Existence, uniqueness and limit behaviour of solutions to a nonlinear boundary-value problem with equivalued surface1

Nonlinear Analysis 34 (1998) 525 – 536

Existence, uniqueness and limit behaviour of solutions to a nonlinear boundary-value problem with equivalued surface1 Xu Zhang, Feng quan Li ∗ Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China Received 1 October 1996; received in revised form 6 January 1997

Keywords: Existence; Uniqueness; Limit behaviour; Nonlinear boundary-value problem; Equivalued surface

1. Introduction Let ⊂ Rn be a bounded domain with the boundary @ = 0 ∪ 1 ( 0 being the interior boundary and 1 6= ∅ the outer boundary with 0 ∩ 1 = ∅). For simplicity, we assume 0 ; 1 ∈ C ∞ . In this paper, we consider the following nonlinear boundary-value problem with equivalued surface:  X @  @u aij (x; u) = b(x) in ; (1.1) Au ≡ − @xi @xj u| 1 = 0;

(1.2)

u| 0 = C (a constant to be determined); Z @u ds = A0 (a known constant); @n A 0

(1.3) (1.4)

where

@u @u X = aij (x; u) ni : @nA @xj

∗ 1

(1.5)

Corresponding author. This work is partially supported by NSF of China and NSF of the Chinese State Education Commission.

0362-546X/98/$19.00 ? 1998 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 7 ) 0 0 6 9 2 - 5

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X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

One can nd many concrete physical examples for problem (1.1) – (1.4), for instance, the static electric eld around a conductor, the stable temperature eld around an electric cable, the torsion problem of elastic bar with multiply-connected cross-section and so on. In the linear case, the existence, uniqueness and limit behaviour of solutions to the corresponding problem is well understood (cf. [2, 6–9]). However, in the nonlinear case, there are only few results so far. In this paper, we shall prove the existence of a weak solution to problem (1.1) – (1.4)  solution by means of the pseudomonotone operator theory and the existence of C 2;  ( ) to problem (1.1)–(1.4) by Leray–Schauder’s xed point theorem. The uniqueness of both, the weak solution and the classical solution to problem (1.1) – (1.4) is proved by a method similar to that of [4]. We also discuss a special class of problems for the limit behaviour of the solution to problem (1.1) – (1.4). The paper is organized as follows. In Section 2, we give some necessary preliminaries and a priori estimates. The existence of solutions for problem (1.1) – (1.4) is studied in Section 3. The uniqueness of solutions to problem (1.1) – (1.4) is discussed in Section 4. Finally, we consider the limit behaviour of solutions to problem (1.1) – (1.4) in Section 5.

2. Some preliminaries and a priori estimates  C 1;  ( );  C 1;  (

 × R); C 2;  ( )  denote the usual spaces (cf. Let L 2 ( ); H01 ( ); C  ( ); [1, 3]). Set V = {u | u ∈ H 1 ( ); u| 1 = 0; u| 0 = constant}:

(2.1)

It is obvious that V is a closed subspace of H 1 ( ). Assume that there is a constant 0 ¿0, such that X aij (x; u)i j ≥ 0 || 2 ; ∀(x; u; ) ∈ ( × R × Rn ):

(2.2)

i; j

Lemma 2.1. For any v ∈ V; we have |v| 0 |≤ C|B0 |1=n−1=2 k∇vkL 2 ( )

(2.3)

kvkL 2 ( ) ≤ Ck∇vkL 2 ( ) ;

(2.4)

and

˜ with

˜ = ∪ B0 ∪ where C = C( )

0;

B0 being the domain surrounded by

0.

Proof. Set v˜ (x) = v(x) + v| 0 B0 ∪ 0 :

(2.5)

X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

527

˜ By Holder’s inequality and Sobolev’s imbedding theorem, It is obvious that v˜ ∈ H01 ( ). we have Z |˜v| dx |v| 0 | = |B0 |−1 B0

−1=2+1=n

Z

≤ |B0 |

≤ |B0 |−1=2+1=n

≤ C|B0 |−1=2+1=n

˜



|˜v| dx

B0

Z

1=q

q

|˜v|q dx

Z

˜

1 1 1 = − q 2 n



1=q

|∇v| 2 dx

= C|B0 |−1=2+1=n k∇vkL 2 ( ) :

1=2

(2.6)

This proves Eq. (2.3). Moreover, kvkL 2 ( ) ≤ k˜vkL 2 ( ) ˜ ˜ ≤ C( )k∇˜ vkL 2 ( ) ˜ ˜ = C( )k∇vk L 2 ( ):

(2.7)

