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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932
Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca jun. 2009
A Compact trace theorem for domains with external cusps
Carlos Zuppa
Abstract. This paper deals with the compact trace theorem in domains
with external cusps. We show that if the power sharpness of the cusp is bellow a critical exponent, then the trace operator
exists and it is compact.
2000 Mathematics Subject Classification. 35J25,46E35
Key words and phrases. Cuspidal domains; Compact trace operator
Up to now, Lipschitz domains make up the most general class of domains where a rich function theory can be developed. However, domains with external cusps could appear at several branches of mathematics and applications. In obstacle problems, for example, the free boundary with external cusps may enter into corner points of the fixed boundary (e.g. [8]). Therefore, it is important to know what kind of results in the theory of Sobolev spaces remain valid in cuspidal domains.
Key tools in harmonic analysis and numerical application are the Rellich's theorem and the compact trace theorem. This paper deals with the compact trace theorem in domains with external cusps. We show that if the power sharpness
of the cusp is bellow a critical exponent
, then the trace operator
exists and it is compact. For cuspidal models in
,
(see [1]).
Several classical results of harmonic analysis can be extended in this context, to begin with the divergence theorem, for example, or the characterization of the spaces via the Steklov eigenfunction expansions [3, 4]. In several branch of harmonic analysis, the compacity of the operators
and
are key tools.
It is worth to remark here that certain classical counterexamples of analysis in cuspidal domains, like those of Friedrichs related to Korn inequality [5], have cusps of power sharpness equal to the critical exponent.
In [7] the authors characterize the traces of the Sobolev spaces , by using some weighted norm on the boundary. In [1] a different kind of trace result was obtained by introducing a weighted Sobolev space in
, such that the restriction to the boundary of functions in that space are in
. We extended the arguments in this work to domains in
with some slight modification in the trace estimate, which is more useful in order to prove the compacity of the trace operator.
We shall consider a family of standard models or especial domains in
which have cusps of power sharpness
. We follow the standard notation for the Sobolev spaces
and Sobolev norms. For simplicity, we consider only the case
, and we do not consider other Sobolev spaces
.
Definition 1. Let ,
. We shall say that
is a
cusp if
![]() |
where the map is defined by
![]() |
![]() |
and
![]() |
is a bounded connected region with Lipschitz boundary such that .
The Jacobian of the desingularizing map is
Trace theorems for domains with external cusps could be obtained in weighted Sobolev spaces [1]. For , we define
![]() |
and we introduce the weighted Sobolev space as the closure of
in the norm
![]() |
In what follows, we use the letter to denote a generic constant which depends only on
.
Theorem 2. Let be a
model. Then, there exists a constant
such that for any
, the trace function
is in
and
![]() | (1) |
The proof of this theorem will be given later in the last section. We shall first explore some consequences of this result.
Let . In the next theorem we will make use of the inclusion
![]() | (2) |
which is a particular case of the results given in [2].
We can obtain the inclusion under appropriate assumptions on the values of
and
.
Definition 3. The cusp
satisfies Condition A1 if
![]() |
Theorem 4. If satisfies Condition A1, then
Proof. We shall follow the arguments in [1]. By Hölder's inequality with an exponent to be chosen below
![]() | (3) |
From (2), if we have
![]() |
On the other hand, is bounded if
.
If , we must take
such that
![]() |
and this is possible only if
For , we have
![]() |
Hence, .
Corollary 5. If satisfies Condition A1, then the trace function
is in
for any
. Furthermore, the trace operator
is compact.
Proof. It only remains to show that is compact. Then, let
be a bounded sequence in
and, since we know that the inclusion
is compact [6], we can also assume that
is a Cauchy sequence in
. We shall see now that
is a Cauchy sequence in the
norm.
For , let
. By (3), we have
![]() |
Since
![]() |
given , we can chose
such that
![]() |
On the other hand,
![]() |
if is chosen such that
![]() |
Then,
Now, the result follows easily by estimate (1).
Remark 6. In the bidimensional case, Theorem 4 for cups was obtained in [1]. The compacity of the trace operator follows by the same arguments given above. The key tool is estimate (1) in this appropriate form.
2. ALMOST LIPSCHITZ DOMAINS WITH EXTERNAL CUSPS
Let denote the open cube
.
Definition 7. A bounded domain satisfies Condition A2 if and only if:
(i) There exists a finite family of open subsets of
such that
.
(ii) A Lipschitz diffeomorhism
such that one of the two possibilities occurs:
(iii) is the image of a standard cusp
in
which satisfies Condition A1.
(iv) There exists a Lipschitz map such that
and
![]() |
When Condition A2 holds, there is an outward unit normal defined at
a.e.point of
, where
represents Hausdorff 2-dimensional measure and functions in
are integrated with respect to this measure . Furthermore, by a partition of unity argument we can obtain the following result.
Theorem 8. Let be a bounded domain which satisfies Condition A2. Then, the trace operator
.
We proceed first with case . Thus,
is defined by
![]() |
and the Jacobian of is
. Let
and
. Then,
![]() |
On the other hand,
![]() |
and
![]() |
Now, let . Then,
is parametrized by
![]() |
Thus,
![]() |
and it follows that
![]() |
Let be a
function such that
in
and
in
, and define
by
.
Setting
![]() |
by Hölder's inequality we have
![]() |
Now, it is clear that . On the other hand,
![]() |
From this, we can easily obtain that
![]() |
and the result follows. The proof for is the same.
Case
We shall explain the main arguments for the curve . It will be clear from the proof that the general case follows along the same lines via a partition of unity.
We consider parametrized by
and the parametrization
given by
![]() |
It follows that
![]() |
![]() |
Thus,
![]() |
For , we want now to estimate
with the same arguments as above. We introduce polar coordinates in
and we define
![]() |
where such that
for
and
for
.
Thus,
First, we have
To complete the proof, we must take into account that
![]() |
Then, calculating for the first derivative, we get
![]() |
Hence,
The same inequality is valid for the second derivative and we get
![]() |
Considering these facts together, it is easy to see that we have concluded the proof of the theorem.
[1] G. Acosta, M. G. Armentano, R. G. Durán, and A. L. Lombardi, Finite Element Approximations in a NonLipschitz Domain. SIAM J. Numer. Anal. 45 (2007), 277-295 [ Links ]
[2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [ Links ]
[3] G. Auchmuty, Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems, Num. Functional Analysis and Optimization, 25, (2004), 321-348. [ Links ]
[4] G. Auchmuty, Spectral Characterization of the Trace Spaces , SIAM J of Math. Anal. 38/3, (2006), 894-905. [ Links ]
[5] K. O. Friedrichs, On certain inequalities and characteristic value problems for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41, 321-364, 1937. [ Links ]
[6] V. G. Maz'ja, Sobolev Spaces, Springer Verlag, New York, 1980. [ Links ]
[7] V. G. Maz'ja, Yu. V. Netrusov and V. Poborchi, Boundary values of functions in Sobolev Spaces on certain non-lipschitzian domains, St. Petersburg Math. J. 11, 107-128, 2000. [ Links ]
[8] H. Shahgholian, When does the free boundary enter into corner points of the fixed boundary?, J. Math. Sci. 132, No. 3, 371-377, 2006. [ Links ]
Carlos Zuppa
Departamento de Matemáticas
Universidad Nacional de San Luis
Chacabuco 985, San Luis. 5700. Argentina
zuppa@unsl.edu.ar
Recibido: 13 de noviembre de 2006
Aceptado: 3 de octubre de 2008