponding to Aj(t,x,u ) and b(t,x,u) linearized about a solution u(0) and frozen at the point p0 = (t0,x0), and where, for generality, we include the forcing term F0(t,x) with components in the class ∞ C b ([0,∞ ) × Σ ). Since the freezing process involves a zoom into a very small neighborhood of p0, we may replace Σ by ℝn for all points p0 lying inside the domain Σ. We are then back into the case of Section 3, and we conclude that a necessary condition for the IBVP (5.1View Equation, 5.2View Equation, 5.3View Equation) to be well posed at (0) u, is that all linearized, frozen coefficient Cauchy problems corresponding to p0 ∈ Σ are well posed. In particular, the equation (5.6View Equation) must be strongly hyperbolic.

Now let us consider a point p ∈ ∂Σ 0 at the boundary. Since ∂Σ is assumed to be smooth, it will be mapped to a plane during the freezing process. Therefore, taking points p0 ∈ ∂Σ, it is sufficient to consider the linear, constant coefficient IBVP (5.6View Equation, 5.7View Equation, 5.8View Equation) on the half space

Σ := {(x1,x2,...,xn ) ∈ ℝn : x1 > 0}, (5.9 )
say. This is the subject of this subsection. Because we are dealing with a constant coefficient problem on the half-space, we can reduce the problem to an ordinary differential boundary problem on the interval [0,∞ ) by employing Fourier transformation in the directions t and y := (x2,...,xn) tangential to the boundary. More precisely, we first exponentially damp the function u (t,x ) in time by defining for η > 0 the function
{ e− ηtu (t,x) for t ≥ 0,x ∈ Σ, uη(t,x) := 0 for t < 0,x ∈ Σ. (5.10 )
We denote by &tidle;u η(ξ, x1,k) the Fourier transformation of uη(t,x1,y) with respect to the directions t, and y tangential to the boundary and define the Laplace–Fourier transformation of u by
∫ ---1--- −st−ik⋅y n−1 &tidle;u(s,x1,k) := ˆuη(ξ,x1,k ) = (2π)n∕2 e u(t,x1,y)dtd y, s := η + iξ, (5.11 )
Then, &tidle;u satisfies the following boundary value problem,
∂-- &tidle; A ∂x1 &tidle;u = B (s, k)&tidle;u + F(s,x1, k), x1 > 0, (5.12 ) b&tidle;u = &tidle;g(s,k) x1 = 0, (5.13 )
where, for notational simplicity, we set 1 A := A and j j B := A, j = 2,...,n, and where B (s,k) := sI − iB2k2 − ...− iBnkn. Here, &tidle;F (s,x1,k) = &tidle;F0(s,x1,k ) + fˆ(x1, k) with &tidle;F0 and fˆ denoting the Laplace–Fourier and Fourier transform, respectively, of F0 and f, and &tidle;g(s,k) is the Laplace–Fourier transform of the boundary data g.

In the following, we assume for simplicity that the boundary matrix A is invertible, and that the equation (5.6View Equation) is strongly hyperbolic. An interesting example with a singular boundary matrix is mentioned in Example 26 below. If A can be inverted, then we rewrite Eq. (5.12View Equation) as the linear ordinary differential equation

-∂-- − 1 &tidle; ∂x &tidle;u = M (s,k )&tidle;u + A F (s,x1,k ), x1 > 0, (5.14 ) 1
where − 1 M (s,k) := A B(s,k ). We solve this equation subject to the boundary conditions (5.13View Equation) and the requirement that u&tidle; vanishes as x1 → ∞. For this, it is useful to have information about the eigenvalues of M (s,k).

Lemma 3 ([258Jump To The Next Citation Point, 259Jump To The Next Citation Point, 228Jump To The Next Citation Point]). Suppose the equation (5.6View Equation) is strongly hyperbolic and the boundary matrix A has q negative and m − q positive eigenvalues. Then, M (s,k ) has precisely q eigenvalues with negative real part and m − q eigenvalues with positive real part. (The eigenvalues are counted according to their algebraic multiplicity.) Furthermore, there is a constant δ > 0 such that the eigenvalues κ of M (s,k) satisfy the estimate

|Re (κ )| ≥ δRe (s), (5.15 )
for all Re (s) > 0 and k ∈ ℝn −1.

Proof. Let Re (s) > 0, β ∈ ℝ and n− 1 k ∈ ℝ. Then

[ ] M (s,k) − iβI = A −1 sI − iβA − ikjBj = A −1[sI − P0(iβ,ik)]. (5.16 )
Since the equation (5.6View Equation) is strongly hyperbolic there is a constant K and matrices S(β,k ) such that (see the comments below Definition 2)
−1 − 1 |S (β,k)| + |S (β,k) | ≤ K, S (β,k) P0 (iβ, ik)S(β,k ) = iΛ (β,k), (5.17 )
for all (β,k) ∈ ℝn, where Λ(β, k) is a real, diagonal matrix. Hence,
−1 −1 M (s,k ) − iβI = A S (β,k) [sI − iΛ (β, k)]S (β, k) , (5.18 )
and since sI − iΛ(β, k) is diagonal and its diagonal entries have real part greater than or equal to Re (s), it follows that
−1 − 1 −1 1 |[M (s,k) − iβI] | ≤ |A ||S (β,k)||S (β,k ) ||[sI − iΛ (β,k)] | ≤ -------, (5.19 ) δRe (s)
with δ := (K2 |A|)−1. Therefore, the eigenvalues κ of M (s,k) must satisfy
|κ − iβ| ≥ δRe (s) (5.20 )
for all β ∈ ℝ. Choosing β := Im (κ ) proves the inequality (5.15View Equation). Furthermore, since the eigenvalues κ = κ(s,k) can be chosen to be continuous functions of (s,k) [252], and since for k = 0, M (s,0) = sA− 1, the number of eigenvalues κ with positive real part is equal to the number of positive eigenvalues of A. □

