Uniqueness and radial symmetry for an inverse elliptic equation

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UNIQUENESS AND RADIAL SYMMETRY FOR

AN INVERSE ELLIPTIC EQUATION

B. EMAMIZADEH and M. H. MEHRABI

Received 7 November 2002

We consider an inverse rearrangement semilinear partial differential equation in a 2-dimensional ball and show that it has a unique maximizing energy solution. The solution represents a confined steady flow containing a vortex and passing over a seamount. Our approach is based on a rearrangement variational principle extensively developed by G. R. Burton.

2000 Mathematics Subject Classification: 35J35, 35J60, 76B03.

1. Introduction. This paper is concerned with the following problem in a bounded domainΩ:

−∆u=φ(u)+h inΩ, u=0 on∂Ω, u >0, −∆u∈Ᏺ+h,

(1.1)

whereΩis some bounded domain inR2_{. In (1.1), thenonlinearityφis unknown,}

andᏲis a family of functions which arerearrangementsof a prescribed func-tion, hence problem (1.1) is named aninverse rearrangement semilinear elliptic equation. Therefore, by a solution for (1.1) we mean a pair(u,φ)which satis-fies all conditions (in some sense) of (1.1). Here we are concerned with special types of solutions for (1.1); namely, theenergy maximizing solutions. To state the definition of such solutions, we first need some preparations.

Henceforthpis a fixed number in(2,∞)andqis its conjugate exponent, so 1/p+1/q=1. The so-calledheightfunctionhis some nonnegative function inLp_{(Ω). We letK:L}p_(Ω)→H1

0(Ω)denote the standard inverse of−∆with

Dirichlet homogeneous boundary conditions inΩ. We recall thatKis continu-ousandpositive; that is,

ΩζKζ >0 ∀ζ∈L

p_{(Ω). (1.2)}

Finally note thatKissymmetric:

ΩζKζ ₌

Ωζ

Now we can set up theenergy functional associated with (1.1). We defineΨ: Lp_{(Ω)→ ∞as follows:}

Ψ(ζ)=1₂

ΩζKζ+

Ωηζ, (1.4)

whereη=Kh. Next we define the variational problem

sup ζ∈ᏲΨ(ζ),

(1.5)

whereᏲdenotes the set of rearrangements of some nonnegative functionζ0∈

Lp_{(Ω). We recall thatζis arearrangementofζ0whenever the sets}

x∈Ω:ζ(x)≥α, x∈Ω:ζ0(x)≥α (1.6)

have the same Lebesgue measures for every positiveα. Note that all members ζ∈Ᏺsatisfy

ζp=ζ0_p, (1.7)

where · p denotes the usual norm inLp(Ω). The solution set for (1.5) is denotedΣ.

Definition1.1. The pair(u,φ)is called a maximizing energy solution of (1.1) whenever the following conditions are satisfied:

(i) u∈K(Σ)+h,

(ii) (u,φ) is a solution of (1.1). In (i) we have

K(Σ)= {Kζ:ζ∈Σ}. (1.8)

The main result of this paper is the following theorem.

Theorem1.2. IfΩis a ball centered at the origin, then there exists a unique uand there exists an increasing functionφsuch that(u,φ)is a maximizing energy solution for (1.1).

We end this section with some history of problem (1.1). This problem was first considered in an unbounded domain, precisely in the whole ofR2_{, by}

UNIQUENESS AND RADIAL SYMMETRY... 3049

2. Preliminary results. In this section, we state some lemmas which will be used in the proof ofTheorem 1.2.

Lemma2.1. LetΦ:Lp_{(B)→ ∞be strictly convex, weakly sequentially} contin-uous, and Gateaux differentiable. Then the variational problem

sup

ζ∈ᏲΦ(ζ) (2.1)

is solvable. Moreover, ifζˆ∈Ᏺis any such solution, then

ˆ

ζ=φ◦Φ_ζˆ _(2.2)

for some increasing functionφunknowna priori.

Ifu∈H1

0(Rn)is nonnegative,u∗ will denote the essentially unique

spher-ically symmetric radially decreasing rearrangement of u; thenu∗_∈H1 0(Rn)

also, and the inequality

Rn∇u

∗2 ≤

Rn|∇u|

2 _(2.3)

is standard. The case of equality has been studied by Brothers and Ziemer [1]; they proved results from which the following lemma can be deduced.

Lemma2.2. Letu∈H01(Rn)be nonnegative and have compact support, and

letM=ess supu(which may be infinite). Suppose that

Rn∇u

∗2 =

Rn|∇u|

2_{. (2.4)}

Then,

(1) for0≤α < M,u−1_{(α,∞)is a translate of the ball(u}∗₎−1_{(α,∞), apart}

from a set of measure zero; (2) if additionally

x∈Rn_:∇u∗_{(x)=0,0< u}∗_{(x) < M (2.5)}

is a set of zero measure, thenuis a translate ofu∗_.

The following lemma is an immediate consequence of [1, Lemma 2.3(v) and the succeeding remark].

