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DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SUPPLEMENT 2007

Website: www.AIMSciences.org pp. 477–486

COMPARISONS OF EIGENVALUES OF SECOND ORDER ELLIPTIC OPERATORS F. HAMEL, N. NADIRASHVILI AND E. RUSS

Franc ¸ ois Hamel and Emmanuel Russ Universit´ e Aix-Marseille III, LATP Facult´ e des Sciences et Techniques, Case cour A Avenue Escadrille Normandie-Niemen F-13397 Marseille Cedex 20, France

Nikolai Nadirashvili CNRS, LATP CMI 39 rue F. Joliot-Curie F-13453 Marseille Cedex 13, France

Abstract. To any second order elliptic operator L = −div(A∇) + v · ∇ + V in a bounded C 2 domain Ω with Dirichlet boundary condition, we associate a second order elliptic operator L∗ in divergence form in the Euclidean ball Ω∗ centered at 0 and having the same Lebesgue measure as Ω. In Ω, the symmetric matrix field A is in W 1,∞ (Ω), the vector field v is in L∞ (Ω, Rn ) and V is a continuous function in Ω. In Ω∗ , the coefficients of L∗ are radial, they preserve some quantities associated to the coefficients of L, and we can construct the operator L∗ in such a way that its principal eigenvalue is not too much larger than that of L. In particular, we generalize the RayleighFaber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization and interesting by itself.

1. Introduction. Let n ≥ 1. If A ⊂ Rn is measurable, |A| stands for the Lebesgue measure of A. For all x ∈ Rn , denote by |x| the Euclidean norm of x and, for all x ∈ Rn \ {0}, set x er (x) = . |x| The word “domain” means a nonempty open connected subset of Rn . If Ω ⊂ Rn is a domain, denote by Ω∗ the Euclidean ball centered at 0 such that |Ω∗ | = |Ω|. Define C as the class of all C 2 bounded domains of Rn . If 1 ≤ p ≤ +∞, Ω ⊂ Rn is a domain of Rn and v : Ω → Rn is measurable, say that v ∈ Lp (Ω, Rn ) if |v| ∈ Lp (Ω), 2000 Mathematics Subject Classification. Primary: 35P15; Secondary: 35J25. Key words and phrases. Principal eigenvalue, second order elliptic operators, rearrangement inequalities. This paper corresponds to a conference given by the third author during the AIMS sixth international conference on dynamical systems, differential equations and applications, in the special session SS13, “Shapes and free boundaries” organized by M. Pierre and P. Cardaliaguet.

477

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F. HAMEL, N. NADIRASHVILI AND E. RUSS

and write kvkp = kvkLp (Ω,Rn ) := k|v|kLp (Ω) . Finally, if u : Ω → R is measurable, define, for all λ ∈ R, µu (λ) = |{x ∈ Ω; u(x) > λ}| . Let Ω ⊂ Rn be a (say) C 1 bounded domain and λ1 (Ω) the principal eigenvalue of −∆ on Ω under Dirichlet boundary condition. Recall that λ1 (Ω) > 0. Let us first ask the following question: Question 1. Let m > 0. Among all the smooth bounded domains Ω ⊂ Rn with |Ω| = m, for which Ω does λ1 (Ω) reach its infimum ? A celebrated conjecture due to Lord Rayleigh states that, when n = 2, the disk should be the (unique) domain which minimizes λ1 (Ω) over all the domains Ω with given area ([13]). It was proved independently by Faber ([2]) and Krahn ([9]) that this conjecture holds true in the plane, and this result was generalized later by Krahn in any dimension ([10, 11]). The result can be formulated as follows: if Ω ⊂ Rn is a C 1 bounded domain, then one has λ1 (Ω) ≥ λ1 (Ω∗ ),