According to this lemma, the Jordan normal form of the matrix M (s,k ) has the following form:

M (s,k ) = T(s,k) [D (s,k) + N (s,k)]T (s,k)−1, (5.21 )
with T (s,k) a regular matrix, N (s,k ) is nilpotent (m N (s,k) = 0) and
D (s,k) = diag(κ1,...,κq,κq+1,...,κm ) (5.22 )
is the diagonal matrix with the eigenvalues of M (s,k ), where κ1,...,κq have negative real part. Furthermore, N (s,k) commutes with D (s,k). Transforming to the variable −1 &tidle;v(s,x, k) := T (s,k) &tidle;u(s,x, k) the boundary value problem (5.12View Equation, 5.13View Equation) simplifies to
-∂- −1 −1 &tidle; ∂x1&tidle;v = [D (s,k ) + N (s,k)]&tidle;v + T (s,k) A F (s, x1,k), x1 > 0, (5.23 ) bT (s, k)&tidle;v = &tidle;g(s,k) x1 = 0. (5.24 )

5.1.1 Necessary conditions for well-posedness and the Lopatinsky condition

Having cast the IBVP into the ordinary differential system (5.23View Equation, 5.24View Equation), we are ready to obtain a simple necessary condition for well-posedness. For this, we consider the problem for F&tidle; = 0 and split &tidle;v = (&tidle;v− ,&tidle;v+) where v&tidle;− := (&tidle;v1,...,&tidle;vq) and v&tidle;+ := (&tidle;vq+1,...,v&tidle;m ) are the variables corresponding to the eigenvalues of M (s,k) with negative and positive real parts, respectively. Accordingly, we split

( D (s,k ) 0 ) ( N (s,k ) 0 ) D (s,k) = − , N (s,k) = − , (5.25 ) 0 D+ (s,k ) 0 N+ (s,k)
and bT(s,k) = (b− (s,k ),b+ (s,k)). When &tidle;F = 0 the most general solution of Eq. (5.23View Equation) is
&tidle;v− (s,x1,k ) = eD −(s,k)x1eN− (s,k)x1σ− (s,k ), D+ (s,k)x1 N+ (s,k)x1 &tidle;v+(s,x1,k ) = e e σ+(s,k ),
with constant vectors σ− (s,k ) ∈ ℂq and σ+ (s,k) ∈ ℂm −q. The expression for &tidle;v+ describes modes that grow exponentially in x1 and do not satisfy the required boundary condition at x1 → ∞ unless σ+ (s,k) = 0; hence, we set σ+ (s,k) = 0. In view of the boundary conditions (5.24View Equation), we then obtain the algebraic equation
b− (s,k )σ − (s,k) = &tidle;g. (5.26 )
Therefore, a necessary condition for existence and uniqueness is that the r × q matrix b− (s,k ) be a square matrix, i.e., r = q, and that
det(b− (s,k)) ⁄= 0 (5.27 )
for all Re(s) > 0 and k ∈ ℝn− 1. Let us make the following observations:

Example 25. Consider the IBVP for the massless Dirac equation in two spatial dimensions (cf. Section 8.4.1 in [259Jump To The Next Citation Point]),

( ) ( ) ( ) 1 0 0 1 u1 ut = 0 − 1 ux + 1 0 uy, t ≥ 0, x ≥ 0, y ∈ ℝ, u = u (5.32 ) 2 u(0,x,y ) = f (x,y), x ≥ 0, y ∈ ℝ, (5.33 ) au1 + bu2 = g(t,y), t ≥ 0, y ∈ ℝ, (5.34 )
where a and b are two complex constants to be determined. Assuming f = 0, Laplace–Fourier transformation leads to the boundary-value problem
( s − ik ) &tidle;ux = M (s,k)&tidle;u, x > 0, M (s,k) = (5.35 ) ik − s a&tidle;u1 + b&tidle;u2 = &tidle;g(s,k ), x = 0. (5.36 )
The eigenvalues and corresponding eigenvectors of the matrix M (s,k) are κ± = ±λ and T e± = (ik, s ∓ λ ), with √ ------- λ := s2 + k2, where the root is chosen such that Re(λ ) > 0 for Re (s) > 0. The solution, which is square integrable on [0,∞ ), is the one associated with κ−; that is,
&tidle;u(s,x,k) = σe −λxe , (5.37 ) −
with σ a constant. Introduced into the boundary condition (5.36View Equation) leads to the condition
[ika + (s + λ)b]σ = &tidle;g(s,k ), (5.38 )
and the Lopatinsky condition is satisfied if and only if the expression inside the square brackets on the left-hand side is different from zero for all Re(s) > 0 and k ∈ ℝ. Clearly, this implies b ⁄= 0, since otherwise this expression is zero for k = 0. Assuming b ⁄= 0 and k ⁄= 0, we then obtain the condition
z + √z2-+--1 ± ia-⁄= 0, (5.39 ) b
for all z := s ∕|k | with Re (z) > 0, which is the case if and only if |a∕b | ≤ 1 or a∕b ∈ ℝ; see Figure 1View Image. The particular case a = 0, b = 1 corresponds to fixing the incoming normal characteristic field u2 to g at the boundary.
View Image

