In this paper we prove sharp regularity for a differential inclusion into a set K ⊂ R 2×2 that ar... more In this paper we prove sharp regularity for a differential inclusion into a set K ⊂ R 2×2 that arises in connection with the Aviles-Giga functional. The set K is not elliptic, and in that sense our main result goes beyondŠverák's regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami equation. In terms of the Aviles-Giga energy, our main result implies that zero energy states coincide (modulo a canonical transformation) with solutions of the differential inclusion into K. This opens new perspectives towards understanding energy concentration properties for Aviles-Giga: quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions. All these reformulations of our results are strong improvements of a recent work by the last two authors Lorent and Peng, where the link between the differential inclusion into K and the Aviles-Giga functional was first observed and used. Our proof relies moreover on new observations concerning the algebraic structure of entropies.
ESAIM: Control, Optimisation and Calculus of Variations, 2005
In this paper we analyse the structure of approximate solutions to the compatible two well proble... more In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let H = σ 0 0 σ −1 for σ > 0. Let 0 < ζ1 < 1 < ζ2 < ∞. Let K := SO (2) ∪ SO (2) H. Let u ∈ W 2,1 (Q1 (0)) be a C 1 invertible bilipschitz function with Lip (u) < ζ2, Lip u −1 < ζ −1 1. There exists positive constants c1 < 1 and c2 > 1 depending only on σ, ζ1, ζ2 such that if ∈ (0, c1) and u satisfies the following inequalities Q 1 (0) d (Du (z) , K) dL 2 z ≤ Q 1 (0) D 2 u (z) dL 2 z ≤ c1, then there exists J ∈ {Id, H} and R ∈ SO (2) such that Qc 1 (0) |Du (z) − RJ| dL 2 z ≤ c2 1 800 .
The Aviles Giga functional is a well known second order functional that forms a model for blister... more The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ IR 2 the functional is I ǫ (u) = 1 2 Ω ǫ −1 1 − |Du| 2 2 + ǫ D 2 u 2 dz where u belongs to the subset of functions in W 2,2 0 (Ω) whose gradient (in the sense of trace) satisfies Du(x) • η x = 1 where η x is the inward pointing unit normal to ∂Ω at x. In [Ja-Ot-Pe 02] Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences u n ∈ W 2,2 0 (Ω) with I ǫn (u n) → 0 as ǫ n → 0. A corollary to their work is that if there exists such a sequence (u n) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u := lim n→∞ u n = dist(•, ∂Ω). Recently [Lo 09] we provided a quantitative generalization of this corollary over the space of convex domains using 'compensated compactness' inspired calculations of [De-Mu-Ko-Ot 01]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B 1 (0) without the requiring the trace condition on Du.
We consider certain subsets of the space of n × n matrices of the form K = ∪ m i=1 SO(n)A i , and... more We consider certain subsets of the space of n × n matrices of the form K = ∪ m i=1 SO(n)A i , and we prove that for p > 1, q ≥ 1 and for connected Ω ′ ⊂⊂ Ω ⊂ IR n , there exists positive constant a < 1 depending on n, p, q, Ω, Ω ′ such that for ε = dist(Du, K) p
This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a d... more This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincaré inequality. The main result shows that the measure theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory study rectifiability issues related to quasiminimal sets in strong A ∞ -weighted Euclidean case.
In this paper we prove sharp regularity for a differential inclusion into a set K ⊂ R 2×2 that ar... more In this paper we prove sharp regularity for a differential inclusion into a set K ⊂ R 2×2 that arises in connection with the Aviles-Giga functional. The set K is not elliptic, and in that sense our main result goes beyondŠverák's regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami equation. In terms of the Aviles-Giga energy, our main result implies that zero energy states coincide (modulo a canonical transformation) with solutions of the differential inclusion into K. This opens new perspectives towards understanding energy concentration properties for Aviles-Giga: quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions. All these reformulations of our results are strong improvements of a recent work by the last two authors Lorent and Peng, where the link between the differential inclusion into K and the Aviles-Giga functional was first observed and used. Our proof relies moreover on new observations concerning the algebraic structure of entropies.
ESAIM: Control, Optimisation and Calculus of Variations, 2005
In this paper we analyse the structure of approximate solutions to the compatible two well proble... more In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let H = σ 0 0 σ −1 for σ > 0. Let 0 < ζ1 < 1 < ζ2 < ∞. Let K := SO (2) ∪ SO (2) H. Let u ∈ W 2,1 (Q1 (0)) be a C 1 invertible bilipschitz function with Lip (u) < ζ2, Lip u −1 < ζ −1 1. There exists positive constants c1 < 1 and c2 > 1 depending only on σ, ζ1, ζ2 such that if ∈ (0, c1) and u satisfies the following inequalities Q 1 (0) d (Du (z) , K) dL 2 z ≤ Q 1 (0) D 2 u (z) dL 2 z ≤ c1, then there exists J ∈ {Id, H} and R ∈ SO (2) such that Qc 1 (0) |Du (z) − RJ| dL 2 z ≤ c2 1 800 .
The Aviles Giga functional is a well known second order functional that forms a model for blister... more The Aviles Giga functional is a well known second order functional that forms a model for blistering and in a certain regime liquid crystals, a related functional models thin magnetized films. Given Lipschitz domain Ω ⊂ IR 2 the functional is I ǫ (u) = 1 2 Ω ǫ −1 1 − |Du| 2 2 + ǫ D 2 u 2 dz where u belongs to the subset of functions in W 2,2 0 (Ω) whose gradient (in the sense of trace) satisfies Du(x) • η x = 1 where η x is the inward pointing unit normal to ∂Ω at x. In [Ja-Ot-Pe 02] Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences u n ∈ W 2,2 0 (Ω) with I ǫn (u n) → 0 as ǫ n → 0. A corollary to their work is that if there exists such a sequence (u n) for a bounded domain Ω, then Ω must be a ball and (up to change of sign) u := lim n→∞ u n = dist(•, ∂Ω). Recently [Lo 09] we provided a quantitative generalization of this corollary over the space of convex domains using 'compensated compactness' inspired calculations of [De-Mu-Ko-Ot 01]. In this note we use methods of regularity theory and ODE to provide a sharper estimate and a much simpler proof for the case where Ω = B 1 (0) without the requiring the trace condition on Du.
We consider certain subsets of the space of n × n matrices of the form K = ∪ m i=1 SO(n)A i , and... more We consider certain subsets of the space of n × n matrices of the form K = ∪ m i=1 SO(n)A i , and we prove that for p > 1, q ≥ 1 and for connected Ω ′ ⊂⊂ Ω ⊂ IR n , there exists positive constant a < 1 depending on n, p, q, Ω, Ω ′ such that for ε = dist(Du, K) p
This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a d... more This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincaré inequality. The main result shows that the measure theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory study rectifiability issues related to quasiminimal sets in strong A ∞ -weighted Euclidean case.
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