(1) ∗

and the equality holds if and only if, up to translation, Ω = Ω (note that the inequality λ1 (Ω) ≥ λ1 (Ω∗ ) actually holds for any domain Ω ⊂ Rn , possibly nonsmooth, but some smoothness assumption on Ω is required to guarantee the equality case). Since λ1 (Ω∗ ) can be computed somewhat explicitly, we get the following lower bound for λ1 (Ω): 2 λ1 (Ω) ≥ |Ω|−2/n αn2/n jn/2−1,1 , (2) where αn is the Lebesgue measure of the Euclidean unit ball in Rn and jm,1 the first positive zero of the Bessel function Jm . Note that the right-hand side of (2) only depends on n and |Ω|. Let us briefly recall what the ingredients of the proof of (1) are. The first one is a variational formula for λ1 (Ω): Z 2 |∇ϕ(x)| dx Ω Z . (3) λ1 (Ω) = min ϕ∈H01 (Ω)\{0} 2 |ϕ(x)| dx Ω

Note that such a formula is available because −∆ is symmetric on L2 (Ω). The second tool in the proof of (1) is the Schwarz symmetrization. If u : Ω → [0, +∞) is measurable, denote by u∗ : Ω∗ → [0, +∞) the radially nonincreasing function such that, for all λ ≥ 0, |{x ∈ Ω; u(x) > λ}| = |{x ∈ Ω∗ ; u∗ (x) > λ}| . The basic properties of this symmetrization are well-known [12]: if u ∈ H01 (Ω), then u∗ ∈ H01 (Ω∗ ) and ku∗ kL2 (Ω∗ ) = kukL2 (Ω) ,

k∇u∗ kL2 (Ω∗ ) ≤ k∇ukL2 (Ω) .

(4)

Then, (1) follows easily from (3) and (4). Many optimization results for the eigenvalues of −∆ under Dirichlet boundary condition (or other boundary conditions) and other references can be found in [7, 8]. It is natural to seek for generalizations of (1) when −∆ is replaced by a more general second order elliptic operator, possibly non-symmetric. Let us first focus on a rather simple (but non-symmetric) situation.

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2. The case of the Laplacian with a drift term. The results of this section are taken from [4, 5]. Let Ω ⊂ Rn be a C 2 nonempty bounded domain (strictly speaking, it is assumed in [4] and [5] that Ω is C 2,α for some 0 < α < 1, but it follows from [6] that taking Ω of class C 2 is enough) and v ∈ L∞ (Ω, Rn ). Denote by λ1 (Ω, v) the principal eigenvalue of −∆ + v · ∇ on Ω under Dirichlet boundary condition. By the maximum principle, λ1 (Ω, v) > 0 (see [1]). The corresponding eigenspace has dimension 1, and we will denote by ϕΩ,v the unique solution of the problem   −∆ϕ + v · ∇ϕ = λ1 (Ω, v)ϕ in Ω, (5)  ϕ > 0 in Ω, ϕ = 0 on ∂Ω, kϕk = 1. L∞ (Ω) By standard elliptic estimates, ϕΩ,v ∈ W 2,p (Ω) for all 1 ≤ p < +∞ and ϕΩ,v ∈ C 1,α (Ω) for all 0 ≤ α < 1. Various optimization results for λ1 (Ω, v) can be given. Fix τ ≥ 0 and consider first the situation when the domain Ω is fixed and the vector field v varies and satisfies the constraint kvk∞ = kvkL∞ (Ω,Rn ) ≤ τ . It turns out that both the corresponding minimization and the maximization problems for λ1 (Ω, v) have a unique solution which can be described as follows. Set λ(Ω, τ ) =