Figure 1: Image of the lines Re (z) = const > 0 under the map ℂ → ℂ, √ -2---- z ↦→ z + z + 1.

Example 26. We consider the Maxwell evolution equations of Example 15 on the half-space x > 0 1, and freeze the incoming normal characteristic fields to zero at the boundary. These fields are the ones defined in Eq. (3.54View Equation), which correspond to negative eigenvalues and k = ∂x;19 hence

μ- E1 + β (W22 + W33 ) = 0, EA + W1A − (1 + α )WA1 = 0, x1 = 0, xA ∈ ℝ, t ≥ 0,(5.40 )
where A = 2,3 label the coordinates tangential to the boundary, and where we recall that √ --- μ = α β, assuming that α and β have the same sign such that the evolution system (3.50View Equation, 3.51View Equation) is strongly hyperbolic. In this example, we apply the Lopatinsky condition in order to find necessary conditions for the resulting IBVP to be well posed. For simplicity, we assume that μ = √ αβ-= 1, which implies that the system is strongly hyperbolic for all values of α ⁄= 0, but symmetric hyperbolic only if − 3∕2 < α < 0; see Example 15.

In order to analyze the system, it is convenient to introduce the variables U1 := W22 + W33, U := W − (1 + α )W A 1A A1, Z := βW − (1 + β∕2)U 11 1, and ¯W := W − δ U ∕2 AB AB AB 1, which are motivated by the form of the characteristic fields with respect to the direction k = − ∂1 normal to the boundary x1 = 0; see Example 15. With these assumptions and definitions, Laplace–Fourier transformation of the system (3.50View Equation, 3.51View Equation) yields

[ ] s &tidle;E = − α ∂ &tidle;U + ikA (1 + α )&tidle;U + α (2 + α ) &tidle;W , 1 1 1 A A1 &tidle; &tidle; B [ &tidle;¯ &tidle;¯ ] [ &tidle; &tidle; ] sEA = − ∂1UA − ik WBA − (1 + α)WAB − αikA αZ + (1 + α)U1 , 1 [ ] s&tidle;U1 = − -- ∂1 &tidle;E1 + (1 + α)ikAE&tidle;A , α sU&tidle;A = − ∂1E&tidle;A + (1 + α)ikAE&tidle;1, 3 +-2-α A s&tidle;Z = 2 α ik &tidle;EA, &tidle; &tidle; sWA1 = − ikAE1, &tidle;¯ &tidle; -i C &tidle; sWAB = − ikAEB + 2 δABk EC ,
where we have used β = 1∕α since μ = 1. The last three equations are purely algebraic and can be used to eliminate the zero speed fields &tidle;Z, W&tidle;A1 and &tidle;¯WAB from the remaining equations. The result is the ordinary differential system
∂ &tidle;E = − αs &tidle;U − (1 + α)ikAE&tidle; , 1 1 [ 1 2] A ∂ &tidle;U = − s-− (2 + α )|k-|- E&tidle; + 1 +-α-ikAU&tidle; , 1 1 α s 1 α A &tidle; &tidle; &tidle; ∂1EA = − sU[A + (1 +] α)ikAE1, &tidle; |k-|2- &tidle; 2kAkB--&tidle; &tidle; ∂1UA = − s + s EA + (1 + α) s EB − α (1 + α )ikA U1.
In order to diagonalize this system, we decompose E&tidle;A and &tidle;UA into their components parallel and orthogonal to k; if ˆk := k∕|k| and ˆl form an orthonormal basis of the boundary x1 = 0,20 then these are defined as
E&tidle; := ˆkAE&tidle; , &tidle;E := ˆlAE&tidle; , &tidle;U := ˆkAU&tidle; , &tidle;U := ˆlAU&tidle; . (5.41 ) || A ⊥ A || A ⊥ A
Then, the system decouples into two blocks, one comprising the transverse quantities (E&tidle;⊥, &tidle;U ⊥) and the other the quantities (E&tidle;1, &tidle;U1,E&tidle;||, &tidle;U||). The first block gives
( ) ( ) ( ) ∂ E&tidle;⊥ = [ 0 2]− s &tidle;E⊥ , (5.42 ) 1 U&tidle;⊥ − s + |k|s- 0 &tidle;U⊥
and the corresponding solutions with exponential decay at x1 → ∞ have the form
( &tidle;E (s,x ,k)) ( s) &tidle;⊥ 1 = σ0e−λx1 , (5.43 ) U⊥ (s,x1,k) λ
where σ0 is a complex constant, and where we have defined ∘ --------- λ := s2 + |k|2 with the root chosen such that Re (λ) > 0 for Re (s) > 0. The second block is
( &tidle; ) ( 0 − αs − i(1 + α)|k| 0 ) ( &tidle; ) E1 s |k|2 1+α E1 ∂ || &tidle;U1 || = || − α-+ (2 + α)-s- 0 0 i-α-|k||| || &tidle;U1 || , (5.44 ) 1 ( &tidle;E||) ( i(1 + α)|k | 0 0 − s ) ( &tidle;E||) &tidle;U 0 − iα (1 + α)|k| − s + α(2 + α)|k|2 0 &tidle;U || s ||
with corresponding decaying solutions
( ) ( ) ( ) E&tidle;1 (s,x1,k) i|k|s isλ | U&tidle; (s,x ,k) | | − i|k |λ | | i(s2∕α − |k|2) | |( &tidle;1 1 |) = σ1e−λx1|( |) + σ2e− λx1 |( |) , (5.45 ) E ||(s,x1,k) 2 sλ 2 |k|s U&tidle;||(s,x1,k) s − α |k | − α|k|λ
with complex constants σ1 and σ2.

