Singular Perturbation on a Subdomain*

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ARTICLE NO. AY97544 Singular Perturbation on a Subdomain* G. Aguilar Departamento de Matematica Aplicada, Centro Politecnico Superior, Uniersidad de Zaragoza, Maria de Luna, 3, 55, Zaragoza, Spain and F. Lisbona Departamento de Matematica Aplicada, Facultad de Ciencias, Uniersidad de Zaragoza, Plaza San Francisco, 59, Zaragoza, Spain Submitted by Hans-Gorg Roos Received May, 995 We consider a kind of singularly perturbed problem with a small positive parameter affecting the second order derivative only in a part of the domain. We analyse the existence and uniqueness of the solution and the asymptotic behaviour as the small parameter goes to zero. 997 Academic Press. INTRODUCTION In this paper we consider the following singular perturbation problem: Find w C Ž,. such that where Ž w. w KŽ w. bž x,w., x Ž,., w Ž., w,. is the piecewise constant function defined by, if x Ž,., Ž x. ½ Ž,.,, if x Ž,., *This research has been supported by the CICYT, Project AMB and by ENRESA Contract address: lisbona@posta.unizar.es. -47X97 $5. Copyright 997 by Academic Press All rights of reproduction in any form reserved. 9

2 SINGULAR PERTURBATION ON A SUBDOMAIN 93 and : R R, K: R R, b:,rr are functions such that C Ž R., Ž t., KC Ž R., strictly monotone, bc Ž, R., bwž x,w.. Problem. may be regarded as the coupling of two second order differential equations, Ž Ž w. wk. Ž w. bž x,w., x Ž,., a. Ž Ž w. wk. Ž w. bž x,w., x Ž,., b. with Dirichlet boundary conditions and continuity of the unknown and its derivative as interface conditions at the point. Similar problems have been considered in 3 in order to find out the transmission conditions for elliptic-hyperbolic or parabolic-hyperbolic problems. In these papers we choose to impose the continuity of the unknown and of the flux as interface conditions: wž. wž. w Ž. w Ž.. 3. Although the interface conditions in. do not have a clear physical sense, they build a smoother transmission problem that a., b., 3., which has a physical sense through. A simpler case of this kind of problem, the linear case, was studied by Gastaldi and Quarteroni 4. In Section we study the existence and uniqueness of the solution of problem.. For the existence we use a method which essentially consists of formulating a modified problem by applying a truncation operator, proving the existence of a solution of the modified problem, and showing that this solution is also a solution of the given problem. The uniqueness is proved by using the concept of a monotone-inverse operator. For this analysis we adapt some techniques presented in, Chap. III. Section 3 is devoted to an asymptotic analysis of problem.. We show that a sequence of solutions w 4 converges, in an appropriate sense, to a function w which is a solution of the reduced equation Ž Ž w. wk. Ž w. bž x,w., x Ž,., KŽ w. bž x,w., x Ž,.,

3 94 AGUILAR AND LISBONA and satisfies the boundary and interface conditions, wž., wž. wž., wž. wž. if K is an increasing function, and wž., wž. wž., w, if K is decreasing. From these conditions the continuity of w in,. follows. However, in the asymptotic analysis of the problem a., b., 3., the transmission and boundary conditions obtained in were wž., wž., wž. wž. if K, and wž., Ž w. wž. KŽ w.ž. KŽ w.ž., w if K. Then, a discontinuity in the unknown w at if K and another in the derivative of w at the interface if K were allowed. If one considers the limit procedure as a technique to find out interface conditions when coupling a second order equation with a first order one, the result obtained in this paper gives smoother conditions.. THE SINGULARLY PERTURBED PROBLEM Let S be the space of all functions w C, C Ž,. such that w, C,, w, C Ž,.. Given, we consider an operator on S defined by w Ž., x, Ž Ž w. w. Ž x. KŽ w. Ž x. bž x,wž x.., x Ž,., MwŽ x. Ž.. Ž Ž w. w. Ž x. KŽ w. Ž x. bž x,wž x.., x Ž,., wž., x,

