Uniqueness of weak solution for nonlinear elliptic equations in divergence form

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UNIQUENESS OF WEAK SOLUTION FOR NONLINEAR

ELLIPTIC EQUATIONS IN DIVERGENCE FORM

XU ZHANG

(Received 3 April 1997 and in revised form 21 June 1997)

Abstract.We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

Keywords and phrases. Uniqueness, weak solution, quasilinear elliptic equation, diver-gence form, comparison theorem.

2000 Mathematics Subject Classiﬁcation. Primary 35A05, 35B05, 35B50.

1. Introduction. In this paper, we demonstrate the uniqueness of weak solution of the Dirichlet problem for divergence structure elliptic equations of the form

L[u]≡ − n

i=1

∂ ∂xi

ai(x,u,∇u)+b(x,u)=0 inΩ, u|∂Ω=0, (1.1)

whereΩis a bounded domain ofRn_{. In [1, 2, 3, 7, 8, 5], the uniqueness of classical} solutions of problem (1.1) is treated under various hypotheses. Here, we consider the same problem for weak solutions. Especially, we give some counterexamples to show that our result cannot be improved in the general case.

To conclude this section, we would like to point out that after this paper had been submitted for publication, it came to our attention that a similar (uniqueness) re-sult had been given in [4]. However, there is no further discussion in [4] as we do in Section 4.

2. Statement of the main results. Suppose that, for any (x,z1,z2,η)∈Ω×R×

R×Rn_,

i,j

∂ai(x,z,η)

∂ηj ξiξj≥ |ξ|

2_, _{for all}_ξ_∈_Rn_, _(2.1)

b(x,z1)−b(x,z2)(z1−z2)≥0, (2.2)

_a_i_(x,z₁_,η)₋_a_i_(x,z₂_,η)_≤_a(z₁_,z₂₎_|_z₁₋_z_2|

1+|η|, (2.3)

wherea∈L∞

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Remark2.1. Unlike the previous works (cf. [1, 5, 7, 8] and so on), we do not assume thatb(·,·)is Lipschitz continuous in its second argument.

We need the following two deﬁnitions, which can be found in many references (cf. [3] and so on).

Deﬁnition2.2. u∈H1₍_Ω₎_∩_L∞₍_Ω₎_{is said to satisfy}_L[u]_≥₍_≤₎_{0 in}_Ω_{in the weak} sense if

Ω

  

n

i=1

ai(x,u,∇u)∂w_∂x

i+b(x,u)w

 

dx≥(≤)0, (2.4)

for anyw∈H1

0(Ω)such thatw≥0 a.e. inΩ.

Deﬁnition2.3. u,v∈H1₍_Ω₎_{are said to satisfy}_u_≤_v_on_∂_Ω_{in the weak sense if}

(u−v)+_≡_max{_u₋_v,_{0} ∈}_H1 0(Ω). Now, we can state our main results.

Theorem2.4(Comparison theorem). Let the hypotheses (2.1), (2.2), and (2.3) hold, and letu1,u2∈H1(Ω)∩L∞(Ω)satisfy

L[u1]≥0, L[u2]≤0 inΩ, u2≤u1 on∂Ω (2.5)

in the weak sense. Then we haveu1≥u2a.e. inΩ. If, furthermore,a∈L∞_(R_×_R)_and_u

1,u2∈H1(Ω)satisfy condition (2.5), then the same conclusion holds.

Theorem2.5(Uniqueness Theorem). Let the hypotheses (2.1), (2.2), (2.3), and (2.4) hold. Then the problem (1.1) admits at most one weak solutionu∈H1

0(Ω)∩L∞(Ω). If, furthermore, a∈L∞_(R_×_R)_{, then (1.1) admits at most one weak solution}_u_∈ H1

0(Ω).