inf

kvk∞ ≤τ

λ1 (Ω, v) and λ(Ω, τ ) =

sup λ1 (Ω, v). kvk∞ ≤τ

Here is the solution of these optimization problems: Theorem 1. Let Ω be a domain in C and τ ≥ 0 be fixed. (a) There exists a unique vector field v ∈ L∞ (Ω) with kvk∞ ≤ τ such that λ1 (Ω, v) = λ(Ω, τ ) (> 0), and this field satisfies |v(x)| = τ almost everywhere in Ω. Moreover, the corresponding principal eigenfunction ϕ = ϕΩ,v is such that v · ∇ϕ = −τ ∇ϕ almost everywhere in Ω. The function ϕ is then a solution of the following nonlinear problem − ∆ϕ − τ ∇ϕ = λ(Ω, τ )ϕ in Ω. (6) Moreover, if ψ is a function of class W 2,p (Ω) (for all 1 ≤ p < +∞) such that ψ > 0 in Ω, ψ = 0 on ∂Ω, kψk∞ = 1 and if µ ∈ R is such that (6) holds with ψ and µ instead of ϕ and λ(Ω, τ ), then ψ = ϕ and µ = λ(Ω, τ ). (b) There exists a unique vector field v ∈ L∞ (Ω) with kvk∞ ≤ τ such that λ1 (Ω, v) = λ(Ω, τ ) (> 0), and this field satisfies |v(x)| = τ almost everywhere in Ω. Moreover, the corresponding principal eigenfunction ϕ = ϕΩ,v is such that v · ∇ϕ = τ |∇ϕ| almost everywhere in Ω. The function ϕ is then a solution of the following nonlinear problem − ∆ϕ + τ |∇ϕ| = λ(Ω, τ )ϕ in Ω.

(7)

Moreover, if ψ is a function of class W 2,p (Ω) (for all 1 ≤ p < +∞) such that ψ > 0 in Ω, ψ = 0 on ∂Ω, kψk∞ = 1 and if µ ∈ R is such that (7) holds with ψ and µ instead of ϕ and λ(Ω, τ ), then ψ = ϕ and µ = λ(Ω, τ ). When Ω is a ball (up to translation, one assumes that it is centered at the origin), one can provide an explicit expression of v and v: n Theorem 2. Assume that Ω = B = BR is the open Euclidean ball of center 0 and radius R > 0, and let τ ≥ 0 be fixed. Then v = τ er and v = −τ er where v and v

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are defined in Theorem 1. One therefore has:   −∆ϕ + τ er · ∇ϕ = λ(Ω, τ )ϕ 

−∆ϕ − τ er · ∇ϕ

= λ(Ω, τ )ϕ

in B, (8) in B.

Moreover, the functions ϕ and ϕ are radially decreasing in B, which means that there are two decreasing functions φ, φ : [0, R] → [0, +∞) such that ϕ(x) = φ(|x|) and ϕ(x) = φ(|x|) for all x ∈ B. In view of these results and of (1), it is natural to focus now on the following optimization problem: Question 2. Let m > 0 and τ ≥ 0 be fixed. Among all the domains Ω ∈ C with |Ω| = m and all the vector fields v ∈ L∞ (Ω, Rn ) such that kvkL∞ (Ω,Rn ) ≤ τ , for which (Ω, v) does λ1 (Ω, v) reach its infimum ? Let us first give a fairly heuristic approach of this issue. Consider the evolution problem  ∂t u = ∆u − v · ∇u in Ω, (9) u(0) = f in Ω, under Dirichlet boundary condition. Then, λ1 (Ω, v) is minimal if and only if the exponential time-decay of the solution u(t) of (9) is as slow as possible. This reasonably requires that Ω should be a ball (to minimize boundary effects) and that −v should point towards the center, in such a way that |v(x)| should be maximal for almost every x ∈ Ω, i.e. |v(x)| = τ for almost every x ∈ Ω. One can definitely prove the following result: Theorem 3. Let Ω ⊂ Rn be a C 2 nonempty bounded domain and v ∈ L∞ (Ω, Rn ) with kvkL∞ (Ω,Rn ) ≤ τ . Then λ1 (Ω, v) ≥ λ1 (Ω∗ , τ er ), and the equality holds if and only if, up to translation, Ω = Ω∗ and v = τ er . Here is a version of (2) in this context. For all n ≥ 1, all m > 0 and all τ ≥ 0, define Fn (m, τ ) = λ1 (Ω∗ , τ er ), where Ω∗ is the Euclidean ball in Rn (centered at 0) with |Ω| = m. Then: Corollary 1. (a) For any n ≥ 1, any C 2 non empty bounded domain Ω ⊂ Rn and any v ∈ L∞ (Ω, Rn ), λ1 (Ω, v) ≥ Fn (|Ω|, kvk∞ )

