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
.
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.
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.
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.
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:
is Hermitian.
for all
.
- 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].