4 SINGULAR PERTURBATION ON A SUBDOMAIN 95 and the following boundary value problem, find w S such that Mw. Ž.. The existence of a solution for Ž.. is proved by using comparison functions and a generalized Nagumo theorem. Recall that a function S such that M is called an upper solution of Mw and a function S such that M is called a lower solution. A lower solution and an upper solution such that are said to be a pair of comparison functions for the problem Mw. We start with the following technical lemma. LEMMA.. For each pair of functions, C, with Ž x. Ž x. for all x, and, there exists a family of pairs of functions Ž x, y., Ž x, y. 4 k k k such that for each w S with w, Ž,wŽ.. wž. Ž,wŽ.., for some,, Ž.3. k k and Ž x, Ž x.. Ž x. Ž x, Ž x.., x Ž,., Ž.4a. k k x,ž x. Ž x. x,ž x., x Ž,., Ž.4b. k k for all k. Proof. For each w S satisfying w, it follows readily that there exists, such that Let 4 wž. max Ž., Ž. Ž min Ž x. x, and max Ž x. x. Obviously,. For each k, let us define functions u, C, by k k Ž. t s k ds t, t,, kž s. u H Ž. t s k ds t, t,, kž s. H

5 96 AGUILAR AND LISBONA with large enough. Then uk and k are positive, monotone func- tions such that k k Ž k. u t u t k u t, k k Ž k. t t k t. The functions ukž y., x, kž x, y. ½ kž y., x, Ž.5a. kž y., x, kž x, y. ½ u kž y., x, Ž.5b. satisfy inequality Ž.3. and if we choose such that and, inequalities Ž.4a. and Ž.4b. hold. For the given comparison functions for the problem Ž.., and in SC,, we now introduce two modified operators by applying a truncation on M. In the first one we consider a truncation on the nonlinear terms of M, replacing w by w* and w by w, and in the second one we truncate only w, replacing it by w, where w*max, minw, 44, 44 wmaxž x, w., minw, Ž x, w. 44, wmax x, w*, min w, x, w*, and, defined in Ž.5a. and Ž.5b. k k with k a positive constant such that y p K y pb x, y k y p y p K y pb x, y k y p, Ž.6. for any x, y, p,x, x y x, p R.

6 SINGULAR PERTURBATION ON A SUBDOMAIN 97 Thus, let M* be an operator on S defined by M*w x w Ž., x, Ž w*. KŽ w*. bž x,w*. w ww* w w, Ž w*. Ž w*. Ž w*. xž,, Ž w*. KŽ w*. bž x,w*. w ww* w w, Ž w*. Ž w*. Ž w*. x Ž,., wž., x, 4 and M be another defined on w S x w x x, x, by w Ž., x, Ž w. w Ž w. w KŽ w. wbž x,w., xž,, MwŽ x. Ž w. w Ž w. Ž w. KŽ w. wbž x,w., x Ž,., wž., x. To prove the existence result we need the following THEOREM.. We consider an operator defined by až x. už x. g x,už x.,už x., x Ž,., HuŽ x. g Ž už.,už.., x, g už.,už., x, with gien functions a: Ž,. R, g: Ž,. RRR, g i: RR R Ž i,. where and gž y, p. gž y, p. g Ž y, p. g Ž y, p. až x. Ž x. for p p Ž y, p, pr.. Let R be the space of all functions u C, C, such that all deriaties exist which occur in Hu.

7 98 AGUILAR AND LISBONA or Let and w be fixed functions in R. Let there exist a function z in R such that HwŽ x. HŽ wz.ž x., for x,, HŽ z.ž x. HŽ x., for x,. Then H Hw w. PROPOSITION.3. Suppose that, S C, is a pair of comparison functions for M. Then, there exists a function w S such that: Ž. i M*w. Ž ii. w and consequently Mw. Ž iii. Ž x, w. w Ž x, w.. Thus, w is a solution of the problem Ž... Proof. We shall assume. Otherwise the existence of a solution to Ž.. is trivial. First, to prove part Ž. i, we define the operator N on S Then we can write w Ž., x, wž x. wž x., xž,, NwŽ x. wž x. wž x., x Ž,., wž., x. M*wŽ x. NwŽ x. FŽ x,w*, w., with FŽ x,w*, w. bounded uniformly in w C,. Let us now consider the operator T defined on C, by H TwŽ x. GŽ x,. FŽ,w* Ž.,wŽ.. d, where G is the Green function which belongs to N. Observe that if w satisfies Tw w, then w is a solution of M*w. Thus, to prove the existence of a solution, we will only show that T has a fixed point. C, is a Banach space with respect to the norm f f f. Since FŽ x,w*, w. is uniformly bounded in w C,, the set TC Ž,. is a

