### 5.1 The Laplace method

Upon linearization and localization, the IBVP (5.1, 5.2, 5.3) reduces to a linear, constant-coefficient problem of the following form,
where , denote the matrix coefficients corresponding to and linearized about a solution and frozen at the point , and where, for generality, we include the forcing term with components in the class . Since the freezing process involves a zoom into a very small neighborhood of , we may replace by for all points lying inside the domain . We are then back into the case of Section 3, and we conclude that a necessary condition for the IBVP (5.1, 5.2, 5.3) to be well posed at , is that all linearized, frozen coefficient Cauchy problems corresponding to are well posed. In particular, the equation (5.6) must be strongly hyperbolic.

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

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 by employing Fourier transformation in the directions and tangential to the boundary. More precisely, we first exponentially damp the function in time by defining for the function
We denote by the Fourier transformation of with respect to the directions , and tangential to the boundary and define the Laplace–Fourier transformation of by
Then, satisfies the following boundary value problem,
where, for notational simplicity, we set and , , and where . Here, with and denoting the Laplace–Fourier and Fourier transform, respectively, of and , and is the Laplace–Fourier transform of the boundary data .

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

where . We solve this equation subject to the boundary conditions (5.13) and the requirement that vanishes as . For this, it is useful to have information about the eigenvalues of .

Lemma 3 ([258, 259, 228]). Suppose the equation (5.6) is strongly hyperbolic and the boundary matrix has negative and positive eigenvalues. Then, has precisely eigenvalues with negative real part and eigenvalues with positive real part. (The eigenvalues are counted according to their algebraic multiplicity.) Furthermore, there is a constant such that the eigenvalues of satisfy the estimate

for all and .

Proof. Let , and . Then

Since the equation (5.6) is strongly hyperbolic there is a constant and matrices such that (see the comments below Definition 2)
for all , where is a real, diagonal matrix. Hence,
and since is diagonal and its diagonal entries have real part greater than or equal to , it follows that
with . Therefore, the eigenvalues of must satisfy
for all . Choosing proves the inequality (5.15). Furthermore, since the eigenvalues can be chosen to be continuous functions of  [252], and since for , , the number of eigenvalues with positive real part is equal to the number of positive eigenvalues of . □

According to this lemma, the Jordan normal form of the matrix has the following form:

with a regular matrix, is nilpotent () and
is the diagonal matrix with the eigenvalues of , where have negative real part. Furthermore, commutes with . Transforming to the variable the boundary value problem (5.12, 5.13) simplifies to

#### 5.1.1 Necessary conditions for well-posedness and the Lopatinsky condition

Having cast the IBVP into the ordinary differential system (5.23, 5.24), we are ready to obtain a simple necessary condition for well-posedness. For this, we consider the problem for and split where and are the variables corresponding to the eigenvalues of with negative and positive real parts, respectively. Accordingly, we split

and . When the most general solution of Eq. (5.23) is
with constant vectors and . The expression for describes modes that grow exponentially in and do not satisfy the required boundary condition at unless ; hence, we set . In view of the boundary conditions (5.24), we then obtain the algebraic equation
Therefore, a necessary condition for existence and uniqueness is that the matrix be a square matrix, i.e., , and that
for all and . Let us make the following observations:
• The condition (5.27) implies that we must specify exactly as many linearly-independent boundary conditions as there are incoming characteristic fields, since is the number of negative eigenvalues of the boundary matrix .
• The violation of condition (5.27) at some with and gives rise to the simple wave solutions
where is a nontrivial solution of the problem (5.23, 5.24) with homogeneous data and . Therefore, an equivalent necessary condition for well-posedness is that no such simple wave solutions exist. This is known as the Lopatinsky condition.
• If such a simple wave solution exists for some , then the homogeneity of the problem implies the existence of a whole family,
of such solutions parametrized by . In particular, it follows that
such that
for all , as . Therefore, one has solutions growing exponentially in time at an arbitrarily large rate.

Example 25. Consider the IBVP for the massless Dirac equation in two spatial dimensions (cf. Section 8.4.1 in [259]),

where and are two complex constants to be determined. Assuming , Laplace–Fourier transformation leads to the boundary-value problem
The eigenvalues and corresponding eigenvectors of the matrix are and , with , where the root is chosen such that for . The solution, which is square integrable on , is the one associated with ; that is,
with a constant. Introduced into the boundary condition (5.36) leads to the condition
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 and . Clearly, this implies , since otherwise this expression is zero for . Assuming and , we then obtain the condition
for all with , which is the case if and only if or ; see Figure 1. The particular case , corresponds to fixing the incoming normal characteristic field to at the boundary.

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

where label the coordinates tangential to the boundary, and where we recall that , assuming that and have the same sign such that the evolution system (3.50, 3.51) 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 , which implies that the system is strongly hyperbolic for all values of , but symmetric hyperbolic only if ; see Example 15.

