Isoperimetric inequalities for the logarithmic potential operator

In this paper we prove that the disc is a maximiser of the Schatten $p$-norm of the logarithmic potential operator among all domains of a given measure in $\mathbb R^{2}$, for all even integers $2\leq p<\infty$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or P{\'o}lya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.


Introduction
Let Ω ⊂ R 2 be an open bounded set. We consider the logarithmic potential operator on L 2 (Ω) defined by where ln is the natural logarithm and |x − y| is the standard Euclidean distance between x and y. Clearly, L Ω is compact and self-adjoint. Therefore, all of its eigenvalues and characteristic numbers are discrete and real. We recall that the characteristic numbers are inverses of the eigenvalues. The characteristic numbers of L Ω may be enumerated in ascending order of their modulus, |µ 1 (Ω)| ≤ |µ 2 (Ω)| ≤ ...
where µ i (Ω) is repeated in this series according to its multiplicity. We denote the corresponding eigenfunctions by u 1 , u 2 , ..., so that for each characteristic number µ i there is a unique corresponding eigenfunction u i , It is known, see for example Mark Kac [Kac51] (see also [KS09]), that the equation where ∂ ∂ny denotes the outer normal derivative at a point y on the boundary ∂Ω, which is assumed piecewise C 1 here.
Spectral properties of the logarithmic potential have been considered in many papers (see [AKL92], [AK92], [BS77], [Kac70], [Tro67], [Tro69]). In this paper we are interested in isoperimetric inequalities of the logarithmic potential L Ω , that is also, in isoperimetric inequalities of the nonlocal Laplacian (1.2)-(1.3). For a recent general review of isoperimetric inequalities for the Dirichlet, Neumann and other Laplacians we refer to Benguria, Linde and Loewe in [BLL12]. Isoperimetric inequalities for Schatten norms for double layer potentials have been recently considered by Miyanishi and Suzuki [MS15].
In Rayleigh's famous book "Theory of Sound" (first published in 1877), by using some explicit computation and physical interpretations, he stated that the disc minimizes (among all domains of the same area) the first eigenvalue of the Dirichlet Laplacian. The proof of this conjecture was obtained about 50 years later, simultaneously (and independently) by G. Faber and E. Krahn. Nowadays, the Rayleigh-Faber-Krahn inequality has been established for many other operators; see e.g. [Hen06] for further references (see also [Ban80] and [PS51]). Among other things, in this paper we also prove the Rayleigh-Faber-Krahn theorem for the integral operator L Ω , i.e. it is proved that the disc is a minimizer of the first eigenvalue of the Laplacian (1.2)-(1.3) among all domains of a given measure in R 2 .
By using the Feynman-Kac formula and spherical rearrangement Luttinger [Lut73] proved that the disc D is a maximizer of the partition function of the Dirichlet Laplacian among all domains of the same area as D for all positive values of time, i.e.
where µ D i , i = 1, 2, . . . , are the eigenvalues of the Dirichlet Laplacian. From here by using the Mellin transform one obtains We prove an analogy of this Luttinger's inequality for the integral operator L Ω . In our note [RS15] we obtained similar results for convolution type integral operators with positive nonincreasing kernels. In the present setting the main difficulty arises from the fact that the logarithmic potential is not positive and that we can not use the Brascamp-Lieb-Luttinger type rearrangement inequalities directly.
In Section 2 we present the main result of this paper. Its proof will be given in Section 4. Among other things, we establish the Rayleigh-Faber-Krahn type inequality in Section 3. In Section 5 we discuss shortly about isoperimetric inequalities for polygons and show that the Schatten p-norm is maximized on the equilateral triangle centred at the origin among all triangles of a given area.

