Regularity of the Eikonal equation with two vanishing entropies

Lorent, Andrew; Peng, Guanying

doi:10.1016/j.anihpc.2017.06.002

Lorent, Andrew ; Peng, Guanying

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 481-516.

Résumé

Let $Ω \subset R^{2}$ be a bounded simply-connected domain. The Eikonal equation $| \nabla u | = 1$ for a function $u : Ω \subset R^{2} \to R$ has very little regularity, examples with singularities of the gradient existing on a set of positive $H^{1}$ measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ∇u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if

\nabla \cdot ({\tilde{Σ}}_{e_{1} e_{2}} (\nabla u^{⊥})) = 0 and \nabla \cdot ({\tilde{Σ}}_{ϵ_{1} ϵ_{2}} (\nabla u^{⊥})) = 0 distributionally in Ω,

where

{\tilde{Σ}}_{e_{1} e_{2}}

and

{\tilde{Σ}}_{ϵ_{1} ϵ_{2}}

are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ∇u is locally Lipschitz continuous outside a locally finite set.

Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if $\lim_{n \to \infty} I_{ϵ_{n}} (u_{n}) = 0$ for some sequence $u_{n} \in W_{0}^{2, 2} (Ω)$ and $u = \lim_{n \to \infty} u_{n}$ then ∇u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation $| \nabla u | = 1$ a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies $\nabla \cdot [Φ (\nabla u^{⊥})] = 0$ distributionally in Ω then ∇u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.

The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion $D F \in K$ where $K \subset M^{2 \times 2}$ is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established.

DOI : 10.1016/j.anihpc.2017.06.002

Classification : 28A75
Mots clés : Eikonal equation, Aviles Giga functional, Entropies, Non-linear Beltrami equation, Differential inclusions

@article{AIHPC_2018__35_2_481_0,
     author = {Lorent, Andrew and Peng, Guanying},
     title = {Regularity of the {Eikonal} equation with two vanishing entropies},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {481--516},
     publisher = {Elsevier},
     volume = {35},
     number = {2},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.06.002},
     mrnumber = {3765550},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2017.06.002/}
}

TY  - JOUR
AU  - Lorent, Andrew
AU  - Peng, Guanying
TI  - Regularity of the Eikonal equation with two vanishing entropies
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2018
SP  - 481
EP  - 516
VL  - 35
IS  - 2
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2017.06.002/
DO  - 10.1016/j.anihpc.2017.06.002
LA  - en
ID  - AIHPC_2018__35_2_481_0
ER  -

%0 Journal Article
%A Lorent, Andrew
%A Peng, Guanying
%T Regularity of the Eikonal equation with two vanishing entropies
%J Annales de l'I.H.P. Analyse non linéaire
%D 2018
%P 481-516
%V 35
%N 2
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2017.06.002/
%R 10.1016/j.anihpc.2017.06.002
%G en
%F AIHPC_2018__35_2_481_0

Lorent, Andrew; Peng, Guanying. Regularity of the Eikonal equation with two vanishing entropies. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 2, pp. 481-516. doi : 10.1016/j.anihpc.2017.06.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2017.06.002/

Bibliographie
Cité par

[1] Alouges, F.; Rivière, T.; Serfaty, S. Neel and cross-tie wall energies for planar micromagnetic configurations. A tribute to J.L. Lions, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 31–68 | DOI | Numdam | MR | Zbl

[2] Ambrosio, L.; DeLellis, C.; Mantegazza, C. Line energies for gradient vector fields in the plane, Calc. Var. Partial Differ. Equ., Volume 9 (1999) no. 4, pp. 327–355 | DOI | MR | Zbl

[3] Ambrosio, L.; Lecumberry, M.; Rivière, T. A viscosity property of minimizing micromagnetic configurations, Commun. Pure Appl. Math., Volume 56 (2003) no. 6, pp. 681–688 | DOI | MR | Zbl

[4] Astala, K.; Iwaniec, T.; Martin, G. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009 | MR | Zbl

[5] Astala, K.; Iwaniec, T.; Saksman, E. Beltrami operators in the plane, Duke Math. J., Volume 107 (2001) no. 1, pp. 27–56 | DOI | MR | Zbl

[6] Aviles, P.; Giga, Y. Miniconference on Geometry and Partial Differential Equations, 2, Proc. Centre Math. Anal. Austral. Nat. Univ., Volume vol. 12, Austral. Nat. Univ., Canberra (1987), pp. 1–16 (Canberra, 1986) | MR

[7] Aviles, P.; Giga, Y. The distance function and defect energy, Proc. R. Soc. Edinb., Sect. A, Volume 126 (1996) no. 5, pp. 923–938 | DOI | MR | Zbl

[8] Bojarski, B. Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, Symposia Mathematica, Volume vol. XVIII, Academic Press, London (1976), pp. 485–499 (INDAM, Rome, 1974) | MR | Zbl

[9] Bojarski, B.; Iwaniec, T. Quasiconformal mappings and non-linear elliptic equations in two variables. I, II, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., Volume 22 (1974), pp. 473–478 | MR | Zbl