In order to analyze the system, it is convenient to introduce the variables , , , and , which are motivated by the form of the characteristic fields with respect to the direction normal to the boundary ; see Example 15. With these assumptions and definitions, Laplace–Fourier transformation of the system (3.50, 3.51) yields

where we have used since . The last three equations are purely algebraic and can be used to eliminate the zero speed fields , and from the remaining equations. The result is the ordinary differential system
In order to diagonalize this system, we decompose and into their components parallel and orthogonal to ; if and form an orthonormal basis of the boundary , then these are defined as
Then, the system decouples into two blocks, one comprising the transverse quantities and the other the quantities . The first block gives
and the corresponding solutions with exponential decay at have the form
where is a complex constant, and where we have defined with the root chosen such that for . The second block is
with corresponding decaying solutions
with complex constants and .

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

Introducing into this solutions (5.43, 5.45) gives
and
In the first case, since , we obtain and there are no simple wave solutions in the transverse sector. In the second case, the determinant of the system is
which is different from zero if and only if for all , where . Since is real, this is the case if and only if ; see Figure 1.

We conclude that the strongly hyperbolic evolution system (3.50, 3.51) with and incoming normal characteristic fields set to zero at the boundary does not give rise to a well-posed IBVP when or . This excludes the parameter range 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 (5.6, 5.7, 5.8) to be well posed. For this, we first transform the problem to trivial initial data by replacing with . Then, we obtain the IBVP

with and replaced by . By applying the Laplace–Fourier transformation to it, one obtains the boundary-value problem (5.12, 5.13), 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 be the bulk and the boundary surface, and introduce the associated norms and defined by

where we have used the definition of as in Eq. (5.10). Using Parseval’s identities we may also rewrite these norms as

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

Definition 6. [258] The IBVP (5.50, 5.51, 5.52) is called strongly well posed in the generalized sense if there is a constant such that each compatible data and gives rise to a unique solution satisfying the estimate

for all .

The inequality (5.55) implies that both the bulk norm and the boundary norm of are bounded by the corresponding norms of and . For a trivial source term, , the inequality (5.55) implies, in particular,

which is an estimate for the solution at the boundary in terms of the norm of the boundary data . In view of Eq. (5.54) this is equivalent to the following requirement.

Definition 7. [259, 267] The boundary problem (5.50, 5.51, 5.52) is called boundary stable if there is a constant such that all solutions of Eqs. (5.12, 5.13) with satisfy

for all and .

Since boundary stability only requires considering solutions for trivial source terms, , it is a much simpler condition than Eq. (5.55). 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. [258, 340] Consider the linear, constant coefficient IBVP (5.50, 5.51, 5.52) on the half space . Assume that equation (5.50) is strictly hyperbolic, meaning that the eigenvalues of the principal symbol are distinct for all . Assume that the boundary matrix 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 , , , such that lies on the half sphere for and . Then, we have,

Theorem 6. [258] Consider the linear, constant coefficient IBVP (5.50, 5.51, 5.52) on the half space . Assume that equation (5.50) is strictly hyperbolic, that the boundary matrix is invertible, and that the problem is boundary stable. Then, there exists a family of complex matrices , , whose coefficients belong to the class , with the following properties:

1. is Hermitian.
2. for all .
3. There is a constant such that
for all and all .

Furthermore, can be chosen to be a smooth function of the matrix coefficients of and .

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

where we have used the fact that in the second step, and the inequality for complex numbers and and any positive constant in the third step. Integrating both sides from to and choosing , we obtain, using (iii),
Since is bounded, there exists a constant such that for all . Integrating over and and using Parseval’s identity, we obtain from this
and the estimate (5.55) follows with .

Example 27. Let us go back to Example 25 of the 2D Dirac equation on the halfspace with boundary condition (5.34) at . The solution of Eqs. (5.35, 5.36) at the boundary is given by , where , and

Therefore, the IBVP is boundary stable if and only if there exists a constant such that
for all and . We may assume , otherwise the Lopatinsky condition is violated. For the left-hand side is . For we can rewrite the condition as
for all , where and . This is satisfied if and only if the function is bounded away from zero, which is the case if and only if ; see Figure 1.

This, together with the results obtained in Example 25, yields the following conclusions: the IBVP (5.32, 5.33, 5.34) gives rise to an ill-posed problem if or if and and to a problem, which is strongly well posed in the generalized sense if and . The case is covered by the energy method discussed in Section 5.2. For the case with , see Section 10.5 in [228].