On the other hand, Laplace–Fourier transformation of the boundary conditions (5.40View Equation) leads to

E&tidle;1 + α&tidle;U1 = 0, &tidle;EA + U&tidle;A = 0, x1 = 0. (5.46 )
Introducing into this solutions (5.43View Equation, 5.45View Equation) gives
(s + λ)σ0 = 0 (5.47 )
and
( ) ( ) |k |(s − α λ) sλ + s2 − α |k|2) σ1 sλ + s2 − α|k|2 |k|(s − αλ ) σ2 = 0. (5.48 )
In the first case, since Re(s + λ) ≥ Re (s) > 0, we obtain σ0 = 0 and there are no simple wave solutions in the transverse sector. In the second case, the determinant of the system is
2[ 2 2 2] − s (s + λ) − (1 + α) |k| , (5.49 )
which is different from zero if and only if ------ z + √ z2 + 1 ⁄= ± (1 + α) for all Re(z) > 0, where z := s∕|k|. Since α is real, this is the case if and only if − 2 ≤ α ≤ 0; see Figure 1View Image.

We conclude that the strongly hyperbolic evolution system (3.50View Equation, 3.51View Equation) with α β = 1 and incoming normal characteristic fields set to zero at the boundary does not give rise to a well-posed IBVP when α > 0 or α < − 2. This excludes the parameter range − 3∕2 < α < 0 for which the system is symmetric hyperbolic. This case is covered by the results in Section 5.2, which utilize energy estimates and show that symmetric hyperbolic problems with zero incoming normal characteristic fields are well posed.

5.1.2 Sufficient conditions for well-posedness and boundary stability

Next, let us discuss sufficient conditions for the linear, constant coefficient IBVP (5.6View Equation, 5.7View Equation, 5.8View Equation) to be well posed. For this, we first transform the problem to trivial initial data by replacing u(t,x) with u (t,x) − e−tf(x ). Then, we obtain the IBVP

∑n j ∂ ut = A ∂xju + F (t,x ), x ∈ Σ, t ≥ 0, (5.50 ) j=1 u(0,x ) = 0, x ∈ Σ, (5.51 ) bu = g(t,x), x ∈ ∂Σ, t ≥ 0, (5.52 )
with ∑n F (t,x) = F0(t,x) − e−t[f (x) + Aj ∂jf (x)] j=1 ∂x and g(t,x ) replaced by g(t,x ) + e− tbf (x ). By applying the Laplace–Fourier transformation to it, one obtains the boundary-value problem (5.12View Equation, 5.13View Equation), which could be solved explicitly, provided the Lopatinsky condition holds. However, in view of the generalization to variable coefficients, one would like to have a method that does not rely on the explicit representation of the solution in Fourier space.

In order to formulate the next definition, let Ω := [0,∞ ) × Σ¯ be the bulk and 𝒯 := [0,∞ ) × ∂ Σ the boundary surface, and introduce the associated norms ∥ ⋅ ∥η,0,Ω and ∥ ⋅ ∥ η,0,𝒯 defined by

∫ ∫ ∥u∥2η,0,Ω := e−2ηt|u(t,x1,y)|2dtdx1dn−1y = |u η(t,x )|2dtdnx, Ω ℝn+1 ∫ ∫ ∥u∥2 := e−2ηt|u(t,0,y)|2dtdn−1y = |u η(t,0,y )|2dtdn−1y, η,0,𝒯 𝒯 ℝn
where we have used the definition of u η as in Eq. (5.10View Equation). Using Parseval’s identities we may also rewrite these norms as
∫ ⌊∫∞ ( ∫ ) ⌋ 2 ⌈ ( 2 n−1 ) ⌉ ∥u∥ η,0,Ω := |&tidle;u(η + iξ,x1,k )| d k dx1 dξ, (5.53 ) ℝ 0 ℝn−1 ∫ ( ∫ ) 2 ( 2 n−1 ) ∥u∥η,0,𝒯 := |&tidle;u(η + iξ,0,k)| d k dξ. (5.54 ) ℝ ℝn−1

The relevant concept of well-posedness is the following one.

