© Hindawi Publishing Corp. UNIQUENESS AND RADIAL SYMMETRY FOR AN INVERSE ELLIPTIC EQUATION

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.


Introduction.
This paper is concerned with the following problem in a bounded domain Ω: where Ω is some bounded domain in R 2 .In (1.1), the nonlinearity φ is unknown, and Ᏺ is a family of functions which are rearrangements of a prescribed function, hence problem (1.1) is named an inverse rearrangement semilinear elliptic equation.Therefore, by a solution for (1.1) we mean a pair (u, φ) which satisfies all conditions (in some sense) of (1.1).Here we are concerned with special types of solutions for (1.1); namely, the energy maximizing solutions.To state the definition of such solutions, we first need some preparations.Henceforth p is a fixed number in (2, ∞) and q is its conjugate exponent, so 1/p + 1/q = 1.The so-called height function h is some nonnegative function in L p (Ω).We let K : L p (Ω) → H 1 0 (Ω) denote the standard inverse of −∆ with Dirichlet homogeneous boundary conditions in Ω.We recall that K is continuous and positive; that is, Finally note that K is symmetric: Now we can set up the energy functional associated with (1.1).We define Ψ : L p (Ω) → ∞ as follows: where η = Kh.Next we define the variational problem where Ᏺ denotes the set of rearrangements of some nonnegative function ζ 0 ∈ L p (Ω).We recall that ζ is a rearrangement of ζ 0 whenever the sets have the same Lebesgue measures for every positive α.Note that all members ζ ∈ Ᏺ satisfy where • p denotes the usual norm in L p (Ω).The solution set for (1.5) is denoted Σ.
Definition 1.1.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 (1.8) The main result of this paper is the following theorem.
Theorem 1.2.If Ω is a ball centered at the origin, then there exists a unique u and there exists an increasing function φ such 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 of R 2 , by Emamizadeh and Nycander [7].Later Emamizadeh and Bahrami [6] considered the problem in the half-plane.In the case of unbounded domain, we usually face the lack of compactness which causes unavailability of the direct method in the analysis.In the present situation, we do not need to worry about the existence of a solution since this will readily be provided using results of Burton [2] about maximization of convex functionals over the sets of rearrangements.However, the point here is the uniqueness that we usually do not obtain when dealing with unbounded domains.The reader could also be referred to [3,4,5] for similar problems in unbounded domains.

Preliminary results.
In this section, we state some lemmas which will be used in the proof of Theorem 1.2.Lemma 2.1.Let Φ : L p (B) → ∞ be strictly convex, weakly sequentially continuous, and Gateaux differentiable.Then the variational problem for some increasing function φ unknown a priori.
) is nonnegative, u * will denote the essentially unique spherically symmetric radially decreasing rearrangement of u; then u * ∈ H 1 0 (R n ) also, and the inequality 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.
Lemma 2.2.Let u ∈ H 1 0 (R n ) be nonnegative and have compact support, and let M = ess sup u (which may be infinite).Suppose that (2.4) The following lemma is an immediate consequence of [1, Lemma 2.3(v) and the succeeding remark].
has zero measure, then also has zero measure.

Proof of the theorem.
We begin by considering the solvability of (1.5).Indeed, using elliptic regularity theory, it is clear that K : L p (B) → W 2,p (B) is a continuous linear operator.Since W 2,p (B) is compactly embedded into C 1 (B), it follows that K : L p (B) → L q (B) is a linear compact operator.Therefore, Ψ turns to be a weakly sequentially continuous functional.Moreover, since K is positive and symmetric, it follows that Ψ is also strictly convex.The Gateaux differentiability of Ψ is straightforward; and it is easy to see that the derivative of Ψ at v can be identified with Kv +η.From all this we can see that Lemma 2.1 is applicable.So (1.5) is solvable, and if ζ is any solution of (1.5), then almost everywhere in B, for an increasing function φ.
We set H 1 0 (B) ≡ Ᏼ, and the norm on Ᏼ is denoted u = ( B |∇u| 2 ) 1/2 .We define a parametrized convex functional by where c is a real parameter.We now consider the conjugate convex functional Recalling the variational setup for K, it is easy to obtain * c (v) = from which, by setting c = 1/2 B hη, and from the symmetry property of K we infer that * c = Ψ .( We fix a nonnegative function v ∈ L p (B).Then the supremum in (3.3) is attained at u ≡ Kv + η.Therefore, from (3.3), we obtain Again, from (3.3), we infer that So from (3.6), (3.7), and a standard rearrangement inequality, it follows that At this stage, we make another assumption; namely, we suppose that v ∈ Σ.Since (1.5) is solvable, Σ is not empty.Thus, from (3.8), we infer that 3), we deduce that u = u * .
Claim.We have u = u * .
Proof of the claim.From the maximum principle and elliptic regularity theory, it follows that u is a positive function in C 1 (B).We fix x 1 ∈ B. The set is a ball according to Lemma 2.1.If x ∈ int S, the interior of S, then, by the maximum principle, u(x) > u(x 1 ); thus x 1 ∈ ∂S, the boundary of S. Now we can apply the Hopf boundary point lemma to deduce that ∂u/∂ν(x 1 ) < 0, where ν is the unit normal to ∂S at x 1 pointing outward.Therefore, the set where M = max Ω u, is empty, so its measure is zero.Hence, from Lemma 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 of u * .However, since u is a positive function, we infer that u = u * as desired.This completes the proof of the claim.
Note that, from (3.1), we have almost everywhere in B, for some increasing function φ.So v is also spherically symmetric and radially decreasing; hence, 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 function u is obviously unique; in fact, Thus the proof of the theorem is completed.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.