In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the trace operator extends the notion of the
restriction of a function to the boundary of its domain to "generalized" functions in a
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
. This is particularly important for the study of
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
s with prescribed boundary conditions (
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s), where
weak solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precise ...
s may not be regular enough to satisfy the boundary conditions in the classical sense of functions.
Motivation
On a bounded, smooth
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
, consider the problem of solving
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
with inhomogeneous Dirichlet boundary conditions:
:
with given functions
and
with regularity discussed in the
application section below. The weak solution
of this equation must satisfy
:
for all
.
The
-regularity of
is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense
can satisfy the boundary condition
on
: by definition,
is an equivalence class of functions which can have arbitrary values on
since this is a null set with respect to the n-dimensional Lebesgue measure.
If
there holds
by
Sobolev's embedding theorem, such that
can satisfy the boundary condition in the classical sense, i.e. the restriction of
to
agrees with the function
(more precisely: there exists a representative of
in
with this property). For
with
such an embedding does not exist and the trace operator
presented here must be used to give meaning to
. Then
with
is called a weak solution to the boundary value problem if the integral equation above is satisfied. For the definition of the trace operator to be reasonable, there must hold
for sufficiently regular
.
Trace theorem
The trace operator can be defined for functions in the Sobolev spaces
with
, see the section below for possible extensions of the trace to other spaces. Let
for
be a bounded domain with Lipschitz boundary. Then
there exists a bounded linear trace operator
:
such that
extends the classical trace, i.e.
:
for all
.
The continuity of
implies that
:
for all
with constant only depending on
and
. The function
is called trace of
and is often simply denoted by
. Other common symbols for
include
and
.
Construction
This paragraph follows Evans,
where more details can be found, and assumes that
has a
-boundary. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo.
On a
-domain, the trace operator can be defined as
continuous linear extension of the operator
:
to the space
. By
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of
in
such an extension is possible if
is continuous with respect to the
-norm. The proof of this, i.e. that there exists
(depending on
and
) such that
:
for all
is the central ingredient in the construction of the trace operator. A local variant of this estimate for
-functions is first proven for a locally flat boundary using the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
. By transformation, a general
-boundary can be locally straightened to reduce to this case, where the
-regularity of the transformation requires that the local estimate holds for
-functions.
With this continuity of the trace operator in
an extension to
exists by abstract arguments and
for
can be characterized as follows. Let
be a sequence approximating
by density. By the proven continuity of
in
the sequence
is a Cauchy sequence in
and
with limit taken in
.
The extension property
holds for
by construction, but for any
there exists a sequence
which converges uniformly on
to
, verifying the extension property on the larger set
.
The case p = ∞
If
is bounded and has a
-boundary then by
Morrey's inequality there exists a continuous embedding
, where
denotes the space of
Lipschitz continuous functions. In particular, any function
has a classical trace
and there holds
:
Functions with trace zero
The Sobolev spaces
for
are defined as the
closure of the set of compactly supported
test functions
with respect to the
-norm. The following alternative characterization holds:
:
where
is the
kernel of
, i.e.
is the subspace of functions in
with trace zero.
Image of the trace operator
For p > 1
The trace operator is not surjective onto
if
, i.e. not every function in
is the trace of a function in
. As elaborated below the image consists of functions which satisfy a
-version of
Hölder continuity.
Abstract characterization
An abstract characterization of the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of
can be derived as follows. By the
isomorphism theorems there holds
:
where
denotes the
quotient space of the Banach space
by the subspace
and the last identity follows from the characterization of
from above. Equipping the quotient space with the quotient norm defined by
:
the trace operator
is then a surjective, bounded linear operator
:
.
Characterization using Sobolev–Slobodeckij spaces
A more concrete representation of the image of
can be given using
Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the
-setting. Since
is a ''(n-1)''-dimensional Lipschitz
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
embedded into
an explicit characterization of these spaces is technically involved. For simplicity consider first a planar domain
. For
define the (possibly infinite) norm
:
:
which generalizes the Hölder condition
. Then
:
equipped with the previous norm is a Banach space (a general definition of
for non-integer
can be found in the article for
Sobolev-Slobodeckij spaces). For the ''(n-1)''-dimensional Lipschitz manifold
define
by locally straightening
and proceeding as in the definition of
.
The space
can then be identified as the image of the trace operator and there holds
that
:
is a surjective, bounded linear operator.