Main result and examples
Let H be a separable Hilbert space. By S ∞ (H) we denote the space of compact operators P : H → H. Recall that the singular numbers {s n } of P ∈ S ∞ (H) are the eigenvalues of the positive operator (P * P ) 1/2 (see [GK69]). The Schatten p-classes are defined as In S p (H) the Schatten p-norm of the operator P is defined by For p = ∞, we can set P ∞ := P to be the operator norm of P on H. As outlined in the introduction, we assume that Ω ⊂ R 2 is an open bounded set and we consider the logarithmic potential operator on L 2 (Ω) of the form It is known that L Ω is a Hilbert-Schmidt operator. By |Ω| we will denote the Lebesque measure of Ω.
Theorem 2.1. Let D be a disc centred at the origin. Then  The right hand sight of the formula (2.4) can be confirmed by a direct calculation of the logarithmic potential eigenvalues in the unit disc, see Theorem 3.1 in [AKL92].
We also obtain when p = ∞: Example 2.3. Let D ≡ U be the unit disc. Then by Theorem 2.1 we have for any bounded open domain Ω with |Ω| = |D|. Here · is the operator norm on the space L 2 .
From Corollary 3.2 in [AKL92] we calculate explicitly the operator norm in the right hand sight of (2.5).

On Rayleigh-Faber-Krahn theorem
In this section we prove the following analogy of the Rayleigh-Faber-Krahn theorem for the integral operator L Ω .
Theorem 3.1. The disc D is a minimizer of the characteristic number of the logarithmic potential L Ω with the smallest modulus among all domains of a given measure, i.e. 0 < |µ 1 (D)| ≤ |µ 1 (Ω)| for an arbitrary bounded open domain Ω ⊂ R 2 with |Ω| = |D|.
We will use this result later in the proof of Theorem 2.1.
Remark 3.2. In Landkof [Lan72] the positivity of the operator L Ω is proved in domains Ω ⊂ U, where U is the unit disc. In general, L Ω is not a positive operator.
For any bounded open domain Ω the logarithmic potential operator L Ω can have at most one negative eigenvalue, see Troutman [Tro67] (see also Kac [Kac70]). Lemma 3.4. The characteristic number µ 1 of the logarithmic potential L Ω with the smallest modulus is simple, and the corresponding eigenfunction u 1 can be chosen nonnegative.
Proof of Lemma 3.4. The eigenfunctions of the logarithmic potential L Ω may be chosen to be real as its kernel is real. First let us prove that u 1 cannot change sign in the domain Ω, that is, In fact, in the opposite case, by virtue of the continuity of the function u 1 (x), there would be neighborhoods U(x 0 , r) ⊂ Ω such that On the other hand we have From here by continuity it is simple to check that there exists ρ > 0 such that

Now let us introduce a new function
Then we obtain where µ 2 1 is the smallest characteristic number of L 2 Ω and u 1 is the eigenfunction corresponding to µ 2 1 , i.e. u 1 = µ 2 1 L 2 Ω u 1 . Therefore, by the variational principle we also have This means that the strong inequality (3.4) contradicts the variational principle (3.5) because Since u 1 is nonnegative it follows that µ 1 is a simple. Indeed, if there were an eigenfunction v 1 linearly independent of u 1 and corresponding to µ 1 , then for all real c the linear combination u 1 + cv 1 also would be an eigenfunction corresponding to µ 1 and therefore, by what has been proved, it could not become negative in Ω. As c is arbitrary, this is impossible.
Proof of Theorem 3.1. Let Ω be a bounded open set in R 2 . Its symmetric rearrangement Ω * ≡ D is an open disc centred at 0 with the measure equal to the measure of Ω, i.e. |D| = |Ω|. Let u be a nonnegative measurable function in Ω, so that all its positive level sets have finite measure, In the definition of the symmetric-decreasing rearrangement of u one can use the socalled layer-cake decomposition (see [LL01]), which expresses a nonnegative function u in terms of its level sets as where χ is the characteristic function of the corresponding domain.
Definition 3.5. Let u be a nonnegative measurable function in Ω. A function is called a symmetric-decreasing rearrangement of a nonnegative measurable function u.
As in the proof of Lemma 3.4 µ 2 1 (Ω) is the smallest characteristic number of L 2 Ω and u 1 is the eigenfunction corresponding to µ 2 1 , i.e. u 1 = µ 2 1 (Ω)L 2 Ω u 1 . By Lemma 3.4 the first characteristic number µ 1 of the operator L Ω is simple; the corresponding eigenfunction u 1 can be chosen positive in Ω. Recall 1 the rearrangement inequality for the logarithmic kernel (cf. Lemma 2 in [CL92]) In addition, for each nonnegative function u ∈ L 2 (Ω) we have (3.9) u L 2 (Ω) = u * L 2 (D) . Therefore, from (3.8), (3.9) and the variational principle for the positive operator L 2 D , we get Finally, note that 0 is not a characteristic number of L D (cf. Corollary 1 in [Tro67]). Therefore, 0 < |µ 1 (D)|. This completes the proof.

