In 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. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions.

A function defined on a rectangle (top figure, in red), and its trace (bottom figure, in red).

Motivation

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On a bounded, smooth domain  , consider the problem of solving Poisson's equation 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

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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[1] 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

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This paragraph follows Evans,[2] where more details can be found, and assumes that   has a  -boundary [a]. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo.[1] On a  -domain, the trace operator can be defined as continuous linear extension of the operator

 

to the space  . By density 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. 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  .

  1. ^   boundary: We say   is   if for each point   there exist   and a   function   such that—upon relabeling and reorienting the coordinate axes if necessary-we have:  

The case p = ∞

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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

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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

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For p > 1

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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 an  -version of Hölder continuity.

Abstract characterization

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An abstract characterization of the image 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

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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 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[1] that

 

is a surjective, bounded linear operator.

For p = 1

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For   the image of the trace operator is   and there holds[1] that

 

is a surjective, bounded linear operator.

Right-inverse: trace extension operator

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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[3]

 ,

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

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Higher derivatives

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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[3] 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[3]

 .

Finally, the spaces  , the completion of   in the  -norm, can be characterized as the kernel of  ,[3] i.e.

 .

Less regular spaces

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No trace in Lp

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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

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Let   denote the distributional divergence 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[4] 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 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

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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

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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

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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

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References

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  1. ^ a b c d Gagliardo, Emilio (1957). "Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili". Rendiconti del Seminario Matematico della Università di Padova. 27: 284–305.
  2. ^ Evans, Lawrence (1998). Partial differential equations. Providence, R.I.: American Mathematical Society. pp. 257–261. ISBN 0-8218-0772-2.
  3. ^ a b c d Nečas, Jindřich (1967). Les méthodes directes en théorie des équations elliptiques. Paris: Masson et Cie, Éditeurs, Prague: Academia, Éditeurs. pp. 90–104.
  4. ^ Sohr, Hermann (2001). The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts Basler Lehrbücher. Basel: Birkhäuser. pp. 50–51. doi:10.1007/978-3-0348-8255-2. ISBN 978-3-0348-9493-7.