8 SINGULAR PERTURBATION ON A SUBDOMAIN 99 relatively compact subset of C,. T is a continuous mapping so that we can apply the Schauder fixed point theorem. Next we show that w. The inequality w follows by similar arguments. Since Ž x,. Ž x,. we have M* M. On the other hand, by the definition of M* it follows that M* Ž. M* Ž., Ž,.. Therefore, we have M*w M* Ž. M* Ž. Ž,.. Hence by Theorem., it follows that w. Finally, we will prove that w Ž x. Žx,w Ž x.., x,. The inequality Žx, w Ž x.. w Ž x. follows by analogous arguments. Suppose for example that x, and,, with from Ž.3. for w. On the set of all functions h C, such that h Ž,. C Ž,. and h C Ž,., we consider the operator A defined by Ž,. hž., x, Ž w. hž w. už w.ž hw. Ž w. h AhŽ x. KŽ w. hbž x,w., xž,, Ž.7. Ž w. hž w. už w.ž hw. Ž w. h KŽ w. hbž x,w., xž,, where h maxž x, w., minh, Ž x, w.44, and u uk as in the proof of Lemma., and k defined in Ž.6.. Since Mw, we have that Aw Ž x., x Ž,. Moreover, by the choice of the constant k, From.3 we get We also have AŽ x, w., x Ž,. A, w Ž. Aw Ž.. x AŽ x,w. AŽ Ž x,w. e., x,.

9 3 AGUILAR AND LISBONA Therefore, we can apply Theorem. in, with až x., gž x, h, h. Ž w. hž w. už w.ž hw. Ž w. h KŽ w. hbž x,w. gž hž., hž.. hž. g Ž hž., h. Ž w. h Ž w. už w.žh zž x. e x. K w h b, w. w w h This theorem yields w Ž x. Ž x,w. x,. If x, and,, to show that w Ž x. Žx,w Ž x.., one uses the operator h Ž., x, AhŽ x. Ž w. hž w. už w.ž hw. Ž w. h KŽ w. hbž x,w., xž,, in place of the operator A in Ž.7.. If x, one proceeds in a similar way. PROPOSITION.4. The problem Ž.. has a solution w S such that cwc, where c max½ bž x,.. 5 Proof. The statement follows from Proposition.3 by verifying that c and c constitute a pair of comparison functions. In order to show the uniqueness of the solution we use the concept of an inverse-monotone operator. DEFINITION.5. Let M denote an operator with domain in a partially ordered linear space Ž V,. and range in a partially ordered linear space Ž U,.. This operator is called inverse-monotone on D V if for all, w D M Mw w.

10 PROPOSITION.6. SINGULAR PERTURBATION ON A SUBDOMAIN 3 Problem Ž.. has a unique solution. s Proof. Let us introduce two functions: defined by s H Ž. t dt and sg a regularization of the Heaviside function sg, that is, a function which satisfies sg C Ž R., sg sg Ž t. if t or t, C sg Ž t., with C independent of. We will show that M is an inverse-monotone operator on S. Let w, w S be two functions such that Mw Mw. Multiplying Mw x and Mw x, x,, by sg ŽŽ w. Ž w.. and integrating the difference by parts, we have H KŽ w. KŽ w. sg Ž Ž w. Ž w.. H b x,w b x,w sg w w Ž. Ž Ž. Ž.. w w sg w w, where we used Ž. sg w w x for x, because of Mw x Mw x for x, and the fact that is an increasing monotone function. As, H by Sacks lemma 9 and then H Ž. K w K w sg w w Ž. b x,w b x,w sg w w Ž Ž. Ž.. w w sg w w. Ž.8. Then, if wž. wž. we have w w by the monotonicity of b. Thus, suppose that wž. wž.. Moreover because of Ž.8., it follows Ž w. Ž w. Ž.. We shall derive a contradiction from this. Define an operator T on C Ž,. by Ž., x, T K b x, Ž x. Ž Ž.., x Ž,., Ž., x.