3. Proof of the main results

Proof of Theorem2.4. Assume thatu1,u2∈H1(Ω)∩L∞(Ω) satisfy condition (2.5) in the weak sense, that is

Ω

  

n

i=1

aix,u1,∇u1∂w_∂x

i+b(x,u1)w

 

dx≥0 (3.1)

and

Ω

  

n

i=1

aix,u2,∇u2_∂x∂w

i+b(x,u2)w

 

dx≤0, (3.2)

for anyw∈H1

0(Ω)such thatw≥0,a.e. inΩ. Put

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We assert that|Ω1| =0 (|Ω1|denotes the Lebesgue measure ofΩ1). In fact, for any

ε >0, we write

Eε=x∈Ω1;u2−u1> ε, vε=minε,(u2−u1)+. (3.4)

Note thatvε=0 and∇vε=0, a.e. inΩ\Ω1, and thatvε=εand∇vε=0, a.e. inEε (see, e.g., [9]).

Lemma3.1. Letu∈H1

0(Ω). Then|u|,u+,min(ε,u)∈H01(Ω)for any nonnegative constantε.

Proof. Letun∈C0∞(Ω)be such thatun→uinH01(Ω). Set

vn=

u2 n+_n1₂

1/2

−_n1. (3.5)

It is easy to see thatvn∈C0∞(Ω). By a direct calculation, we can check that{vn}is a Cauchy sequence inH1

0(Ω). Thus, there is a functionv∈H01(Ω)such thatvn→v inH1

0(Ω). On the other hand, it is obvious that for an appropriate subsequence (still denoted by itself)un→u, a.e. inΩand, hence,

vn → |u|, a.e. inΩ. (3.6)

So, we obtain that|u| =v∈H1

0(Ω).Similarly, we can prove thatu+, min(ε,u)∈H10(Ω) for any nonnegative constantε. This completes the proof of the lemma.

Now, we can return to the proof of Theorem 2.4. Sinceu1,u2∈H1(Ω)andu2≤u1 on∂Ωin the weak sense, we have(u2−u1)+∈H01(Ω)by Deﬁnition 2.3 andvε∈H10(Ω) by the above lemma. By (3.1) and (3.2), we then obtain

0≥

Ω

i

aix,u2,∇u2−aix,u1,∇u1_∂x∂w idx

+

Ω

bx,u2−bx,u1w dx

=

Ω

i,j

₁

0

∂aix,u2,∇u1+s∇u2−∇u1

∂ηj ds

∂u2−u1

∂xj

∂w ∂xidx

+

Ω

i

aix,u2,∇u1−aix,u1,∇u1_∂x∂w idx

+

Ω

bx,u2−bx,u1w dx,

(3.7)

for anyw∈H1

0(Ω)satisfyingw≥0 a.e. inΩ. Take w=vε in the above inequality. Note thatvε≥0 and thatu2> u1whenevervε>0. By condition (2.2), we see that

b(x,u2)−b(x,u1)≥0 whenevervε>0.Thus,

Ω

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So, we have 0≥ Ω i,j 1 0

∂aix,u2,∇u1+s∇u2−∇u1

∂ηj ds

∂u2−u1

∂xj

∂vε

∂xidx

+ Ω i

aix,u2,∇u1−aix,u1,∇u1∂v_∂xε idx.

(3.9)

Sincevε=u2−u1a.e. inΩ1\Eεand∇vε=0 a.e. inΩ\{Ω1\Eε}, by the above inequality, we get

0≥

Ω1\Eε

i,j

1 0

∂aix,u2,∇u1+s∇u2−∇u1

∂ηj ds

∂vε

∂xj

∂vε

∂xidx

+

Ω1\Eε

i

aix,u2,∇u1−aix,u1,∇u1∂v_∂xε idx

≥

Ω1\Eε|∇vε|

2_dx₊

Ω1\Eε

i

=

Ω|∇vε|

2_dx₊

Ω1\Eε

i

aix,u2,∇u1−aix,u1,∇u1∂v_∂xε i dx,

(3.10)

where we have used condition (2.1) at the second step. By (2.3) and the above inequal-ity, we get

