Brezis–Gallouët

Brezis–Gallouët
Name	Brezis–Gallouët inequality
Area	Functional analysis, Partial differential equations
Introduced	1980
Authors	Haïm Brezis, Thierry Gallouët
Statement	Control of sup-norm by Sobolev norms in two dimensions

Brezis–Gallouët

The Brezis–Gallouët inequality is a two-dimensional functional inequality introduced by Haïm Brezis and Thierry Gallouët that quantifies sup-norm control of functions in terms of Sobolev norms. It plays a central role in the analysis of nonlinear evolution equations and elliptic regularity, influencing work by researchers affiliated with Courant Institute, École Normale Supérieure, Institute for Advanced Study, Princeton University, and Université Paris-Sud. The inequality connects techniques from Sobolev embedding theorem, Fourier analysis, Littlewood–Paley theory, and Calderón–Zygmund theory.

Introduction

The inequality addresses the failure of a direct embedding from H^1 into L^\infty in two dimensions by introducing a logarithmic correction term; this complements results by Sobolev, Gagliardo, Nirenberg, Trudinger, and Moser. It has been invoked in studies of the Navier–Stokes equations, Nonlinear Schrödinger equation, Ginzburg–Landau equation, and in the regularity theory developed by figures connected to Jean-Michel Bismut, Cédric Villani, and Luis Caffarelli. The inequality is particularly useful on domains like the 2-torus, the unit disk, and compact manifolds such as 2-sphere and Riemann surfaces.

Statement of the Inequality

Let Ω be a bounded domain in the plane, or consider periodic functions on the 2-torus. For u in H^s(Ω) with s>0 and often s=1, the Brezis–Gallouët inequality states that there exists a constant C depending on Ω such that - in its classical form, ||u||_{L^\infty} ≤ C (1 + ||u||_{H^1} sqrt{log(1 + ||u||_{H^s}/||u||_{H^1})}), with formulations involving the H^1 and H^2 norms or equivalently the L^2 norm of derivatives. Variants replace Ω by R^2 with decay conditions, or by compact manifolds where constants depend on geometric quantities studied by Michael Gromov and Richard Hamilton. This statement refines embeddings related to the Trudinger–Moser inequality and is compatible with scale-invariant estimates like those in Kato's inequality contexts studied by Tosio Kato and L. Nirenberg.

Proofs and Techniques

Proofs employ a mixture of spectral and harmonic analysis: decompositions via Fourier series, Littlewood–Paley decomposition, and Bernstein inequalities as developed in work connected to Elias Stein, Charles Fefferman, and Jean Bourgain. Another classical route uses energy estimates, integration by parts, and elliptic regularity rooted in techniques from Sergiu Klainerman and Luis Caffarelli. A typical argument splits u into low and high frequency components, bounds low frequencies by Cauchy–Schwarz inequality and high frequencies using summation with logarithmic weights akin to arguments by Jean-Pierre Aubin and Isadore M. Singer. Alternative proofs adapt methods of Calderón–Zygmund estimates, interpolation inequalities from Gagliardo–Nirenberg, and concentration-compactness ideas due to Pierre-Louis Lions.

Sharpness and Counterexamples

The logarithmic factor is essentially sharp: classical counterexamples are built from rescaled bump functions, radial profiles, or concentrated Fourier modes inspired by constructions in work by Ennio De Giorgi, Jürgen Moser, and Nicolaas Kuiper. On domains like the unit disk or noncompact R^2 without decay, sequences exhibit growth matching the logarithmic term, a phenomenon reminiscent of extremals for the Trudinger–Moser inequality found by Florian Robert and Miguel del Pino. Refinements showing necessity of the logarithm draw on examples related to eigenfunctions of the Laplacian studied by Shmuel Agmon and Peter Li.

Applications

The inequality is widely used in global existence and regularity arguments for evolution equations such as the Nonlinear Schrödinger equation, the Korteweg–de Vries equation, and the Navier–Stokes equations in two dimensions; it appears in work by researchers at CNRS, Institut Henri Poincaré, Massachusetts Institute of Technology, and University of California, Berkeley. It enables control of nonlinearities in proofs by Terence Tao, Benoît Perthame, Jean-Yves Chemin, and Pierre-Louis Lions and underpins scattering and blow-up criteria in studies by Craig Kenig, Carlos Kenig, Stephen Klainerman, and Walter Strauss. In geometric analysis it contributes to bounds for harmonic maps connected to Richard Hamilton and in the study of vortices in Ginzburg–Landau theory linked to François Trèves and Elliott H. Lieb.

Extensions and Related Results

Extensions include vector-valued and weighted versions, versions on manifolds with boundary influenced by techniques from Michael Taylor and László Fehér, and endpoint results connecting to the Trudinger–Moser inequality and refinements by Adimurthi, Ovidiu Savin, and Guy David. Related inequalities include logarithmic Sobolev inequalities of Leonard Gross, endpoint embeddings studied by James Serrin and E. M. Stein, and multilinear analogues used in harmonic analysis by Terence Tao and Béla Bollobás. Contemporary research explores stochastic and nonlocal analogues in works affiliated with Courant Institute, Imperial College London, and University of Cambridge.

Category:Functional inequalities Category:Partial differential equations Category:Sobolev spaces