11 3 AGUILAR AND LISBONA Obviously T Ž w. T Ž w.. Since T is an inverse-monotone operator on C Ž,. Žsee. 6 the inequality Ž w. Ž w. holds. Thus, by the monotonicity of a contradiction is derived. 3. THE LIMIT PROBLEM We first obtain some bounds that provide sufficient conditions for the existence of a sequence which converges Ž as., in an appropriate sense, to a solution of the reduced problem. From now on, we denote by C any positive constant independent of the parameter. LEMMA 3.. Let w sole problem Ž... There is a constant C such that w C, Ž 3.. L Ž,. w C, Ž 3.. H Ž,. w C, 3.3 ' L Ž,. w C. Ž 3.4. Proof. We can prove Ž 3.. using the same techniques as Lorenz 6 and Niijima 8 to obtain a similar result for the solutions of nonlinear singularly perturbed problems. To prove Ž 3.. let us consider the function z H Ž,. solution of the boundary value problem Ž w. z, in Ž,., z Ž., z w Ž.. The H Ž,. norm of z is bounded uniformly with respect to, as well as the value of z Ž.. Moreover, the function d w z H Ž,. satisfies Ž w. d KŽ w. d f, Ž 3.5. where f bž x,w. KŽ w. z, is bounded in L Ž,. uniformly with respect to. Multiplying Eq. Ž 3.5. by d and integrating on Ž,. it follows that the H Ž,. norm of d is bounded. Consequently Ž 3.. holds. Next, we multiply Ž 3.5. by Ž w. d where is a C Ž,. function such that Ž. and, and integrate on,, obtaining that d Ž. is bounded uniformly in, which yields Ž Bound Ž 3.4. can be proven by multiplying the differential equation in Ž,. by w and integrating by parts on Ž,..

12 SINGULAR PERTURBATION ON A SUBDOMAIN 33 LEMMA 3.. Ž. a Suppose that K is an increasing monotone function. Then the set KŽ w. 4 is bounded in L Ž,., for any with. Ž b. If K is a decreasing monotone function, KŽ w. 4 is bounded in L Ž,.. Proof. Let us introduce a function C Ž,. such that,,, and. Consider the L Ž,. inner product of both sides of the differential equation b. by Ž w. with as in the proof of Proposition.6. Integrating by parts in the first term, we get H Ž w. Ž. KŽ w. Ž w. H H Ž w. bž x,w. Ž w.. From the bounds 3., 3.3, and 3.4 we get Hence H H KŽ w. Ž w. C. H K w C K w w C H KŽ w. Ž w. C, and part Ž. a of the lemma is proved. To show part Ž. b, we proceed in an analogous way replacing Ž w. by Ž w.. PROPOSITION 3.3. Let w sole problem Ž... There exists a sequence w 4 which conerges a.e., as to a function w such that w CŽ,., w C Ž,., KŽ w. C Ž,. and Ž,. Ž,. wž. Ž 3.6a. Ž Ž w. wk. Ž w. bž x,w., x Ž,., Ž 3.6b. wž. wž. Ž 3.6c. KŽ w. bž x,w., x Ž,., Ž 3.6d. w if K. Ž 3.6e. Proof. From Proposition.4 we have w L Ž,. c,. Thus, there exist a function w L Ž,. and a subsequence that we note in the

13 34 AGUILAR AND LISBONA same way such that w w weakly in L Ž,.. Ž 3.7. Moreover by Ž 3.. we have Ž upon extracting a subfamily. w w weakly in L Ž,.. Ž 3.8. From Ž 3.. and Rellich s theorem we also get Ž upon extracting a subfamily. w w in L Ž,. and pointwise. Ž 3.9. In the same way, from Lemma 3. we can show the existence of a function Ž H Ž,. if K and H Ž,. if K. such that Ž upon extracting a subfamily. ½ if K, K Ž w. in L Ž,. and pointwise if K, KŽ w. in L Ž,. and pointwise. Ž 3.. Because of the continuity and the strict monotonicity of K wk Ž., wcž,. if K and w C, if K Ž 3.. ww everywhere in Ž,. if K and in Ž,. if K. Ž 3.. The boundary conditions Ž 3.6a., Ž 3.6e., and the interface condition Ž 3.6c. follow from Ž 3.9. and Ž 3... Multiplying the differential equation in Ž,. by C Ž,., integrating by parts, and considering the limit as in the obtained expression, we have Thus H H H Ž w. w KŽ w. bž x,w.. Ž Ž w. wkž w.. bž x,w. in DŽ,.. Since w is a continuous function in,, we conclude that it satisfies Ž 3.6b.. Similarly equality Ž 3.6d. can be proved. PROPOSITION 3.4. Suppose that K k. The function w defined in Proposition 3.3 satisfies wž. wž.. Ž 3.3.