Lemma2.3. Letu∈H1

0(Rn)be nonnegative andM=ess supu. If

x∈Rn_{:∇u(x)=0,0< u(x) < M (2.6)}

has zero measure, then

x∈Rn_:∇u∗_{(x)=0,0< u}∗_{(x) < M (2.7)}

3. Proof of the theorem. We begin by considering the solvability of (1.5). Indeed, using elliptic regularity theory, it is clear thatK:Lp_(B)→W2,p_{(B)is a} continuous linear operator. SinceW2,p_{(B)is compactly embedded intoC}1_(B),

it follows that K:Lp_(B)→Lq_{(B)is a linear compact operator. Therefore, Ψ} turns to be a weakly sequentially continuous functional. Moreover, sinceKis positive and symmetric, it follows thatΨis also strictly convex. The Gateaux differentiability ofΨis straightforward; and it is easy to see that the derivative ofΨatvcan be identified withKv+η. From all this we can see thatLemma 2.1

is applicable. So (1.5) is solvable, and if ˆζis any solution of (1.5), then

ˆ

ζ=φK_ζˆ_{+η, (3.1)}

almost everywhere inB, for an increasing functionφ. We setH1

0(B)≡Ᏼ, and the norm onᏴis denotedu =(

B|∇u|2)1/2. We define a parametrized convex functional᏷by

᏷c(u)=     

1 2u−

Bhu+c, u∈Ᏼ,

∞, u∈Lq_(B)\Ᏼ, (3.2)

wherecis a real parameter. We now consider the conjugate convex functional ᏷∗

c of᏷cdefined by

᏷∗

c(v)= sup u∈Lq_(B)

Buv−᏷c(u)

, v∈Lp_{(B). (3.3)}

Recalling the variational setup forK, it is easy to obtain

᏷∗ c(v)=

1 2

B(v+h)K(v+h)−c, (3.4)

from which, by settingc=1/2_Bhη, and from the symmetry property ofKwe infer that

᏷∗

c =Ψ. (3.5)

We fix a nonnegative functionv∈Lp_{(B). Then the supremum in (3.3) is} at-tained atu≡Kv+η. Therefore, from (3.3), we obtain

Ψ(v)+᏷c(u)=

Buv. (3.6)

Again, from (3.3), we infer that

Ψv∗_+᏷ cu∗≥

Bu

∗_v∗_{. (3.7)}

So from (3.6), (3.7), and a standard rearrangement inequality, it follows that

Ψv∗_+᏷

UNIQUENESS AND RADIAL SYMMETRY... 3051

At this stage, we make another assumption; namely, we suppose thatv∈Σ. Since (1.5) is solvable,Σis not empty. Thus, from (3.8), we infer that

᏷cu∗≥᏷c(u). (3.9)

Therefore,

1 2u

∗2 −

Bhu ∗_≥1

2u

2₋

Bhu. (3.10)

Since_Bhu≤Bhu∗, it follows from (3.10) thatu∗ ≥ u. So, in view of (2.3), we deduce thatu = u∗_.

Claim. We haveu=u∗_.

Proof of the claim. From the maximum principle and elliptic regularity theory, it follows thatuis a positive function inC1_{(B). We fixx}

1∈B. The set

S≡x∈B:u(x)≥ux1

=u−1_ux 1

,∞ (3.11)

is a ball according toLemma 2.1. Ifx∈intS, the interior ofS, then, by the maximum principle,u(x) > u(x1); thusx1∈∂S, the boundary ofS. Now we

can apply the Hopf boundary point lemma to deduce that ∂u/∂ν(x1) <0,

whereνis the unit normal to∂Satx1pointing outward. Therefore, the set

x∈B:∇u(x)=0,0< u(x) < M, (3.12)

whereM=max_Ωu, is empty, so its measure is zero. Hence, fromLemma 2.3, the set{x∈B:∇u∗_{(x)=0,0< u}∗_{(x) < M}also has zero measure. Therefore,} by Lemma 2.2, it follows that u is a translate ofu∗_{. However, sinceu is a} positive function, we infer thatu=u∗_{as desired. This completes the proof of} the claim.

Note that, from (3.1), we have

v=φ(u), (3.13)

almost everywhere inB, for some increasing functionφ. Sovis also spherically symmetric and radially decreasing; hence,v=v∗_=ζ∗

0. Since−∆u=v+h, it

follows that

−∆u=φ(u)+h, (3.14)

which is the differential equation in (1.1). It is easy to check that(u,φ)satisfies all other conditions in (1.1), so(u,φ)is a maximizing energy solution of (1.1) as desired. The functionuis obviously unique; in fact,

u=Kζ∗

0. (3.15)

References

[1] J. E. Brothers and W. P. Ziemer,Minimal rearrangements of Sobolev functions, J. reine angew. Math.384(1988), 153–179.

[2] G. R. Burton,Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann.276(1987), no. 2, 225–253.

[3] G. R. Burton and B. Emamizadeh,A constrained variational problem for steady vortices in a shear flow, Comm. Partial Differential Equations 24(1999), no. 7-8, 1341–1365.

[4] B. Emamizadeh,Steady vortex in a uniform shear flow of an ideal fluid, Proc. Roy. Soc. Edinburgh Sect. A130(2000), no. 4, 801–812.

[5] ,Existence of a steady flow with a bounded vortex in an unbounded domain, J. Sci. Islam. Repub. Iran12(2001), no. 1, 57–63.

[6] B. Emamizadeh and F. Bahrami,Steady vortex flows obtained from an inverse prob-lem, Bull. Austral. Math. Soc.66(2002), no. 2, 213–226.

[7] J. Nycander and B. Emamizadeh, Variational problem for vortices attached to seamount, to appear in Nonlinear Analysis Theory, Methods and Applica-tions.

B. Emamizadeh: Department of Mathematics, Iran University of Science and Technol-ogy, Narmak 16844, Tehran, Iran

Current address: Institute for Studies in Theoretical Physics and Mathematics, Ni-avaran Square, Tehran, Iran

E-mail address:[email protected]

M. H. Mehrabi: Department of Mathematics, Iran University of Science and Technol-ogy, Narmak 16844, Tehran, Iran