This proves Eq. (2.4). The second conclusion of Lemma 2.1 means that k∇vkL 2 ( ) is an equivalent norm in V .  solution to problem (1.1) – (1.3), it is In order to study the existence of C 2;  ( ) necessary to consider the following linear problem:  X @  @u aij (x) = b(x) in ; (2.8) Au ≡ − @xi @xj u| 1 = 0;

(2.9)

(2.10) u| 0 = C (a constant to be determined); Z @u ds = A0 (a known constant); (2.11) @nA 0 P where @u=@nA = aij (x)ni @u=@xj , and aij (i; j = 1; : : : ; n) is assumed to satisfy the following elliptic condition: there is a constant ¿0, such that X (2.12) aij i j ≥ || 2 ; ∀(x; ) ∈ × Rn : The following lemma is a revised form of some known results (for the proof cf. [2, 6–9]).

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X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

Lemma 2.2. Under the previous assumptions; let b ∈ V 0 ; where V 0 denotes the dual space of V . Then problem (2.8)–(2.11) admits a unique weak solution u ∈ V such that Z X @u @w aij (x) dx − A0 w| 0 =hb; wiV 0 ; V ; ∀w ∈ V: (2.13) @xj @xi

i; j Furthermore; we have k∇ukL 2 ( ) ≤ C(kbkV 0 + |A0 |);

(2.14)

where C¿0 is a constant depending only on n; and . Assume furthermore that aij ∈ W k+1; ∞ ( ); b ∈ W k; 2 ( ); where k is a nonnegative integer; then u ∈ W k+2; 2 ( ) ∩ V . Remark 2.1. When aij depend on u and Eq. (2.2) is satis ed; Eq. (2.14) for the solution u to problem (1.1)–(1.4) still holds.  satisfy the elliptic condition (2.12), b ∈ C  ( ).  Then Lemma 2.3. Let aij ∈ C 1;  ( ) 2;   problem (2.8)–(2.11) admits a unique solution u ∈ C ( ); moreover; |u|2; ; ≤ C(|b|; + |A0 |);

(2.15)

where C¿0 is a constant depending only on n; ; and |aij |1; ; (i; j = 1; : : : ; n).  solution follows easily from Proof. The existence and uniqueness of C 2;  ( ) Lemma 2.2 and the result on linear elliptic equations (cf. [3]). In order to prove  such that ’1 = 0 on 1 , ’1 = 1 on 0 . Eq. (2.15), construct a function ’1 ∈ C ∞ ( ) Denote ’ = a’1 , where a = u| 0 . By Schauder’s estimate and the maximum principle, we get |u|2; ; ≤ C(|b|; + |a||’1 |2; ; + |u|0; ) ≤ C(|b|; + |a|):

(2.16)

By Lemma 2.1, we have |a| ≤ |u| 0 | ≤ Ck∇ukL 2 ( ) :

(2.17)

By Eq. (2.14), we have k∇ukL 2 ( ) ≤ C(kbkV 0 + |A0 |) ≤ C(kbk; + |A0 |):

(2.18)

Now, by Eqs. (2.16) and (2.17), we obtain Eq. (2.15). This completes the proof.  × R) and u ∈ C 2 ( )  is a solution to the problem Lemma 2.4. If aij ∈ C 1 (

(1.1) – (1.4), then there exists a positive constant C1 ; such that sup |u| ≤ C1 :



(2.19)

X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

529

Proof. The proof follows easily from Theorem 9.7 in [3] and estimates (2.3) and (2.14). We omit the details. Lemma 2.5. Under the same assumptions of Lemma 2.4, suppose that aij ∈  × [−M; M ]) for each M ¿0. Let u ∈ C 2 ( )  is a solution to the problem (1.1) – C 1;  (

(1.4). Then there exist two constants C2 and 0¡ ¡1 depending only n; 0 ; ;  and A0 ; such that |u|1; ; ≤ C2 :

(2.20)

Proof. The proof is similar to [3, 5]. 3. Existence of solutions 3.1. Existence of weak solutions Suppose that aij (x; u) (i; j = 1; : : : ; n) are Caratheodory functions satisfying that aij (· ; 0) ∈ L∞ ( ) and for any given M ¿0 there exists hM (·) ∈ L∞ ( ) such that |aij (x; u1 ) − aij (x; u2 )| ≤ hM (x)|u1 − u2 | ∀uk ∈ R; |uk | ≤ M;

a.e x ∈ ;

k = 1; 2 (i; j = 1; 2; : : : ; n):

(3.1)