(10)

and the equality holds if and only if, up to translation, Ω = Ω∗ and v = kvk∞ er . (b) F1 (m, τ ) ∼ τ 2 e−τ m/2 when τ → +∞, (c) log Fn (m, τ ) ∼ −τ (m/αn )1/n when τ → +∞. Since (10) provides a lower bound of λ1 (Ω, v) only involving |Ω| and kvkL∞ (Ω,Rn ) , we call (10) a Faber-Krahn inequality. Note also that, except the fact that er is not C 1 at 0, the estimates in (b) and (c) of Corollary 1 are consequences of a more general result proved by Friedman by probabilistic arguments in [3]. Since the operator −∆ + v · ∇ is not symmetric on L2 (Ω), there is no variational formula for λ1 (Ω, v) in the spirit of (3) and the proof of Theorem 3 does not rely

EIGENVALUES OF SECOND ORDER ELLIPTIC OPERATORS

481

on the Schwarz symmetrization but on a different rearrangement technique. Before going further into this, let us ask the following question: more generally, can analogous results be given for elliptic operators of the form −div(A∇) + v · ∇ + V ? 3. The case of general second order elliptic operators in divergence form. In the sequel, we deal with operators of the form L = −div(A∇) + v · ∇ + V on a domain Ω ∈ C under Dirichlet boundary condition. We always assume in the sequel that A ∈ W 1,∞ (Ω, Sn (R)), so that, up to a modification on a negligible set, A can be assumed to be continuous in Ω. Assume also that there exists γ > 0 such that, for all x ∈ Ω and all ξ ∈ Rn , 2

A(x)ξ · ξ ≥ γ |ξ| , that v ∈ L∞ (Ω, Rn ) and V ∈ C(Ω). Denote by λ1 (Ω, A, v, V ) the principal eigenvalue of L = −div(A∇)+v·∇+V on Ω under Dirichlet boundary condition. Let us recall basic properties of λ1 (Ω, A, v, V ) (see [1]): – if λ 6= λ1 (Ω, A, v, V ) is an eigenvalue of L, then Re λ > λ1 (Ω, A, v, V ), – the eigenspace of L associated with λ1 (Ω, A, v, V ) has dimension 1 and, if ϕ is “the” corresponding eigenfunction, one has ±ϕ > 0 in Ω and ϕ = 0 on ∂Ω, – if ψ is any eigenfunction for L associated with the eigenvalue λ and if ψ > 0 in Ω and ψ = 0 on ∂Ω, then λ = λ1 (Ω, A, v, V ). Let Λ ∈ L∞ (Ω). Say that Λ ∈ L∞ + (Ω) if there exists γ > 0 such that Λ(x) ≥ γ for almost every x ∈ Ω. If A ∈ W 1,∞ (Ω, Sn (R)) is uniformly elliptic and Λ ∈ L∞ + (Ω), say that A ≥ Λ Id in Ω if, for almost every x ∈ Ω and all ξ ∈ Rn , 2

A(x)ξ · ξ ≥ Λ(x) |ξ| .

