Generalized logarithmic Hardy-Littlewood-Sobolev inequality

This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy-Littlewood-Sobolev inequality. The second regime corresponds to a reverse inequality, with the opposite sign in the convolution term, that allows us to bound the free energy of a drift-diffusion-Poisson system from below. Our method is based on an extension of an entropy method proposed by E. Carlen, J. Carrillo and M. Loss, and on a nonlinear diffusion equation.

We shall denote by L 1 + (R 2 ) the set of a.e. nonnegative functions in L 1 (R 2 ). Our main result is the following generalized logarithmic Hardy-Littlewood-Sobolev inequality. Theorem 1.1. For any α ≥ 0, we have that f (x) f (y) log |x − y| d x d y (1) for any function f ∈ L 1 + (R 2 ) with M = R 2 f d x > 0. Moreover, the equality case is achieved by f ⋆ = M µ and f ⋆ is the unique optimal function for any α > 0.
With α = 0, the inequality is the classical logarithmic Hardy-Littlewood-Sobolev inequality In that case f ⋆ is an optimal function as well as all functions generated by a translation and a scaling of f ⋆ . As long as the parameter α is in the range 0 ≤ α < 1, the coefficient of the right-hand side of (1) is negative and the inequality is essentially of the same nature as the one with α = 0. It can indeed be written as For reasons that will be made clear below, we shall call this range the attractive range.
Now, let us consider the repulsive range α > 1. It is clear that the inequality is no more the consequence of a simple interpolation. We can also observe that the coefficient (α−1) in the right-hand side of (1) is now positive. Since is the Green function associated with − ∆ on R 2 , so that we can define it is interesting to write (1) as If f has a sufficient decay as |x| → +∞, for instance if f is compactly supported, we know that (−∆) −1 f (x) ∼ − M 2 π log |x| for large values of |x| and as a consequence, In a minimization scheme, this prevents the runaway of the left-hand side in (4). On the other hand, R 2 f log f d x prevents any concentration, and this is why it can be heuristically expected that the lefthand side of (4) indeed admits a minimizer.
Inequality (2) was proved in [8] by E. Carlen and M. Loss (also see [2]). An alternative method based on nonlinear flows was given by E. Carlen, J. Carrillo and M. Loss in [7]: see Section 2 for a sketch of their proof. Our proof of Theorem 1.1 relies on an extension of this approach which takes into account the presence of the external potential V . A remarkable feature of this approach is that it is insensitive to the sign of α − 1.
One of the key motivations for studying (4) arises from entropy methods applied to drift-diffusion-Poisson models which, after scaling out all physical parameters, are given by with a nonlinear coupling given by the Poisson equation Here V = − log µ is the external confining potential and we choose it as in the statement of Theorem 1.1, while β ≥ 0 is a coupling parameter with V , which measures the strength of the external potential. We shall consider more general potentials at the end of this paper. The coefficient ε in (6) is either ε = −1, which corresponds to the attractive case, or ε = +1, which corresponds to the repulsive case. In terms of applications, when ε = −1, (6) is the equation for the mean field potential obtained from Newton's law of attraction in gravitation, for applications in astrophysics, or for the Keller-Segel concentration of chemo-attractant in chemotaxis. The case ε = +1 is used for repulsive electrostatic forces in semiconductor physics, electrolytes, plasmas and charged particle models.
In view of entropy methods applied to PDEs (see for instance [15]), it is natural to consider the free energy functional because, if f > 0 solves (5)-(6) and is smooth enough, with sufficient decay properties at infinity, then so that F β is a Lyapunov functional. Of course, a preliminary question is to establish under which conditions F β is bounded from below. The answer is given by the following result.
, the functional F β is bounded from below and admits a minimizer on the set of the functions f ∈ L 1 If ε = +1, the minimizer is unique.
As we shall see in Section 3.1, Corollary 1.2 is a simple consequence of Theorem 1.1. In the case of the parabolic-elliptic Keller-Segel model, that is, with ε = −1 and β = 0, this has been used in [12,4] to provide a sharp range of existence of the solutions to the evolution problem. In [6], the case ε = −1 with a potential V with quadratic growth at infinity was also considered, in the study of intermediate asymptotics of the parabolic-elliptic Keller-Segel model.
Concerning the drift-diffusion-Poisson model (5)-(6) and considerations on the free energy, in the electrostatic case, we can quote, among many others, [14,13] and subsequent papers. In the Euclidean space with confinig potentials, we shall refer to [10,11,3,1]. However, as far as we know, these papers are primarily devoted to dimensions d ≥ 3 and the sharp growth condition on V when d = 2 has not been studied so far. The goal of this paper is to fill this gap. The specific choice of V has been made to obtain explicit constants and optimal inequalities, but the confining potential plays a role only at infinity if we are interested in the boundedness from below of the free energy. In Section 3.3, we shall give a result for general potentials on R 2 : see Theorem 3.4 for a statement.