Theorem 3.1. Let Eqs. (2.2) and (3.1) hold. Assume; furthermore; that b ∈ Lq ( ); q¿ max{n=2; 2}. Then problem (1.1)–(1.4) admits a weak solution u ∈ V ∩ L∞ ( ) in the following sense: Z X @u @w aij (x; u) dx − A0 w| 0 = hb; wiV 0 ; V ; ∀w ∈ V: (3.2) @xj @xi

ij Furthermore; we have max{kukL∞ ( ) ; k∇ukL 2 ( ) } ≤ C(kbkLq + |A0 |);

(3.3)

where C¿0 is a constant depending only on n and . Proof. (1) For any xed m¿0, let  if u¿m,  aij (x; m) (x; u) if |u| ≤ m, a am (x; u) = ij ij  a (x; −m) if u¡ − m. ij De ne the operator A : V → V 0 in the following way: for any u; w ∈ V , Z X @u @w 0 am dx − A0 w| 0 − hb; wiV 0 ; V : hAu; wiV ; V = ij (x; u) @x j @xi

i; j

(3.4)

(3.5)

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X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

It is easy to check that (i) A is bounded; (ii) hAw; wiV 0 ; V →∞ kwkV

as kwkV → ∞;

(iii) A is pseudomonotone, i.e., if wn ∈ V such that wn → w

weakly in V

(3.6)

and lim hAwn ; wn − wiV 0 ; V ≤ 0;

(3.7)

n→∞

then we have lim hAwn ; wn − viV 0 ; V ≥ hAw; w − viV 0 ; V ;

n→∞

∀v ∈ V:

(3.8)

In fact, by Eqs. (3.5) and (3.1), we have hAwn ; wn − wi   Z X @w @(wn − w) @wn am (x; w ) − dx = n ij @xj @xj @xi

ij +

Z X

ij

am ij (x; wn )

@w @(wn − w) dx @xj @xi

− A0 (wn − w)| 0 − hb; wn − wi Z |∇(wn − w)| 2 dx ≥ 0 ×

+

XZ

am ij (x; wn ) ×

ij

@w @(wn − w) dx @xj @xi

− A0 (wn − w)| 0 − hb; wn − wi:

(3.9)

Set Hi (x; v) =

X

am ij (x; v)

j

@w(x) : @xj

(3.10)

It is easy to check that Hi is a Caratheodory function and satis es |Hi (x; v)| ≤ C|Dw(x)|:

(3.11)

X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

531

Hence, Hi ∈ L 2 and set fi (v(x)) = Hi (x; v(x));

x ∈ :

(3.12)

Hence, fi : L 2 ( ) → L 2 ( )

is continuous:

(3.13)

By Eq. (3.6) and Sobolev’s imbedding theorem, we have wn → w

strongly in L 2 ( ):

(3.14)

Hence, fi (wn ) → fi (w)

strongly in L 2 ( ):

(3.15)

By Eq. (3.6), the trace theorem and the imbedding theorem, we have wn → w

in L 2 ( 0 ):

(3.16)

By Eq. (3.7), Lemma 2.1 and Eqs. (3.14)–(3.16), taking n → ∞ in Eq. (3.9), we get wn → w

strongly in V:

(3.17)

Then Eq. (3.8) holds. Thus, by the theory of the pseudomonotone operator (cf. Theorem 27.A in [10]), there exists um ∈ V such that Aum = 0

in V 0 ;

(3.18)

moreover, by Remark 2.1 we have k∇um kL 2 ( ) ≤ C(kbkLq + |A0 |);

(3.19)

where C is independent of m. By the remark of Theorem 9.7 in [3], Lemma 2.1 and Remark 2.1, we have kum kL∞ ≤ C(kbkLq + |A0 |);

(3.20)

where C is independent of m. (2) By Eq. (3.19), there exists a subsequence (still denote by {um }) such that um → u

weakly in V;

(3.21)

um → u

strongly in L1 :

(3.22)

then

By Eqs. (3.20) and (3.22), we have um → u

weak∗ in L∞ :

Therefore, we get kukL∞ ≤ c(kbkLq + |A0 |):

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X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

Now, consider the following identity: XZ @um @w am dx − A0 w| 0 − hb; wiV 0 ; V = 0; ij (x; um ) @xj @xi

ij Taking m¿C(kbkLq + |A0 |) in Eq. (3.23), we obtain XZ @um @w aij (x; um ) dx − A0 w| 0 − hb; wiV 0 ; V = 0; @xj @xi

ij Hence, taking m → ∞, we have XZ @u @w aij (x; u) dx − A0 w| 0 − hb; wiV 0 ; V = 0; @xj @xi

ij

∀w ∈ V:

(3.23)