(11)

The symbol Id denotes the identity matrix field. Note that (11) is satisfied for instance when Λ(x) is the smallest eigenvalue of A(x) for all x ∈ Ω. The spirit of our results, which are detailed in [6], is the following one: if L = −div(A∇) + v · ∇ + V on Ω under Dirichlet boundary condition, we associate to L an operator L∗ = −div(A∗ ∇) + v ∗ · ∇ + V ∗ on Ω∗ under Dirichlet boundary condition, with radial coefficients, preserving some quantities depending on the coefficients (averages, uniform bounds, distribution functions, determinant and other symmetric function of the eigenvalues of the diffusion), in such a way that the principal eigenvalue of L∗ is not too much larger as the principal eigenvalue of L. 3.1. A Faber-Krahn inequality. Here is first a version of Theorem 3 in this framework: Theorem 4. Let Ω ∈ C, M A > 0, mΛ > 0, τ1 ≥ 0 and τ2 ≥ 0 be given. Assume ∞ n that Ω is not a ball. Consider A ∈ W 1,∞ (Ω, Sn (R)), Λ ∈ L∞ + (Ω), v ∈ L (Ω, R ) ∞ and V ∈ L (Ω) satisfying ( A ≥ Λ Id a.e. in Ω, kAkW 1,∞ (Ω,Sn (R)) ≤ M A , ess inf Λ ≥ mΛ , Ω

kvkL∞ (Ω,Rn ) ≤ τ1 and kV kL∞ (Ω) ≤ τ2 . Then there exists a positive constant η = η(Ω, n, M A , mΛ , τ1 ) > 0 depending only on Ω, n, M A , mΛ and τ1 , and there exists a radially symmetric C ∞ (Ω∗ ) field Λ∗ > 0

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F. HAMEL, N. NADIRASHVILI AND E. RUSS

such that

(

ess inf Λ ≤ Λ∗ (x) ≤ ess sup Λ for all x ∈ Ω∗ , Ω

Ω

k(Λ∗ )−1 kL1 (Ω∗ ) = kΛ−1 kL1 (Ω) ,

(12)

and λ1 (Ω∗ , Λ∗ Id, τ1 er , −τ2 ) ≤ λ1 (Ω, A, v, V ) − η.

(13)

Note that, when Λ is equal to a constant γ > 0 on Ω, then Λ∗ (x) = γ for all x ∈ Ω∗ , so that, as it can be seen easily, we recover the conclusion of Theorem 3 when Ω is not a ball. 3.2. Operators whose coefficients have given averages or distribution functions. In the following two theorems, some integral quantities depending on Λ and v are preserved by symmetrization, as well as the distribution function of V − . ∞ n Theorem 5. Let Ω ∈ C, A ∈ W 1,∞ (Ω, Sn (R)), Λ ∈ L∞ + (Ω), v ∈ L (Ω, R ) and V ∈ C(Ω) be given. Assume that A ≥ Λ Id a.e. in Ω, and that λ1 (Ω, A, v, V ) ≥ 0. Then, for all ε > 0, there exist three radially symmetric C ∞ (Ω∗ ) fields Λ∗ > 0, ∗ ω ∗ ≥ 0 and V ≤ 0 such that, for v ∗ = ω ∗ er in Ω∗ \{0},  ∗ ∗   essΩinf Λ ≤ Λ (x) ≤ essΩsup Λ for all x ∈ Ω ,   k(Λ∗ )−1 kL1 (Ω∗ ) = kΛ−1 kL1 (Ω) , (14)  kv ∗ kL∞ (Ω∗ ,Rn ) ≤ kvkL∞ (Ω,Rn ) , k |v ∗ |2 (Λ∗ )−1 kL1 (Ω∗ ) = k |v|2 Λ−1 kL1 (Ω) ,    µ ∗ ≤ µ −, V (V )−