For p = 1
For
the image of the trace operator is
and there holds
that
:
is a surjective, bounded linear operator.
Right-inverse: trace extension operator
The trace operator is not injective since multiple functions in
can have the same trace (or equivalently,
). The trace operator has however a well-behaved right-inverse, which extends a function defined on the boundary to the whole domain. Specifically, for
there exists a bounded, linear trace extension operator
:
,
using the Sobolev-Slobodeckij characterization of the trace operator's image from the previous section, such that
:
for all
and, by continuity, there exists
with
:
.
Notable is not the mere existence but the linearity and continuity of the right inverse. This trace extension operator must not be confused with the
whole-space extension operators which play a fundamental role in the theory of Sobolev spaces.
Extension to other spaces
Higher derivatives
Many of the previous results can be extended to
with higher differentiability
if the domain is sufficiently regular. Let
denote the exterior unit normal field on
.
Since
can encode differentiability properties in tangential direction only the normal derivative
is of additional interest for the trace theory for
. Similar arguments apply to higher-order derivatives for
.
Let
and
be a bounded domain with
-boundary. Then
there exists a surjective, bounded linear higher-order trace operator
:
:
with Sobolev-Slobodeckij spaces
for non-integer
defined on
through transformation to the planar case
for
, whose definition is elaborated in the article on
Sobolev-Slobodeckij spaces. The operator
extends the classical normal traces in the sense that
:
for all
Furthermore, there exists a bounded, linear right-inverse of
, a higher-order trace extension operator
:
.
Finally, the spaces
, the completion of
in the
-norm, can be characterized as the kernel of
,
i.e.
:
.
Less regular spaces
No trace in ''Lp''
There is no sensible extension of the concept of traces to
for
since any bounded linear operator which extends the classical trace must be zero on the space of test functions
, which is a dense subset of
, implying that such an operator would be zero everywhere.
Generalized normal trace
Let
denote the distributional
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of a
vector field . For
and bounded Lipschitz domain
define
:
which is a Banach space with norm
:
.
Let
denote the exterior unit normal field on
. Then
there exists a bounded linear operator
:
,
where
is the
conjugate exponent to
and
denotes the
continuous dual space to a Banach space
, such that
extends the normal trace
for
in the sense that
:
.
The value of the normal trace operator
for
is defined by application of the
divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the ...
to the vector field
where
is the trace extension operator from above.
''Application.'' Any weak solution
to
in a bounded Lipschitz domain
has a normal derivative in the sense of
. This follows as
since
and
. This result is notable since in Lipschitz domains in general
, such that
may not lie in the domain of the trace operator
.
Application
The theorems presented above allow a closer investigation of the boundary value problem
:
on a Lipschitz domain
from the motivation. Since only the Hilbert space case
is investigated here, the notation
is used to denote
etc. As stated in the motivation, a weak solution
to this equation must satisfy
and
:
for all
,
where the right-hand side must be interpreted for
as a duality product with the value
.
Existence and uniqueness of weak solutions
The characterization of the range of
implies that for
to hold the regularity
is necessary. This regularity is also sufficient for the existence of a weak solution, which can be seen as follows. By the trace extension theorem there exists
such that
. Defining
by
we have that
and thus
by the characterization of
as space of trace zero. The function
then satisfies the integral equation
:
for all
.
Thus the problem with inhomogeneous boundary values for
could be reduced to a problem with homogeneous boundary values for
, a technique which can be applied to any linear differential equation. By the
Riesz representation theorem there exists a unique solution
to this problem. By uniqueness of the decomposition
, this is equivalent to the existence of a unique weak solution
to the inhomogeneous boundary value problem.
Continuous dependence on the data
It remains to investigate the dependence of
on
and
. Let
denote constants independent of
and
. By continuous dependence of
on the right-hand side of its integral equation, there holds
:
and thus, using that
and
by continuity of the trace extension operator, it follows that
:
and the solution map
:
is therefore continuous.
See also
*
Trace class
*
Nuclear operators between Banach spaces
References
[
]
[
]
[
]
[
]
*Leoni, Giovanni (2017).
A First Course in Sobolev Spaces: Second Edition'.
Graduate Studies in Mathematics Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats.
List of books
*1 ''The General ...
. 181. American Mathematical Society. pp. 734. {{ISBN, 978-1-4704-2921-8
Sobolev spaces
Operator theory
de:Sobolev-Raum#Spuroperator