∀w ∈ V:

∀w ∈ V

(3.24)

This proves the existence. Inequality (3.3) can be easily checked. So the proof is completed.  solution 3.2. Existence of C 2;  ( )  × R) ∩ C 1;  (

 × [−M; M ]) for each M ¿0; Theorem 3.2. Suppose that aij (· ; ·) ∈ C 1 (

  b ∈ C ( ); 0¡¡1. Suppose; furthermore; that Eq. (2.2) holds. Then the problem  (1.1) – (1.4) admits a solution u ∈ C 2;  ( ). Proof. In order to use Leray–Schauder’s xed point theorem, for  ∈ [0; 1], we consider the following problem:  X    @ @u  A u≡−  aij (x; u) + (1 − )
u = b(x) in ; (3.25) @xi @xj u| 1 = 0;

(3.26)

(3.27) u| 0 = C (a constant to be determined); Z @u ds = A0 (a known constant); (3.28) @n A 0 P where @u=@nA = (aij (x; u) + (1 − )ij )ni @u=@xj . By Lemma 2.5, we conclude that there exists two constants M0 ; ∈ (0; 1) independent of , such that |u|1; ; ≤ M0 :

(3.29)

Set  X = C 1; ( );

D = {u | u ∈ X; u| 1 = 0; u| 0 = constant; |u|1; ; ≤ M0 + 1}: (3.30)

It is easy to check that D is a closed set in X .

X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

533

 we consider the following linear Now, for each  ∈ [0; 1] and any given w ∈ D, problem:  X    @ @u A u ≡ −  aij (x; w) + (1 − )
u = b(x) in ; (3.31) @xi @xj u| 1 = 0;

(3.32)

u| 0 = C (a constant to be determined); (3.33) Z @u ds = A0 (a known constant); (3.34) @n A 0 P where @u=@nA = (aij (x; w) + (1 − )ij )ni @u=@xj .  so by Lemma 2.3, problem (3.31) – The coecients of Eq. (3.31) belong to C 1;  ( ), 2;    It is easy to see that u ∈ X . (3.34) admits a unique solution u ∈ C ( ) ⊂ C 1; ( ).  Thus, u = T (w; ) de nes a map T : D × [0; 1] → X . Similar to the proof of the existence  solution to quasilinear elliptic equations [1, 3, 5], we can check that this of C 2; ( ) map satis es all the conditions in Leray–Schauder’s xed point theorem. Thus, T (· ; 1)  and there is an estimate similar to Eq. (2.15) for admits a xed point u ∈ C 2;  ( ) 1;    and then u is this u. Thus, aij (x; u(x)) ∈ C ( ). By Lemma 2.3, we get u ∈ C 2; ( ), a solution to problem (1.1)–(1.4). This completes the proof.

4. Uniqueness of solution Theorem 4.1. Suppose that hypotheses (2.2) and (3.1) hold; then problem (1.1) – (1.4) admits at most one weak solution u ∈ V ∩ L∞ ( ). Proof. Assume that u1 ; u2 ∈ V ∩ L∞ ( ) are two weak solutions to problem (1.1) – (1.4), i.e., Z X n

i=1

aij (x; uk )

@uk @w dx − A0 w| 0 = hb; wi; @xj @xi

k = 1; 2; ∀w ∈ V:

(4.1)

Denote

1 = {x ∈ ; u1 (x)¡u2 (x)}:

(4.2)

We assert that | 1 | = 0. In fact, for any ¿0, let E = {x ∈ 1 ; u2 − u1 ¿}

(4.3)

v = min(; (u2 − u1 )+ ):

(4.4)

and

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X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

It is easy to see that v ∈ V . By Eq. (4.1), we obtain 0=

=

Z X

ij

Z X

i; j

+

@u2 @u1 − aij (x; u1 ) aij (x; u2 ) @xj @xj

aij (x; u2 )



@(u2 − u1 ) @w dx @xj @xi

Z X

@u1 @u1 − aij (x; u1 ) aij (x; u2 ) @xj @xj

ij

@w dx @xi



@w dx; @xi

∀w ∈ V:

(4.5)

Taking w =˙ v in Eq. (4.5), by Eqs. (2.2) and (3.1), we have Z

Z X

|∇v | 2 dx ≤

aij (x; u1 )

ij

@u1 @u1 − aij (x; u2 ) @xj @xj



@v dx @xi

Z  X @u1 @u1 @v − aij (x; u2 ) dx aij (x; u1 ) ≤ 1 −E @xj @xj @xi ij

Z ≤ C

1 −E

(|∇u1 |) 2 dx

1=2 k∇v kL 2 ( ) :

