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Revista de la Unión Matemática Argentina
versión impresa ISSN 00416932
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 2dimensional 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), 277295 [ 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), 321348. [ Links ]
[4] G. Auchmuty, Spectral Characterization of the Trace Spaces , SIAM J of Math. Anal. 38/3, (2006), 894905. [ 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, 321364, 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 nonlipschitzian domains, St. Petersburg Math. J. 11, 107128, 2000. [ Links ]
[8] H. Shahgholian, When does the free boundary enter into corner points of the fixed boundary?, J. Math. Sci. 132, No. 3, 371377, 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