Definition 6. [258Jump To The Next Citation Point] The IBVP (5.50View Equation, 5.51View Equation, 5.52View Equation) is called strongly well posed in the generalized sense if there is a constant K > 0 such that each compatible data ∞ F ∈ C 0 (Ω) and ∞ g ∈ C0 (𝒯 ) gives rise to a unique solution u satisfying the estimate

( ) 2 2 2 1- 2 2 η∥u ∥η,0,Ω + ∥u∥η,0,𝒯 ≤ K η ∥F ∥η,0,Ω + ∥g ∥η,0,𝒯 , (5.55 )
for all η > 0.

The inequality (5.55View Equation) implies that both the bulk norm ∥ ⋅ ∥ η,0,Ω and the boundary norm ∥ ⋅ ∥η,0,𝒯 of u are bounded by the corresponding norms of F and g. For a trivial source term, F = 0, the inequality (5.55View Equation) implies, in particular,

∥u ∥η,0,𝒯 ≤ K ∥g∥η,0,𝒯, η > 0, (5.56 )
which is an estimate for the solution at the boundary in terms of the norm of the boundary data g. In view of Eq. (5.54View Equation) this is equivalent to the following requirement.

Definition 7. [259Jump To The Next Citation Point, 267Jump To The Next Citation Point] The boundary problem (5.50View Equation, 5.51View Equation, 5.52View Equation) is called boundary stable if there is a constant K > 0 such that all solutions &tidle;u(s,⋅,k ) ∈ L2 (0,∞ ) of Eqs. (5.12View Equation, 5.13View Equation) with F&tidle;= 0 satisfy

|&tidle;u(s,0,k)| ≤ K |&tidle;g (s,k )| (5.57 )
for all Re (s) > 0 and k ∈ ℝn −1.

Since boundary stability only requires considering solutions for trivial source terms, F = 0, it is a much simpler condition than Eq. (5.55View Equation). Clearly, strong well-posedness in the generalized sense implies boundary stability. The main result is that, modulo technical assumptions, the converse is also true: boundary stability implies strong well-posedness in the generalized sense.

Theorem 5. [258Jump To The Next Citation Point, 340] Consider the linear, constant coefficient IBVP (5.50View Equation, 5.51View Equation, 5.52View Equation) on the half space Σ = {(x1,x2,...,xn ) ∈ ℝn : x1 > 0}. Assume that equation (5.50View Equation) is strictly hyperbolic, meaning that the eigenvalues of the principal symbol P0(ik) are distinct for all k ∈ Sn− 1. Assume that the boundary matrix A = A1 is invertible. Then, the problem is strongly well posed in the generalized sense if and only if it is boundary stable.

Maybe the importance of Theorem 5 is not so much its statement, which concerns only the linear, constant coefficient case for which the solutions can also be constructed explicitly, but rather the method for its proof, which is based on the construction of a smooth symmetrizer symbol, and which is amendable to generalizations to the variable coefficient case using pseudo-differential operators.

In order to formulate the result of this construction, define ∘ ---------- ρ := |s|2 + |k|2, s′ := s∕ρ, k′ := k∕ρ, such that (s′,k′) ∈ Sn + lies on the half sphere Sn := {(s′,k′) ∈ ℂ × ℝn : |s′|2 + |k ′|2 = 1,Re(s′) > 0} + for Re (s) > 0 and n−1 k ∈ ℝ. Then, we have,

Theorem 6. [258Jump To The Next Citation Point] Consider the linear, constant coefficient IBVP (5.50View Equation, 5.51View Equation, 5.52View Equation) on the half space Σ. Assume that equation (5.50View Equation) is strictly hyperbolic, that the boundary matrix 1 A = A is invertible, and that the problem is boundary stable. Then, there exists a family of complex m × m matrices H (s′,k), (s′,k) ∈ Sn+, whose coefficients belong to the class C ∞ (Sn+ ), with the following properties:

  1. ′ ′ ′ ′∗ H (s,k ) = H (s,k ) is Hermitian.
  2. H (s′,k ′)M (s′,k′) + M (s′,k′)∗H (s′,k′) ≥ 2Re (s′)I for all (s′,k ′) ∈ Sn+.
  3. There is a constant C > 0 such that
    &tidle;u∗H (s′,k′)u&tidle;+ C|b&tidle;u|2 ≥ |u&tidle;|2 (5.58)
    for all m &tidle;u ∈ ℂ and all ′ ′ + (s ,k ) ∈ Sn.

Furthermore, H can be chosen to be a smooth function of the matrix coefficients of j A and b.