14 SINGULAR PERTURBATION ON A SUBDOMAIN 35 Proof. From Eq. a. and bound Ž 3.. we have that Ž w. is bounded in L Ž,.. Then, there exists a subsequence, that we note in the same way, such that Ž w. Ž w. in L Ž,. and everywhere. Ž Thus, w converges to w.. Therefore, it is enough to show that w is bounded in L Ž,. for some :. It follows from Lemma 3. that the set w 4 is bounded in L Ž,. for any,. Multiplying the differential equation b. by Ž w. sgžž w.. and integrating by parts in Ž,., we obtain that 3 H Ž. Ž. 3 3 w H Ž K. w w sg w 3 C K w w sg w Ž. Ž. H bž x,w. Ž w. sg Ž w.. Then, since K k, we obtain that Ž w. 3 is bounded in L Ž,. for small enough. Similar calculations and the above bound yield that Ž w. 4 is bounded in L Ž,.. Now, let us consider the L Ž,. inner product of both sides of the differential equation by Ž w., and we obtain H Ž w. H Ž K. Ž Ž w.. ž Ž w. / H bž x,w. Ž w.. Integrating by parts in 3.4 it follows that ' Ž w. L Ž,. C. Ž 3.4. We finally differentiate the differential equation, multiply the result by Ž w., and integrate on Ž,., with C Ž,. such that,,, and. From the above results it follows that L Ž,. ' w C. Ž. Consequently w w.

15 36 AGUILAR AND LISBONA PROPOSITION 3.5. Suppose that K. There exists a unique function w satisfying Ž 3.6a. Ž 3.6e.. Proof. Let us suppose that there exist two functions w, w satisfying Ž 3.6a. Ž 3.6e.. Multiplying the differential equations in Ž,. by Ž. sg K w K w and integrating on,, as it follows that Ž. Ž. Ž Ž. Ž.. K w K w sg K w K w H bž x,w. bž x,w. sg Ž KŽ w. KŽ w.., consequently H b x,w b x,w, and w w in,. Likewise we can prove that w w in,. Now, consider the inverse-monotone operator u Ž., x, u K Ž u. ub x, Ž u., x Ž,., u wž., x. TuŽ x. Since T w T w, we have that w w in,. ACKNOWLEDGMENTS Thanks are due to the referees for their very useful remarks which have substantially improved this paper. REFERENCES. G. Aguilar and F. Lisbona, Interface conditions for a kind of nonlinear elliptic-hyperbolic problem, in Domain Decomposition Methods in Science and Engineering ŽA. Quateroni, J. Periaux, Y. A. Kuznetsov, and O. B. Widlund, Eds.., pp. 8996, Amer. Math. Soc., Providence, G. Aguilar and F. Lisbona, On the coupling of elliptic and hyperbolic nonlinear differential equations, Math. Model. Numer. Anal. 8 Ž 994., G. Aguilar, F. Lisbona, and M. Madaune-Tort, Analysis of a nonlinear parabolic-hyperbolic problem, Ad. Math. Sci. Appl., in press. 4. F. Gastaldi and A. Quateroni, On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach, Appl. Numer. Math. 6 Ž 989., V. I. Istratescu, Fixed Point Theory: An Introduction, Reidel, Dordrecht, 98.

16 SINGULAR PERTURBATION ON A SUBDOMAIN J. Lorenz, Nonlinear Singular Perturbation Problems and the Engquist-Osher Difference Scheme, Report 85, Univ. Nijmegen, J. Lorenz, Nonlinear boundary value problems with turning points and properties of difference schemes, in Lecture Notes in Mathematics ŽW. Eckhaus and E. M. de Jager, Eds.., Vol. 94, Springer-Verlag, BerlinHeidelbergNew York, K. Niijima, A uniformly convergent difference scheme for a semilinear singular perturbation problem, Numer. Math. 43 Ž 984., P. E. Sacks, The initial and boundary value problem for a class of degenerate parabolid equations, Comm. Partial Differential Equations 8, No. 7 Ž 983., J. Schroder, Operator Inequalities, Academic Press, New York, 98.