A PDE-based approach to Borell-Brascamp-Lieb inequality
Qing LiuOkinawa Institute of Science and Technology

In this talk, we provide a new PDE perspective for the celebrated Borell-BrascampLieb inequality, which is stated as follows.
Theorem 1. Let n ≥ 1, λ ∈ (0, 1), α ∈ [−1/n, ∞]. Assume that f , g, h ∈ L1 (Rn ) arenonnegative and satisfyh((1 − λ)y + λz) ≥ Mα (f (y), g(z); λ),Then,(ˆˆRnαh(x) dx ≥ M nα+1∀y, z ∈ Rn .

ˆf (x) dx,Rn)g(x) dx; λ .

RnIn this theorem, Mα (a, b; λ) denotes the power mean associated to given α ∈ [−∞, +∞],a, b ≥ 0 and λ ∈ (0, 1), that is, we set Mα (a, b; λ) = 0 if ab = 0, and
max{a, b}
α = +∞1 ((1 − λ)aα + λbα ) α α ̸= 0, ±∞Mα (a, b; λ) =
a1−λ bλα=0 min{a, b}
α = −∞
if ab > 0.
This fundamental functional inequality in its general form was studied by [2, 3]. The
special case with α = 0 is known as the Prékopa-Leindler inequality, which is closely related
to the Brunn-Minkowski inequality and has signiﬁcant applications in convex geometry.
See [4] for an excellent survey concerning related topics on these inequalities.
Previously known proofs of Theorem 1 involve techniques from convex analysis oroptimal transport. In contrast, our new proof is based on properties of diﬀusion equationsof porous medium type∂t u −1∆(um ) = 0,mwhere m ∈ R is chosen asm = 2α + 1
in accordance with the exponent α appearing in the Borell-Brascamp-Lieb inequality. We
focus on the case of α < 0, which consequently requires m to lie in the fast diﬀusion
regime m < 1. The main tools we use in our proof are a generalized concavity maximum
principle and large time asymptotics for the equation.

Our approach reveals a deep connection between the Borell-Brascamp-Lieb inequality
and nonlinear parabolic equations. An analogous idea was introduced in [1] to prove thespecial case of the Prékopa-Leindler inequality by using the heat equation. Our analysis
serves as a nonlinear generalization and improvement of this PDE approach.
Furthermore, our work also demonstrates the possiblity of using PDEs to understandequality conditions for functional inequalities. While it is not completely clear to us how toemploy fast diﬀusions to retrieve the equality condition for the general Borell-BrascampLieb inequality, we manage to recover the equality condition for the Prékopa-Leindler
inequality by further exploiting additional properties of the heat equation including theeventual log-concavity and backward uniqueness of solutions.
This talk is based on recent joint work with Kazuhiro Ishige (Univ. Tokyo) and Paolo
Salani (Univ. Florence).

References[1] F. Barthe, D. Cordero-Erausquin, Inverse Brascamp-Lieb inequalities along the heatequation, Geometric aspects of functional analysis, Lecture Notes in Math., 1850(2004), 65–71.
[2] C. Borell, Convex set functions in d-space, Period. Math. Hungar. 6 (1975), 111–136.
[3] H. J. Brascamp, E. H. Lieb, On extensions of the Brunn-Minkowski and PrékopaLeindler theo- rems, including inequalities for log concave functions, and with anapplication to the diﬀusion equation, J. Functional Analysis 22 (1976), 366–389.
[4] R. J. Gardner. The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.), 39(2002), 355–405.

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