Let us show how the existence of the symmetrizer H (s′,k ′) implies the estimate (5.55View Equation). First, using Eq. (5.14View Equation) and properties (i) and (ii) we have

( ) ∂ ∗ ′ ′ ∂ &tidle;u ∗ ′ ′ ∗ ′ ′ ∂&tidle;u ----[&tidle;u H (s,k )&tidle;u] = ---- H (s ,k )&tidle;u + &tidle;u H (s,k )---- ∂x1 ∂x1 ∂x1 ( ) = ρ&tidle;u∗ [H (s′,k′)M (s′,k ′) + M (s′,k′)∗H (s′,k′)] &tidle;u + 2Re &tidle;u∗H (s′,k ′)A −1 &tidle;F 2 2 1-- ′ ′ −1 &tidle; 2 ≥ 2Re (s)|&tidle;u| − C1|&tidle;u| − C1 |H (s ,k)A F |,
where we have used the fact that ′ ′ M (s,k) = ρM (s ,k ) in the second step, and the inequality 2Re (a∗b) ≤ 2|a||b| ≤ C1|a|2 + C −11|b|2 for complex numbers a and b and any positive constant C1 > 0 in the third step. Integrating both sides from x1 = 0 to ∞ and choosing C1 = Re(s), we obtain, using (iii),
∞ ∞ ∫ 1 ∫ Re(s) |&tidle;u|2dx1 ≤ − [&tidle;u∗H (s′,k ′)&tidle;u]x1=0 + ------ |HA −1 &tidle;F |2dx1 0 Re(s) 0 ∫∞ 2| 2 1 − 1 2 ≤ − |&tidle;u| |x1=0 + C |&tidle;g| + ------ |HA F&tidle;| dx1. (5.59 ) Re (s)0
Since H is bounded, there exists a constant C2 > 0 such that −1 &tidle; &tidle; |HA F | ≤ C2|F | for all (s′,k′) ∈ Sn+. Integrating over ξ = Im (s) ∈ ℝ and k ∈ ℝn −1 and using Parseval’s identity, we obtain from this
2 2 C22 2 2 η∥u ∥η,0,Ω + ∥u ∥η,0,𝒯 ≤ ---∥F ∥η,0,Ω + C ∥g∥η,0,𝒯, (5.60 ) η
and the estimate (5.55View Equation) follows with K2 := max {C22,C }.

Example 27. Let us go back to Example 25 of the 2D Dirac equation on the halfspace with boundary condition (5.34View Equation) at x = 0. The solution of Eqs. (5.35View Equation, 5.36View Equation) at the boundary is given by T &tidle;u (s,0, k) = σ(ik,s + λ), where √--2---2 λ = s + k, and

σ = ----&tidle;g(s,k)----. (5.61 ) ika + (s + λ)b
Therefore, the IBVP is boundary stable if and only if there exists a constant K > 0 such that
∘ ------------- k2 + |s + λ|2 ----------------≤ K (5.62 ) |ika + (s + λ )b|
for all Re(s) > 0 and k ∈ ℝn− 1. We may assume b ⁄= 0, otherwise the Lopatinsky condition is violated. For k = 0 the left-hand side is 1∕|b|. For k ⁄= 0 we can rewrite the condition as
1 ∘1--+-|ψ(z)|2 -------------a-- ≤ K, (5.63 ) |b| |ψ(z) ± ib|
for all Re(z) > 0, where ψ (z) := z + √z2-+-1 and z := s∕ |k|. This is satisfied if and only if the function a |ψ(z) ± ib| is bounded away from zero, which is the case if and only if |a∕b| < 1; see Figure 1View Image.

This, together with the results obtained in Example 25, yields the following conclusions: the IBVP (5.32View Equation, 5.33View Equation, 5.34View Equation) gives rise to an ill-posed problem if b = 0 or if |a∕b | > 1 and a∕b∈∕ ℝ and to a problem, which is strongly well posed in the generalized sense if b ⁄= 0 and |a ∕b| < 1. The case |a| = |b| ⁄= 0 is covered by the energy method discussed in Section 5.2. For the case |a ∕b| > 1 with a∕b ∈ ℝ, see Section 10.5 in [228Jump To The Next Citation Point].

Before discussing second-order systems, let us make a few remarks concerning Theorem 5:

5.1.3 Second-order systems

It has been shown in [267Jump To The Next Citation Point] that certain systems of wave problems can be reformulated in such a way that they satisfy the hypotheses of Theorem 6. In order to illustrate this, we consider the IBVP for the wave equation on the half-space n Σ := {(x1, x2,...,xn) ∈ ℝ : x1 > 0}, n ≥ 1,

vtt = Δv + F (t,x), x ∈ Σ, t ≥ 0, (5.65 ) v(0,x ) = 0, v (0,x) = 0, x ∈ Σ, (5.66 ) t Lv = g(t,x), x ∈ ∂Σ, t ≥ 0, (5.67 )
where F ∈ C ∞ ([0,∞ ) × Σ ) 0 and g ∈ C ∞([0,∞ ) × ∂Σ ) 0, and where L is a first-order linear differential operator of the form
∑n L := a ∂--− b-∂--− c -∂--, (5.68 ) ∂t ∂x1 j∂xj j=2
where a, b, c2, …, cn are real constants. We ask under which conditions on these constants the IBVP (5.65View Equation, 5.66View Equation, 5.67View Equation) is strongly well posed in the generalized sense. Since we are dealing with a second-order system, the estimate (5.55View Equation) in Definition 6 has to be replaced with
( 1 ) η∥v ∥2η,1,Ω + ∥v∥2η,1,𝒯 ≤ K2 --∥F ∥2η,0,Ω + ∥g ∥2η,0,𝒯 , (5.69 ) η
where the norms ∥ ⋅ ∥2 η,1,Ω and ∥ ⋅ ∥2 η,1,𝒯 control the first partial derivatives of v,
∫ ∑n ||∂v ||2 ∥v ∥2η,1,Ω := e−2ηt ||--μ(t,x1,y)|| dtdx1dn −1y, Ω μ=0 ∂x ∫ n | |2 2 −2ηt∑ ||∂v-- || n− 1 ∥v∥ η,1,𝒯 := e |∂xμ(t,0,y)| dtd y, 𝒯 μ=0
with (xμ ) = (t,x ,x ,...,x ) 1 2 n. Likewise, the inequality (5.57View Equation) in the definition of boundary stability needs to be replaced by
|&tidle;g(s,k)| |&tidle;u(s,0,k)| ≤ K ∘-----------. (5.70 ) |s|2 + |k|2