Ω

_∇_v_ε2_dx_≤

Ω1\Eε

i

≤εC1

Ω1\Eε

1+∇u12dx

1/2

∇vεL2₍_Ω₎

(3.11)

for some constantC1(sinceu1,u2are bounded). Thus,

_∇_v_ε_L₂₍_Ω₎_≤_εC₁

Ω1\Eε

1+∇u12dx

1/2

. (3.12)

But also, using the Sobolev-Poincaré inequality onΩ, we obtain

_E_ε₌_ε−2 Eεε

2_dx_≤_ε−2

Ω

_v_ε2_dx

≤Cε−2

Ω

_∇_v_ε2_dx_≤_CC₂ 1

Ω1\Eε

1+∇u12dx.

(3.13)

But 0< u2−u1≤εinΩ1\Eε. So, obviously, lim ε→0

Ω1\Eε= ∅. (3.14)

Thus,|Eε| →0 asε→0. On the other hand, clearly,|Eε|is nondecreasing. Thus,|Eε| =0 for anyε >0 and in turnu2≤u1, a.e. inΩ.

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Proof of Theorem2.5. Assume thatu1,u2∈H01(Ω)∩L∞(Ω)are two weak solu-tions of problem (1.1). By Theorem 2.4, we must have

u1≥u2, u2≥u1, a.e. inΩ. (3.15) So,u1=u2a.e. inΩ, which proves the ﬁrst conclusion.

We omit the details of the proof of the second statement since it is similar to that of the ﬁrst part.

4. Some remarks and counterexamples

Remark4.1. Uniqueness does not hold for equations of nondivergence form,

− n i,j=1

aijx,u,∇u ∂ 2_u

∂xi∂xj+b(x,u)=0 inΩ, (4.1)

even whenb≡0 andaijis independent of∇u, as shown by Meyers [6]. In particular, he gave an example of a nondivergence equation with analytic coeﬃcients, which is uniformly elliptic and which has nonunique analytic solutions in a bounded domain with analytic boundary.

Remark4.2. Condition (2.3) essentially says that ai(x,z,η) is locally Lipschitz continuous with respect to its second argument and, furthermore, that the Lipschitz constant is independent ofx. This condition also cannot be removed. In fact, we have the following counterexamples.

(1) Consider the equation

−_∂x∂ i

a(x,u)δij_∂x∂u j

=1 inΩ, u|∂Ω=0, (4.2)

wherea(x,u)=2−sin{πsgn(u−u1)}and whereu1∈H01(Ω)solves the problem −2∆u=1 inΩ, u|∂Ω=0. (4.3)

It is easy to check that u=u1and u=2u1 are both solutions for problem (4.2), and that the minimum principle holds for (4.2) (for all of the following examples, maximum/minimum principle also holds). (Note thata(x,u)is not continuous in its second argument.)

(2) In the previous example, we take

a(x,u)=

  

2−sinu−2u1sinu−u11/2 ifu≥u1,

2 ifu < u1. (4.4)

It is easy to check that bothu=u1andu=2u1are solutions of problem (4.2). (Note thata(x,u)is not locally Lipschitz continuous in its second argument.)

(3) Take

a(x,u)=2−sin

_π

2

u(u−u1)

u2 1

(4.5)

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Remark4.3. Ifb(x,u)is replaced byb(x,u,∇u)in (1.1), we cannot obtain any satisfactory uniqueness result except for special cases such as∂b(·,·,·)/∂z 0 and where other restrictive conditions are assumed. In particular, consider the equation

d2_u

dx2+

xdu

dx

=0 inΩ=(−1,1), u(−1)=u(1)=0. (4.6)

We can check directly thatu=0 and

u(x)=

₁

xe s2_/2

ds χ[0,1](x)+

_x

−1e s2_/2

ds χ[−1,0)(x) (4.7)

are both weak solutions of this problem.

Acknowledgements. This work is partially supported by Chinese State Educa-tion Commission Science FoundaEduca-tion. The author gratefully acknowledges Professor James Serrin for his careful comments and useful help.

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Zhang: Institute of Mathematics, Fudan University, Shanghai200433, China