On the equation 𝑑𝑖𝑣𝑌=𝑓 and application to control of phases

Bourgain, Jean ¹   ; Brezis, Haïm ^{2 ,  3}

¹ Institute for Advanced Study, Princeton, New Jersey 08540
² Analyse Numérique, Université P. et M. Curie, B.C. 187, 4 Pl. Jussieu, 75252 Paris Cedex 05, France
³ Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854

Journal of the American Mathematical Society, Tome 16 (2003) no. 2, pp. 393-426

Résumé

The main result is the following. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^{d}$, $d\geq 2$. Then for every $f\in L^{d}(\Omega )$ with $\int f =0$, there exists a solution $u\in C^{0}(\bar \Omega )\cap W^{1, d}(\Omega )$ of the equation div $u=f$ in $\Omega$, satisfying in addition $u=0$ on $\partial \Omega$ and the estimate \begin{equation*}\Vert u\Vert _{L^{\infty }}+\Vert u\Vert _{W^{1, d}}\leq C\Vert f\Vert _{L^{d}} \end{equation*} where $C$ depends only on $\Omega$. However one cannot choose $u$ depending linearly on $f$. Our proof is constructive, but nonlinear—which is quite surprising for such an elementary linear PDE. When $d=2$ there is a simpler proof by duality—hence nonconstructive.

Détail

DOI : 10.1090/S0894-0347-02-00411-3

Bibliographie

[1] Adams, Robert A. Sobolev spaces 1975

[2] Arnold, Douglas N., Scott, L. Ridgway, Vogelius, Michael Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1988

[3] Bourgain, Jean, Brezis, Haim, Mironescu, Petru Lifting in Sobolev spaces J. Anal. Math. 2000 37 86

[4] Bourgain, Jean, Brezis, Haïm, Mironescu, Petru On the structure of the Sobolev space 𝐻^{1/2} with values into the circle C. R. Acad. Sci. Paris Sér. I Math. 2000 119 124

[5] Bourgain, Jean, Brezis, Haïm, Mironescu, Petru On the structure of the Sobolev space 𝐻^{1/2} with values into the circle C. R. Acad. Sci. Paris Sér. I Math. 2000 119 124

[6] Burago, D., Kleiner, B. Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps Geom. Funct. Anal. 1998 273 282

[7] Dacorogna, Bernard, Moser, Jürgen On a partial differential equation involving the Jacobian determinant Ann. Inst. H. Poincaré C Anal. Non Linéaire 1990 1 26

[8] Duvaut, G., Lions, J.-L. Les inéquations en mécanique et en physique 1972

[9] Fefferman, C., Stein, E. M. 𝐻^{𝑝} spaces of several variables Acta Math. 1972 137 193

[10] Gromov, M. Asymptotic invariants of infinite groups 1993 1 295

[11] Magenes, Enrico, Stampacchia, Guido I problemi al contorno per le equazioni differenziali di tipo ellittico Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 1958 247 358

[12] Mcmullen, C. T. Lipschitz maps and nets in Euclidean space Geom. Funct. Anal. 1998 304 314

[13] Nirenberg, L. Topics in nonlinear functional analysis 1974

[14] Ornstein, Donald A non-equality for differential operators in the 𝐿₁ norm Arch. Rational Mech. Anal. 1962 40 49

[15] Rivière, Tristan, Ye, Dong Une résolution de l’équation à forme volume prescrite C. R. Acad. Sci. Paris Sér. I Math. 1994 25 28

[16] Rivière, Tristan, Ye, Dong Resolutions of the prescribed volume form equation NoDEA Nonlinear Differential Equations Appl. 1996 323 369

[17] Rubio De Francia, José L. A Littlewood-Paley inequality for arbitrary intervals Rev. Mat. Iberoamericana 1985 1 14

[18] Temam, Roger Navier-Stokes equations 1979

[19] Uchiyama, Akihito A constructive proof of the Fefferman-Stein decomposition of BMO (𝑅ⁿ) Acta Math. 1982 215 241

[20] Wang, Xiao Ming A remark on the characterization of the gradient of a distribution Appl. Anal. 1993 35 40

[21] Wojtaszczyk, P. Banach spaces for analysts 1991

[22] Ye, Dong Prescribing the Jacobian determinant in Sobolev spaces Ann. Inst. H. Poincaré C Anal. Non Linéaire 1994 275 296