Laplace–Fourier transformation of Eqs. (5.65View Equation, 5.67View Equation) leads to the second-order differential problem

-∂2- 2 2 &tidle; ∂x2 &tidle;v = (s + |k|)&tidle;v − F , x1 > 0, (5.71 ) 1 b--∂-&tidle;v = (as − ic(k))&tidle;v − &tidle;g, x1 = 0, (5.72 ) ∂x1
where we have defined ∑n c(k) := j=2 cjkj and where &tidle; F and &tidle;g denote the Laplace–Fourier transformations of F and g, respectively. In order to apply the theory described in Section 5.1.2, we rewrite this system in first-order pseudo-differential form. Defining
( ) ( ) ρ&tidle;v 0 &tidle;u := -∂v&tidle; , &tidle;f := − &tidle;F , (5.73 ) ∂x1
where ∘ ---------- ρ := |s|2 + |k |2, we find
-∂-- &tidle; ∂x u&tidle;= M (s,k)&tidle;u + f , x1 > 0, (5.74 ) 1 L(s,k)&tidle;u = g&tidle;, x1 = 0, (5.75 )
where we have defined
( ) M (s,k) := ρ 0 1 , L(s,k) := (as′ − ic(k′),− b) , (5.76 ) s′2 + |k′|2 0
with ′ s := s∕ρ, ′ k := k∕ρ. This system has the same form as the one described by Eqs. (5.14View Equation, 5.13View Equation), and the eigenvalues of the matrix M (s,k ) are distinct for Re (s) > 0 and k ∈ ℝn− 1. Therefore, we can construct a symmetrizer H (s′,k ′) according to Theorem 6 provided that the problem is boundary stable. In order to check boundary stability, we diagonalize M (s,k) and consider the solution of Eq. (5.74View Equation) for &tidle; f = 0, which decays exponentially as x1 → ∞,
( ρ ) &tidle;u(s,x1,k) = σ− e−λx1 , (5.77 ) − λ
where σ− is a complex constant and ∘ --------- λ := s2 + |k|2 with the root chosen such that Re (λ) > 0 for Re (s) > 0. Introduced into the boundary condition (5.75View Equation), this gives
′ ′ ′ &tidle;g- [as + bλ − ic(k )]σ− = ρ, (5.78 )
and the system is boundary stable if and only if the expression inside the square parenthesis is different from zero for all Re(s′) ≥ 0 and k′ ∈ ℝn−1 with |s′|2 + |k′|2 = 1. In the one-dimensional case, n = 1, this condition reduces to (a + b)s′ = 0 with |s′| = 1, and the system is boundary stable if and only if a + b ⁄= 0; that is, if and only if the boundary vector field L is not proportional to the ingoing null vector at the boundary surface,
-∂-+ -∂-. (5.79 ) ∂t ∂x1
Indeed, if a + b = 0, Lu = a(u + u ) t x1 is proportional to the outgoing characteristic field, for which it is not permitted to specify boundary data since it is completely determined by the initial data.

When n ≥ 2 it follows that b must be different from zero since otherwise the square parenthesis is zero for purely imaginary s′ satisfying as′ = ic(k′). Therefore, one can choose b = 1 without loss of generality. It can then be shown that the system is boundary stable if and only if a > 0 and ∑n |cj|2 < a2 j=2; see [267Jump To The Next Citation Point], which is equivalent to the condition that the boundary vector field L is pointing outside the domain, and that its orthogonal projection onto the boundary surface 𝒯,

∑n T := a ∂-− c --∂-, (5.80 ) ∂t j∂xj j=2
is future-directed time-like. This includes as a particular case the “Sommerfeld” boundary condition ut − ux = 0 1 for which L is the null vector obtained from the sum of the time evolution vector field ∂t and the normal derivative N = − ∂x1. While N is uniquely determined by the boundary surface 𝒯, ∂t is not unique, since one can transform it to an arbitrary future-directed time-like vector field T, which is tangent to 𝒯 by means of an appropriate Lorentz transformation. Since the wave equation is Lorentz-invariant, it is clear that the new boundary vector field ˆL = T + N must also give rise to a well-posed IBVP, which explains why there is so much freedom in the choice of L.

For a more geometric derivation of these results based on estimates derived from the stress-energy tensor associated to the scalar field v, which shows that the above construction for L is sufficient for strong well-posedness; see Appendix B in [263Jump To The Next Citation Point]. For a generalization to the shifted wave equation; see [369Jump To The Next Citation Point].