(4.6)

Thus, Z k∇v kL 2 ( ) ≤ C But |E | = 

Z

−2

E

≤ C

−2

Z ≤C

2

1 −E

 dx ≤ 

Z



1 −E

−2

|∇u1 | 2 dx

Z

1=2 :

(4.7)

|v | 2 dx

|∇v | 2 dx |∇u1 | 2 dx → 0;

as  → 0:

(4.8)

Since |E | → | 1 | as  → 0, hence we get | 1 | = 0. Hence, u1 (x) ≥ u2 (x) a.e. x ∈ . Interchanging the role of u1 and u2 , the uniqueness follows. This completes the proof. Remark 4.1. If b(x) is replaced by b(x; u) and suppose that b(x; ·) is decreasing with respect to the second argument; we can obtain a similar uniqueness result.

X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

535

5. Limit behaviour of solutions In this section, we consider   n  X @   @u aij (x; u ) = b(x) in  ; Au ≡ − @xi @xj

(5.1)

i; j=1

u | 1 = 0;

(5.2)

u |  = C  (a constant to be determined); (5.3) Z  @u ds = A (a known constant); (5.4)  @nA P where @u =@nA = i; j aij (x; u )ni @u =@xj , 1 is the xed outer boundary of  and n  = {x ∈ R ; kxk = } is the interior boundary of  . We will prove that under some conditions, as  → 0, the limit problem of problem (5.1) – (5.4) is as follows:   n X @u @ aij (x; u) = b(x) in ; (5.5) − @xi @xj i; j=1

u| 1 = 0;

(5.6)

where =  ∪ {x ∈ Rn ; kxk ≤ }. Theorem 5.1. (i) Suppose that Eqs. (2.2) and (3.1) hold. Suppose; furthermore; that   n  ;2 ; (5.7) n ≥ 3; q¿ max A = O((n−2)=2 ); b(x) ∈ Lq ( ) 2 then u˜ → u

weakly in H01 ( );

(5.8)

where u˜ = u   + u |   −  :

(5.9)

(ii) Assume; furthermore; that b(x) = 0; then u˜ → 0

strongly in H01 ( ):

(5.10)

Proof. Similar to estimate (2.14), we obtain that k∇u˜ kL 2 ( ) ≤ C;

(5.11)

where C¿0 is a constant independent of . Thus, the desired result is obtained in a way similar to the linear case. Remark 5.1. If n = 2; A = O(ln 1 )−1=2 ); we have a similar result for the limit behaviour of weak solutions to problem (5.1) – (5.4). This result can be applied to the torsion problem of elastic bars with multiply-connected cross-sections.

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X. Zhang, F.q. Li / Nonlinear Analysis 34 (1998) 525 – 536

Acknowledgements The problem considered in this paper was posed by Professor Li Ta-tsien. The authors gratefully acknowledge Professors Li Ta-tsien, Li Xunjing and Yong Jiongmin for their help. References [1] Y.Z. Chen, L.C. Wu, Elliptic Partial Dierential Equations and Elliptic Partial Dierential Systems of Second Order, Science Press, Beijing, 1991 (in Chinese). [2] A. Damlamian, T.T. Li, Comportements limites des solutions de certains problemes mixtes pour des e quations paraboliques, J. Math. Pure Appl. 61 (1982) 113–130. [3] D. Gilbarg, N.S. Trudinger, Elliptic Partial Dierential Equations of Second Order, Springer, Heidelberg, 1977. [4] I. Hlavac ek, M. Krizek, On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type, J. Math. Anal. Appl. 184 (1994) 168–188. [5] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations (English Translation from Russian) Academic Press, New York, 1968. [6] T.T. Li, Elastic torsion of bars with multiply-connected cross section, Comm. Appl. Math. Comput. 1 (1987) 19–24 (in Chinese). [7] T.T. Li, A class of non-local boundary value problems for partial dierential equation and its applications in numerical analysis, J. Comput. Appl. Math. 28 (1989) 49–62. [8] T.T. Li, S.X. Chen, On the asymptotic behaviour of solutions of equivalued surface boundary value problems for the second order self-adjoint elliptic equation, J. Fudan. Univ. (Nat. Sci.) (4) (1978) 6–14 (in Chinese). [9] T.T. Li et al., Boundary value problems with equivalued surface boundary conditions for self-adjoint elliptic dierential equations (I), J. Fudan Univ. (Nat. Sci.) 1 (1976) 61–71 (in Chinese). [10] E. Zeidler, Nonlinear Functional Analysis and Its Applications II=B, Springer, New York, 1990.