and

∗

λ1 (Ω∗ , Λ∗ Id, v ∗ , V ) ≤ λ1 (Ω, A, v, V ) + ε. (15) There also exists a nonpositive radially symmetric field V ∗ ∈ L∞ (Ω∗ ) such that ∗ ∗ µV ∗ = µ−V − , V ∗ ≤ V ≤ 0 in Ω∗ and λ1 (Ω∗ , Λ∗ Id, v ∗ , V ∗ ) ≤ λ1 (Ω∗ , Λ∗ Id, v ∗ , V ) ≤ λ1 (Ω, A, v, V ) + ε. If the function Λ is equal to a constant γ > 0 in Ω, then there exist two radially symmetric bounded functions ω0∗ ≥ 0 and V0∗ ≤ 0 in Ω∗ such that, for v0∗ = ω0∗ er ,  ∗ kvkL∞ (Ω,Rn ) , kv0∗ kL2 (Ω∗ ,Rn ) ≤ kvkL2 (Ω,Rn ) ,   kv0 kL∞ (Ω−∗ ,Rn ) ≤ ∗ −max V ≤ V0 ≤ 0 a.e. in Ω∗ , (16) Ω   ∗ − kV0 kLp (Ω∗ ) ≤ kV kLp (Ω) for all 1 ≤ p ≤ +∞, and λ1 (Ω∗ , γId, v0∗ , V0∗ ) ≤ λ1 (Ω, A, v, V ).

(17)

When Ω is not a ball, the previous inequalities are strict, and more precisely: Theorem 6. Under the notation of Theorem 5, assume that Ω ∈ C is not a ball and let M A > 0, mΛ > 0, M v ≥ 0 and M V ≥ 0 be such that kAkW 1,∞ (Ω,Sn (R)) ≤ M A , ess inf Λ ≥ mΛ , kvk∞ ≤ M v and kV k∞ ≤ M V . Ω

(18)

Then there exists a positive constant θ = θ(Ω, n, M A , mΛ , M v , M V ) > 0, such that if λ1 (Ω, A, v, V ) > 0, then there exist three radially symmetric C ∞ (Ω∗ ) fields Λ∗ > 0, ∗ ω ∗ ≥ 0, V ≤ 0 and a nonpositive radially symmetric L∞ (Ω∗ ) field V ∗ , which ∗ satisfy (14), µV ∗ = µ−V − , V ∗ ≤ V ≤ 0 and are such that, for v ∗ = ω ∗ er , ∗