Proof of Theorem 2.1
First we prove the Brascamp-Lieb-Luttinger type rearrangement inequality for the logarithmic kernel (cf. [BLL74]).   1 For the proof of the rearrangement inequality for the logarithmic kernel (3.8) see Lemma 5.3. The proof is the same with the difference that is in this case the symmetric decreasing rearrangement is used instead of the Steiner symmetrization.
Proof of Lemma 4.1. Here we prove it for p = 2 and the proof is based on the proof of Lemma 2 in [CL92]. The proof for arbitrary p is essentially the same as the case p = 2. Let us fix r 0 > 0 and consider the function 1+s 2 ds, r > r 0 . Let us see that the function f (r) is strictly decreasing and has a limit as r → ∞. If r ≤ r 0 then for r 1 < r 2 . If r > r 0 then Thus f (r 1 ) > f (r 2 ) for r 1 < r 2 , that is, f (r) is strictly decreasing. From (4.3) it is easy to see that We use the notation Ω Ω h 2 (|y 1 − y 2 |)h 2 (|y 2 − y 1 |)dy 1 dy 2 ≤ D D h 2 (|y 1 − y 2 |)h 2 (|y 2 − y 1 |)dy 1 dy 2 .
Since h 1 and q R are positive and nonincreasing by the Brascamp-Lieb-Luttinger rearrangement inequality. Noting that for any bounded open domain Ω ⊂ R 2 with |Ω| = |D|.

As a consequence of Lemma 4.2 we obtain:
Lemma 4.3. We have for any even integer p ≥ 2.
As a consequence of Lemma 4.3 we have the following inequality: Lemma 4.4.
for any bounded open domain Ω with |Ω| = |D| and any even integer p ≥ 2.
Proof of Lemma 4.4. Let L be the bilateral Laplace transform Using the inverse bilateral Laplace transform (frequency shift of unit step) we have We obtain (4.18) by acting on both sides of (4.17) by L −1 .
Proof of Theorem 2.1. For p = ∞, see Remark 3.3, so that we can assume 2 ≤ p < ∞.
for any real l > 1 and any even integer p ≥ 2. Since l is an arbitrary real number > 1, from (4.19) we obtain for any real p > 2 and arbitrary bounded open set Ω with |Ω| = |D|. In addition, by Lemma 4.2 we confirm that the inequality is also true when p = 2. This completes the proof of Theorem 2.1.
Remark 4.5. It follows from the properties of the kernel that the Schatten p-norm of the operator L Ω is finite when p > 1 see e.g. the criteria for Schatten classes in terms of the regularity of the kernel in [DR14]. The above techniques do not allow us to prove Theorem 2.1 for 1 < p < 2. In view of the Dirichlet Laplacian case, it seems reasonable to conjecture that the Schatten p-norm is still maximized on the disc also for 1 < p < 2.