Proof of the main result
As an introduction to the key method, we briefly sketch the proof of (2) given by E. Carlen, J. Carrillo and M. Loss in [7]. The main idea is to use the nonlinear diffusion equation with a nonnegative initial datum f 0 . The equation preserves the mass M = R 2 f d x and is such that According to [9], the Gagliardo-Nirenberg inequality applied to g = f 1/4 guarantees that the right-hand side is nonpositive. By the general theory of fast diffusion equations (we refer for instance to [17]), we know that the solution behaves for large values of t like a self-similar solution, the so-called Barenblatt solution, which is given by As a consequence, we find that After an elementary computation, we observe that the above inequality is exactly (2) written for f = f 0 .
The point is now to adapt this strategy to the case with an external potential. This justifies why we have to introduce a nonlinear diffusion equation with a drift. As we shall see below, the method is insensitive to α and applies when α > 1 exactly as in the case α ∈ (0, 1). A natural question is whether solutions are regular enough to perform the computations below and in particular if they have a sufficient decay at infinity to allow all kinds of integrations by parts needed by the method. The answer is twofold. First, we can take an initial datum f 0 that is as smooth and decaying as |x| → +∞ as needed, prove the inequality and argue by density. Second, integrations by parts can be justified by an approximation scheme consisting in a truncation of the problem in larger and larger balls. We refer to [17] for regularity issues and to [15] for the truncation method. In the proof, we will therefore leave these issues aside, as they are purely technical.
Proof of Theorem 1.1. By homogeneity, we can assume that M = 1 without loss of generality and consider the evolution equation

1) Using simple integrations by parts, we compute
As a consequence, we obtain that 2) By elementary considerations again, we find that where, in the last line, we exchanged the variables x and y and took the half sum of the two expressions. This proves that 3) We observe that and, as a consequence, Let us define Collecting (10), (11) and (13), we find that Notice that and that ϕ is a strictly convex function on R + such that ϕ(1) = ϕ ′ (1) = 0, so that ϕ is nonnegative. On the other hand, by (9), we know that as in the proof of [7]. Altogether, this proves that t → F [ f (t , ·)] is monotone nonincreasing. Hence This completes the proof of (1).

Proof of Corollary 1.2
To prove the result of Corollary 1.2, we have to establish first that the free energy functional F β is bounded from below. Instead of using standard variational methods to prove that a minimizer is achieved, we can rely on the flow associated with (5)-(6).
Proof. With g = f M and α = 1 + M 8 π , this means that according to Theorem 1.1; the condition β ≥ α is enough to prove that F β [ f ] is bounded from below. Reciprocally, let us assume that β < 1 + M 8 π and let f ε (x) := ε 2 f ⋆ (ε x). It is then straightforward to check that F β is not bounded from below because Proof of Corollary 1.2 with ε = +1. Let us consider a smooth solution of (5)- (6). We refer to [16] for details and to [1] for similar arguments in dimension d ≥ 3. According to (8), f converges as t → +∞ to a solution of ∇ log f + β ∇V + ∇φ = 0 .
Notice that this already proves the existence of a stationary solution. The equation can be solved as after taking into account the conservation of the mass. With (6), the problem is reduced to solving Such a functional is strictly convex as, for instance, in [10,11]. We conclude that ψ is unique up to an additional constant.

Lemma 3.2.
Let ε = −1. Then F β is bounded from below on the set of the functions f ∈ L 1 It is not bounded from below if M > 8 π.
proves that the free energy is bounded from below if M ≤ 8 π and β ≥ α. On the other hand, if f ε (x) := ε −2 f (ε −1 x) and M > 8 π, then which proves that F β is not bounded from below.
Proof of Corollary 1.2 with ε = −1. The proof goes as in the case β = 0. We refer to [4] and leave details to the reader.

Remark 3.3.
Let us notice that F β is unbounded from below if β < 0. This follows from the observation that lim |y |→∞ F β [ f y ] = − ∞ where f y (x) = f (x + y) for any admissible f .

Duality
When α > 1, we can write a first inequality by considering the repulsive case in the proof of Corollary 1.2 and observing that J M ,γ [ψ] ≥ minJ M ,γ where ψ ∈ W 2,1 loc (R 2 ) is such that R 2 (∆ψ) d x = 0 and the minimum is taken on the same set of functions. When α ∈ [0, 1), it is possible to argue by duality as in [5,Section 2]. Since f ⋆ realizes the equality case in (1), we know that