λ1 (Ω∗ , Λ∗ Id, v ∗ , V ∗ ) ≤ λ1 (Ω∗ , Λ∗ Id, v ∗ , V ) ≤

λ1 (Ω, A, v, V ) . 1+θ

EIGENVALUES OF SECOND ORDER ELLIPTIC OPERATORS

483

3.3. Operators with constraints on the trace and the determinant of the diffusion. We prove a similar result with constraints on the trace and the determinant of A: Theorem 7. Assume n ≥ 2. Let A ∈ W 1,∞ (Ω, Sn (R)), v ∈ L∞ (Ω, Rn ), V ∈ C(Ω), ω > 0, σ > 0. Assume that λ1 (Ω, A, v, V ) ≥ 0 and det(A(x)) ≥ ω, tr(A(x)) ≤ σ for all x ∈ Ω. Then, there are two positive numbers 0 < a1 ≤ a2 which only depend on n, ω and σ, such that, for all ε > 0, there exist a matrix field A∗ ∈ C ∞ (Ω∗ \{0}, Sn (R)), a radially symmetric C ∞ (Ω∗ ) field ω ∗ ≥ 0 and a radially symmetric L∞ (Ω∗ ) field V ∗ ≤ 0, such that, for v ∗ = ω ∗ er in Ω∗ \{0},   A ≥ a1 Id in Ω, A∗ ≥ a1 Id in Ω∗ , (19) det(A∗ (x)) = ω, tr(A∗ (x)) = σ for all x ∈ Ω∗ \{0},  kv ∗ kL∞ (Ω∗ ,Rn ) ≤ kvkL∞ (Ω,Rn ) , kv ∗ kL2 (Ω∗ ) = kvkL2 (Ω) , µV ∗ = µ−V − , and λ1 (Ω∗ , A∗ , v ∗ , V ∗ ) ≤ λ1 (Ω, A, v, V ) + ε. Furthermore, the matrix field A∗ is given explicitely, for all x ∈ Ω∗ \{0}, by: A∗ (x)x · x = a1 |x|2 and A∗ (x)y · y = a2 |y|2 for all y ⊥ x. 3.4. Some comments on these results. As stated in Theorems 4, 5 and 6, if there exists γ > 0 such that Λ(x) = γ for almost every x ∈ Ω, then one can choose Λ∗ (x) = γ for all x ∈ Ω∗ . When Λ is constant almost everywhere in Ω, inequality (13) in Theorem 4, without η, can be derived implicitly from a result due to Talenti (Theorem 1 in [14]), with some extra arguments involving other results (see [6]). But a very important feature of Theorems 4, 5 and 6 is the fact that Λ may not be constant, and the function Λ∗ is not constant in general, which means that the principal part div(Λ∗ ∇) is not equal to a constant times the Laplacian. When Λ is not constant, the conclusion of Theorem 4 does not follow from Talenti’s result. Strict inequalities when Ω is not a ball are also completely new. In the same way, Theorems 5 and 6 are new, require brand new techniques and do not follow implicitly from previous results, even when the operator is symmetric and even if both the first and zeroth order terms of the operator are zero. Actually, these results are also new even in dimension n = 1. Moreover, optimization problems with constraints on the determinant and another symmetric function of the eigenvalues of the diffusion do not seem to have been considered hitherto (see Theorem 7). 4. Rearrangement inequalities and outline of the proofs. The proofs of Theorems 4, 5, 6 and 7 rely on a new rearrangement technique, different from the Schwarz symmetrization, and which we now briefly describe. Given Ω, A, Λ, v and V as in Section 3, let ϕ be the eigenfunction corresponding to λ := λ1 (Ω, A, v, V ) such that kϕk∞ = 1. One has  −div(A∇ϕ) + v · ∇ϕ + V ϕ = λϕ in Ω,    ϕ > 0 in Ω, ϕ = 0 on ∂Ω,    maxΩ ϕ = 1. Moreover, we recall that ϕ ∈ W 2,p (Ω) for all 1 ≤ p < +∞ and ϕ ∈ C 1,α (Ω) for all 0 ≤ α < 1.

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Define f := λϕ − v · ∇ϕ − V ϕ, so that −div(A∇ϕ) = f. Assume that λ ≥ 0, that A and Λ are of class C 1 in Ω, that v is continuous in Ω and that ϕ is of class C 2 (Ω). Assume furthermore that the set Z of critical values of ϕ in Ω is finite, and let Y = [0, 1]\Z be the set of non-critical values of ϕ. For all a ∈ [0, 1), define Ωa = {x ∈ Ω, ϕ(x) > a} and, for all a ∈ [0, 1],  Σa = x ∈ Ω, ϕ(x) = a . Assume that, for all 0 ≤ a ≤ 1, |Σa | = 0.

(20)

∗

Let R be the radius of Ω . For all a ∈ [0, 1), define ρ(a) ∈ (0, R] such that |Ωa | = |Bρ(a) | = αn ρ(a)n , Define also ρ(1) = 0. It follows easily from (20) that Lemma 1. The function ρ : [0, 1] → [0, R] is decreasing, one-to-one and onto. Let E = {x ∈ Ω∗ , |x| ∈ ρ(Y )}. For all r ∈ ρ(Y ), set Z

−1

|∇ϕ(y)| G(r) = Z

dσρ−1 (r) (y)

Σρ−1 (r)

> 0, −1

Λ(y)−1 |∇ϕ(y)|

(21)

dσρ−1 (r) (y)

Σρ−1 (r)

where dσρ−1 (r) denotes the surface measure on Σρ−1 (r) . For all x ∈ E, let b Λ(x) = G(|x|).