On the case of polygons
We can ask the same question of maximizing the Schatten p-norms in the class of polygons with a given number n of sides. We denote by P n the class of plane polygons with n edges. We would like to identify the maximizer for Schatten p-norms of the logarithmic potential L Ω in P n . According to the Section 2, it is natural to conjecture that it is the n-regular polygon. Currently, we can prove this only for n = 3: Theorem 5.1. The equilateral triangle centred at the origin has the largest Schatten p-norm of the operator L Ω for any 2 ≤ p ≤ ∞ among all triangles of a given area. More precisely, if ∆ is the equilateral triangle centred at the origin, we have Let u be a nonnegative, measurable function on R 2 , and let l be a line through the origin of R 2 . Choose an orthogonal coordinate system in R 2 such that the x 1 -axis is perpendicular to l.
Definition 5.2 ([BLL74]). A nonnegative, measurable function u ⋆ (x|l) on R 2 is called a Steiner symmetrization with respect to l of the function u(x), if u ⋆ (x 1 , x 2 ) is a symmetric decreasing rearrangement with respect to x 1 of u(x 1 , x 2 ) for each fixed x 2 .
The Steiner symmetrization (with respect to the x 1 -axis) Ω ⋆ of a measurable set Ω is defined in the following way: if we write (x 1 , z) with z ∈ R, and let Ω z = {x 1 |(x 1 , z) ∈ Ω}, then where Ω * is a symmetric rearrangement of Ω (see the proof of Theorem (3.1)). We obtain: Lemma 5.3. For a positive function u and a measurable Ω ⊂ R 2 we have where Ω ⋆ and u ⋆ are Steiner symmetrizations of Ω and u, respectively.
Proof of Lemma 5.3. The proof is based on the proof of Lemma 2 in [CL92]. Let us fix r 0 > 0 and consider the function Since f (r) is strictly decreasing 1 2π ln 1 where h 1 is positive strictly decreasing function and h 2 is decreasing. Hence by the Brascamp-Lieb-Luttinger rearrangement inequality for the Steiner symmetrization (see Lemma 3.2 in [BLL74]) we have Thus it remains to show that (5.5) which does not follow directly from the Brascamp-Lieb-Luttinger rearrangement inequality since h 2 is not positive. Define for R > 0 and note that by monotone convergence we have with the notations (5.9) I Ω (u, g) = Since q R is positive and nonincreasing and noting that by the Brascamp-Lieb-Luttinger rearrangement inequality. Therefore, This proves the inequality (5.5). Similarly, now let us show that the inequality (5.6) is valid. We have Since both q R and h 1 are positive and nonincreasing by the Brascamp-Lieb-Luttinger rearrangement inequality. Therefore, we obtain This proves the inequality (5.6).
Lemma 5.3 implies the following analogy of the Pólya theorem [Pól55] for the operator L Ω .
Remark 5.5. In other words Theorem 5.4 says that the operator norm of L Ω is maximized in an equilateral triangle among all triangles of a given area.
Proof of Theorem 5.4. By Theorem 3.1 and Lemma 3.4 the first characteristic number µ 1 of the operator L Ω is positive and simple; the corresponding eigenfunction u 1 can be chosen positive in Ω. Using the fact that by applying a sequence of the Steiner symmetrizations with respect to the mediator of each side, a given triangle converges to an equilateral one (see Figure 3.2. in [Hen06]), from (5.2) we obtain (5.17) Ω Ω Ω u 1 (y) 1 2π ln 1 |y − z| Therefore, from (5.17) and the variational principle for the positive operator L 2 ∆ , we get µ 2 1 (Ω) = Ω |u 1 (x)| 2 dx Ω Ω Ω u 1 (y) 1 2π ln 1 |y−z| Here we have used the fact that the Steiner symmetrization preserves the L 2 -norm.
Proof of Theorem 5.1. The proof of Theorem 5.1 relies on the same technique as the proof of Theorem 2.1 with the difference that now the Steiner symmetrization is used. Since the Steiner symmetrization has the same property (4.14) (cf. Lemma 3.2 in [BLL74]) as the symmetric-decreasing rearrangement, it is clear that any Steiner symmetrization increases (or at least does not decrease) the Schatten p-norms (cf. Lemma 4.2) for integer p ≥ 2. Thus, for the proof we only need to recall the fact that a sequence of Steiner symmetrizations with respect to the mediator of each side, a given triangle converges to an equilateral one. The rest of the proof is exactly the same as the proof of Theorem 2.1.
Remark 5.6. A sequence of three Steiner symmetrizations allows us to transform any quadrilateral into a rectangle (see Figure 3.3. in [Hen06]). Therefore, it suffices to look at the maximization problem among rectangles for P 4 . Unfortunately, for P 5 (pentagons and others), the Steiner symmetrization increases, in general, the number of sides. This prevents us from using the same technique for general polygons.