As pointed out in [267Jump To The Next Citation Point], the advantage of obtaining a strong well-posedness estimate (5.69View Equation) for the scalar-wave problem is the fact that it allows the treatment of systems of wave equations where the boundary conditions can be coupled in a certain way through terms involving first derivatives of the fields. In order to illustrate this with a simple example, consider a system of two wave equations,

( v ) ( v ) ( F (t,x)) 1 = Δ 1 + 1 , x ∈ Σ, t ≥ 0, (5.81 ) v2 tt v2 F2(t,x)
which is coupled through the boundary conditions
( ) ( ) ( ) ( ) ∂--− -∂-- v1 = N v1 + g1(t,x) , x ∈ ∂Σ, t ≥ 0, (5.82 ) ∂t ∂x1 v2 v2 g2(t,x)
where N has the form
( 0 0 ) ∂ ∂ ∂ N = , X = X0 ---+ X1 ----+ ...+ Xn ----, (5.83 ) 0 X ∂t ∂x1 ∂xn
with (X0, X1, ...,Xn ) ∈ ℂn+1 any vector. Since the wave equation and boundary condition for v 1 decouples from the one for v 2, we can apply the estimate (5.69View Equation) to v 1, obtaining
( 1 ) η∥v1∥2η,1,Ω + ∥v1∥2η,1,𝒯 ≤ K2 -∥F1∥2η,0,Ω + ∥g1∥2η,0,𝒯 . (5.84 ) η
If we set g3(t,x) := g2(t,x ) + Xv1 (t,x), t ≥ 0, x ∈ ∂ Σ, we have a similar estimate for v2,
( 1 ) η∥v2∥2η,1,Ω + ∥v2∥2η,1,𝒯 ≤ K2 -∥F2∥2η,0,Ω + ∥g3∥2η,0,𝒯 . (5.85 ) η
However, since the boundary norm of v 1 is controlled by the estimate (5.84View Equation), one also controls
2 ∥g ∥2 ≤ 2∥g ∥2 + C2 ∥v ∥2 ≤ (CK--)-∥F ∥2 + (CK )2∥g ∥2 + 2∥g ∥2 (5.86 ) 3 η,0,𝒯 2 η,0,𝒯 1 η,1,𝒯 η 1 η,0,Ω 1 η,0,𝒯 2 η,0,𝒯
with some constant C > 0 depending only on the vector field X. Therefore, the inequalities (5.84View Equation,5.85View Equation) together yield an estimate of the form (5.69View Equation) for v = (v1,v2), F = (F1, F2) and g = (g1,g2), which shows strong well-posedness in the generalized sense for the coupled system. Notice that the key point, which allows the coupling of v1 and v2 through the boundary matrix operator N, is the fact that one controls the boundary norm of v 1 in the estimate (5.84View Equation). The result can be generalized to larger systems of wave equations, where the matrix operator N is in triangular form with zero on the diagonal, or where it can be brought into this form by an appropriate transformation [267Jump To The Next Citation Point, 264Jump To The Next Citation Point].

Example 28. As an application of the theory for systems of wave equations, which are coupled through the boundary conditions, we discuss Maxwell’s equations in their potential formulation on the half space Σ [267Jump To The Next Citation Point]. In the Lorentz gauge and the absence of sources, this system is described by four wave equations μ ∂ ∂μA ν = 0 for the components (At, Ax,Ay, Az ) of the vector potential A μ, which are subject to the constraint C := ∂μA μ = 0, where we use the Einstein summation convention.

As a consequence of the wave equation for Aν, the constraint variable C also satisfies the wave equation, μ ∂ ∂ μC = 0. Therefore, the constraint is correctly propagated if the initial data is chosen such that C and its first time derivative vanish, and if C is set to zero at the boundary. Setting C = 0 at the boundary amounts in the following condition for A ν at x = 0:

∂At-= ∂Ax- + ∂Ay- + ∂Az-, (5.87 ) ∂t ∂x ∂y ∂z
which can be rewritten as
( ) ( ) ∂--− -∂- (At + Ax ) = − ∂--+ -∂- (At − Ax ) + 2 ∂-Ay + 2-∂-Az. (5.88 ) ∂t ∂x ∂t ∂x ∂y ∂z
Together with the boundary conditions
( ) ∂ ∂ ∂t-− ∂x- (At − Ax) = 0, ( ) -∂- ∂-- -∂- ∂t − ∂x Ay = ∂y (At − Ax ), ( ) -∂-− ∂-- Az = -∂-(At − Ax ), ∂t ∂x ∂z
this yields a system of the form of Eq. (5.82View Equation) with N having the required triangular form, where v is the four-component vector function v = (At − Ax, Ay,Az, At + Ax ). Notice that the Sommerfeld-like boundary conditions on Ay and Az set the gauge-invariant quantities Ey + Bz and Ez − By to zero, where E and B are the electric and magnetic fields, which is compatible with an outgoing plane wave traveling in the normal direction to the boundary.

For a recent development based on the Laplace method, which allows the treatment of second-order IBVPs with more general classes of boundary conditions, including those admitting boundary phenomena like glancing and surface waves; see [262].


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