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

• The boundary stability condition (5.57) is often called the Kreiss condition. Provided the eigenvalues of the matrix are suitably normalized, it can be shown [258, 228, 241] that the determinant in Eq. (5.27) can be extended to a continuous function defined for all and , and condition (5.57) can be restated as the following algebraic condition:
for all and . This is a strengthened version of the Lopatinsky condition, since it requires the determinant to be different from zero also for on the imaginary axis.
• As anticipated above, the importance of the symmetrizer construction in Theorem 6 relies on the fact that, based on the theory of pseudo-differential operators, it can be used to treat the linear, variable coefficient IBVP [258]. Therefore, the localization principle holds: if all the frozen coefficient IBVPs are boundary stable and satisfy the assumptions of Theorem 5, then the variable coefficient problem is strongly well posed in the generalized sense.
• If the problem is boundary stable, it is also possible to estimate higher-order derivatives of the solutions. For example, if we multiply both sides of the inequality (5.59) by , integrate over and and use Parseval’s identity as before, we obtain the estimate (5.55) with , and replaced by their tangential derivatives , and , respectively. Similarly, one obtains the estimate  (5.55) with , and replaced by their time derivatives , and if we multiply both sides of the inequality (5.59) by and assume that for all . Then, a similar estimate follows for the partial derivative, , in the -direction using the evolution equation (5.6) and the fact that the boundary matrix is invertible. Estimates for higher-order derivatives of follow by an analogous process.
• Theorem 5 assumes that the initial data is trivial, which is not an important restriction since one can always achieve by transforming the source term and the boundary data , as described below Eq. (5.52). Since the transformed involves derivatives of , this means that derivatives of would appear on the right-hand side of the inequality (5.55), and at first sight it looks like one “loses a derivative” in the sense that one needs to control the derivatives of to one degree higher than the ones of . However, the results in [341, 342] improve the statement of Theorem 5 by allowing nontrivial initial data and by showing that the same hypotheses lead to a stronger concept of well-posedness (strong well-posedness, defined below in Definition 9 as opposed to strong well-posedness in the generalized sense).
• The results mentioned so far assume strict hyperbolicity and an invertible boundary matrix, which are too-restrictive conditions for many applications. Unfortunately, there does not seem to exist a general theory, which removes these two assumptions. Partial results include [5], which treats strongly hyperbolic problems with an invertible boundary matrix that are not necessarily strictly hyperbolic, and [293], which discusses symmetric hyperbolic problems with a singular boundary matrix.

#### 5.1.3 Second-order systems

It has been shown in [267] 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 , ,

where and , and where is a first-order linear differential operator of the form
where , , , …, are real constants. We ask under which conditions on these constants the IBVP (5.65, 5.66, 5.67) is strongly well posed in the generalized sense. Since we are dealing with a second-order system, the estimate (5.55) in Definition 6 has to be replaced with
where the norms and control the first partial derivatives of ,
with . Likewise, the inequality (5.57) in the definition of boundary stability needs to be replaced by

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

where we have defined and where and denote the Laplace–Fourier transformations of and , respectively. In order to apply the theory described in Section 5.1.2, we rewrite this system in first-order pseudo-differential form. Defining
where , we find
where we have defined
with , . This system has the same form as the one described by Eqs. (5.14, 5.13), and the eigenvalues of the matrix are distinct for and . Therefore, we can construct a symmetrizer according to Theorem 6 provided that the problem is boundary stable. In order to check boundary stability, we diagonalize and consider the solution of Eq. (5.74) for , which decays exponentially as ,
where is a complex constant and with the root chosen such that for . Introduced into the boundary condition (5.75), this gives
and the system is boundary stable if and only if the expression inside the square parenthesis is different from zero for all and with . In the one-dimensional case, , this condition reduces to with , and the system is boundary stable if and only if ; that is, if and only if the boundary vector field is not proportional to the ingoing null vector at the boundary surface,
Indeed, if , 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 it follows that must be different from zero since otherwise the square parenthesis is zero for purely imaginary satisfying . Therefore, one can choose without loss of generality. It can then be shown that the system is boundary stable if and only if and ; see [267], which is equivalent to the condition that the boundary vector field is pointing outside the domain, and that its orthogonal projection onto the boundary surface ,

is future-directed time-like. This includes as a particular case the “Sommerfeld” boundary condition for which is the null vector obtained from the sum of the time evolution vector field and the normal derivative . While is uniquely determined by the boundary surface , is not unique, since one can transform it to an arbitrary future-directed time-like vector field , which is tangent to by means of an appropriate Lorentz transformation. Since the wave equation is Lorentz-invariant, it is clear that the new boundary vector field must also give rise to a well-posed IBVP, which explains why there is so much freedom in the choice of .

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

As pointed out in [267], the advantage of obtaining a strong well-posedness estimate (5.69) 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,

which is coupled through the boundary conditions
where has the form
with any vector. Since the wave equation and boundary condition for decouples from the one for , we can apply the estimate (5.69) to , obtaining
If we set , , , we have a similar estimate for ,
However, since the boundary norm of is controlled by the estimate (5.84), one also controls
with some constant depending only on the vector field . Therefore, the inequalities (5.84,5.85) together yield an estimate of the form (5.69) for , and , which shows strong well-posedness in the generalized sense for the coupled system. Notice that the key point, which allows the coupling of and through the boundary matrix operator , is the fact that one controls the boundary norm of in the estimate (5.84). The result can be generalized to larger systems of wave equations, where the matrix operator is in triangular form with zero on the diagonal, or where it can be brought into this form by an appropriate transformation [267, 264].

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  [267]. In the Lorentz gauge and the absence of sources, this system is described by four wave equations for the components of the vector potential , which are subject to the constraint , where we use the Einstein summation convention.

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

which can be rewritten as
Together with the boundary conditions
this yields a system of the form of Eq. (5.82) with having the required triangular form, where is the four-component vector function . Notice that the Sommerfeld-like boundary conditions on and set the gauge-invariant quantities and to zero, where and 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].