(22)

Note that, by the co-area formula, Z Z b −1 dx = Λ(x) Λ(x)−1 dx. Ω∗

Ω

Define ϕ e ∈ C(Ω∗ ) ∩ H01 (Ω∗ ) ∩ W 1,∞ (Ω∗ ) as the radially decreasing function, positive in Ω∗ such that Z Z b ϕ(x))dx div(A∇ϕ)(x)dx = div(Λ∇ e Ωa

Bρ(a)

for all 0 ≤ a < 1. The function ϕ e is C in E ∪ {0} and C 2 in E ∩ Ω∗ . We also define a rearranged vector field vb and a rearranged potential Vb . For all x ∈ E with |x| = r, let vb(x) = ω b (r)er (x) 1

EIGENVALUES OF SECOND ORDER ELLIPTIC OPERATORS

485

with ω b (r) ≥ 0 and Z ω b (r)2 =

2

|v(y)| Λ(y)−1 |∇ϕ(y)|−1 dσρ−1 (r) (y)

Σρ−1 (r)

.

Z

−1

Λ(y)−1 |∇ϕ(y)|

(23)

dσρ−1 (r) (y)

Σρ−1 (r)

By the co-area formula again, Z Z 2 b 2 −1 |b v (x)| Λ(x) dx = |v(x)| Λ(x)−1 dx. Ω∗

Ω

Finally, for all x ∈ E with |x| = r, define Z −1 − V − (y) |∇ϕ(y)| dσρ−1 (r) (y) Σρ−1 (r) Z . Vb (x) = −1

|∇ϕ(y)|

(24)

dσρ−1 (r) (y)

Σρ−1 (r)

The distribution functions of V − and Vb are not equal, but it follows from the co-area formula that, for all 0 ≤ a < b ≤ 1, Z Z − V − (x)dx = Vb (y)dy. x∈Ω, a≤ϕ(x)≤b

y∈Ω∗ , ρ(b)≤|y|≤ρ(a)

It turns out that we can compare ϕ and ϕ e on the sets Σa and ∂Bρ(a) : Theorem 8. Let x ∈ Ω∗ and a ∈ [0, 1] such that |x| = ρ(a). Then, for all y ∈ Σa (i.e. ϕ(y) = a), ϕ(x) e ≥ ϕ(y) = a. This essential rearrangement inequality, which has not been known before, relies, in particular, on the classical isoperimetric inequality in Rn . b vb and Vb : There is also a partial differential inequality on ϕ, ϕ, e Λ, Theorem 9. Let x ∈ E ∩ Ω∗ . There exists y ∈ Ω such that ϕ(y) = ρ−1 (|x|), and b ϕ)(x)+b −div(Λ∇ e v (x)·∇ϕ(x)+ e Vb (x)ϕ(x) e ≤ −div(A∇ϕ)(y)+v(y)·∇ϕ(y)+V (y)ϕ(y). Note that vb(x) · ∇ϕ(x) e = −|b v (x)| × |∇ϕ(x)|. e As a consequence, for all x ∈ E, b ϕ)(x) − div(Λ∇ e + vb(x) · ∇ϕ(x) e + Vb (x)ϕ(x) e ≤ λϕ(y) = λρ−1 (|x|) ≤ λϕ(x) e

(25) 1 b since λ is assumed to be nonnegative. If Λ and ϕ e were of class C (Ω), it would then follow from (25), from the strong maximum principle and from Hopf lemma that b Id, vb, Vb ) ≤ λ, λ1 (Ω∗ , Λ which is close to the required conclusion in Theorem 5. Of course, in general, the set of critical values of ϕ in Ω is not finite (the finiteness holds for instance if ϕ is real analytic in Ω) and the coefficients A, Λ, v, and the function ϕ, are not smooth enough. We argue by smooth approximations of the coefficients of the operator, which involves many technicalities. When Ω is not a ball, an improved version of Theorem 8 can be given and replaces Theorem 8 in the proof of Theorem 6. We refer to [6] for complete proofs of the results stated in the present survey and further statements and comments.

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Received September 2006; revised June 2007. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]