[10] Cannarsa, P.; Mennucci, A.; Sinestrari, C. Regularity results for solutions of a class of Hamilton–Jacobi equations, Arch. Ration. Mech. Anal., Volume 140 (1997) no. 3, pp. 197–223 | DOI | MR | Zbl

[11] Cannarsa, P.; Sinestrari, C. Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and Their Applications, vol. 58, Birkhäuser, Boston, MA, 2004 | MR | Zbl

[12] Constantin, P.; E, W.; Titi, E. Onsager's conjecture on the energy conservation for solutions of Euler's equation, Commun. Math. Phys., Volume 165 (1994) no. 1, pp. 207–209 | DOI | MR | Zbl

[13] Conti, S.; DeSimone, A.; Dolzmann, G.; Müller, S.; Otto, F. Multiscale modeling of materials—the role of analysis, Trends in Nonlinear Analysis, Springer, Berlin, 2003, pp. 375–408 | DOI | MR | Zbl

[14] Crandall, M.G.; Evans, L.C.; Lions, P.-L. Some properties of viscosity solutions of Hamilton–Jacobi equations, Trans. Am. Math. Soc., Volume 282 (1984) no. 2, pp. 487–502 | DOI | MR | Zbl

[15] DeLellis, C.; Ignat, R. A regularizing property of the 2D-Eikonal equation, Commun. Partial Differ. Equ., Volume 40 (2015) no. 8, pp. 1543–1557 | MR

[16] DeLellis, C.; Otto, F. Structure of entropy solutions to the Eikonal equation, J. Eur. Math. Soc., Volume 5 (2003) no. 2, pp. 107–145 | MR | Zbl

[17] DeLellis, C.; Otto, F.; Westdickenberg, M. Structure of entropy solutions for multi-dimensional scalar conservation laws, Arch. Ration. Mech. Anal., Volume 170 (2003) no. 2, pp. 137–184 | MR | Zbl

[18] DeSimone, A.; Müller, S.; Kohn, R.; Otto, F. A compactness result in the gradient theory of phase transitions, Proc. R. Soc. Edinb., Sect. A, Volume 131 (2001) no. 4, pp. 833–844 | DOI | MR | Zbl

[19] DeSimone, A.; Müller, S.; Kohn, R.; Otto, F. A reduced theory for thin-film micromagnetics, Commun. Pure Appl. Math., Volume 55 (2002) no. 11, pp. 1408–1460 | DOI | MR | Zbl

[20] Faraco, D.; Kristensen, J. Compactness versus regularity in the calculus of variations, Discrete Contin. Dyn. Syst., Ser. B, Volume 17 (2012) no. 2, pp. 473–485 | MR | Zbl

[21] Gioia, G.; Ortiz, M. The morphology and folding patterns of buckling-driven thin-film blisters, J. Mech. Phys. Solids, Volume 42 (1994) no. 3, pp. 531–559 | MR | Zbl

[22] Girault, V.; Raviart, P-A. Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986 | DOI | MR | Zbl

[23] Ignat, R. Two-dimensional unit-length vector fields of vanishing divergence, J. Funct. Anal., Volume 262 (2012) no. 8, pp. 3465–3494 | DOI | MR | Zbl

[24] Iwaniec, T., Symposia Mathematica, Volume vol. XVIII, Academic Press, London (1976), pp. 501–517 | MR | Zbl

[25] Jabin, P-E.; Otto, F.; Perthame, B. Line-energy Ginzburg–Landau models: zero-energy states, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 1 (2002) no. 1, pp. 187–202 | Numdam | MR | Zbl

[26] Jin, W.; Kohn, R.V. Singular perturbation and the energy of folds, J. Nonlinear Sci., Volume 10 (2000) no. 3, pp. 355–390 | MR | Zbl

[27] Lorent, A. A quantitative characterisation of functions with low Aviles Giga energy on convex domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 13 (2014) no. 1, pp. 1–66 | MR | Zbl

[28] Modica, L.; Mortola, S. Un esempio di Γ-convergenza, Boll. Unione Mat. Ital., B (5), Volume 14 (1977) no. 1, pp. 285–299 | MR | Zbl

[29] Müller, S. Calculus of Variations and Geometric Evolution Problems, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Volume vol. 1713, Springer, Berlin (1999), pp. 85–210 (Cetraro, 1996) | DOI | MR | Zbl

[30] Rivière, T.; Serfaty, S. Limiting domain wall energy for a problem related to micromagnetics, Commun. Pure Appl. Math., Volume 54 (2001) no. 3, pp. 294–338 | DOI | MR | Zbl

[31] Rivière, T.; Serfaty, S. Compactness, kinetic formulation, and entropies for a problem related to micromagnetics, Commun. Partial Differ. Equ., Volume 28 (2003) no. 1–2, pp. 249–269 | MR | Zbl

[32] Sverak, V. On Tartar's conjecture, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 10 (1993) no. 4, pp. 405–412 | DOI | Numdam | MR | Zbl

Cité par Sources :