Archive for Rational Mechanics and Analysis, Jul 11, 2019
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
Given any strictly convex norm • on R 2 that is C 1 in R 2 \ {0}, we study the generalized Aviles... more Given any strictly convex norm • on R 2 that is C 1 in R 2 \ {0}, we study the generalized Aviles-Giga functional for Ω ⊂ R 2 and m : Ω → R 2 satisfying ∇ • m = 0. Using, as in the euclidean case • = | • |, the concept of entropies for the limit equation m = 1, ∇ • m = 0, we obtain the following. First, we prove compactness in L p of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in BV , we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case • = | • |, and the last two points are sensitive to the anisotropy of the norm • .
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional who... more The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose Γ-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on H 1 -rectifiable sets in 2D, is arguably the key missing part for a proof of the full Γ-convergence of the Aviles-Giga functional. In the first part of this work, for p ∈ 1, 4 3 we establish an L p version of the main theorem of [GL20]. Specifically we show that if m is a solution to the Eikonal equation, then m ∈ B 1 3 3p,∞,loc is equivalent to all entropy productions of m being in L p loc . Given the main result of [GL20], this result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in L p loc are rigid) -the rigidity/flexibility threshold of the Eikonal equation is exactly the space B 1 3 3,∞,loc . In the second part of this paper, under the assumption that all entropy productions are in L p loc , we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the L p setting -this is a strong extension of the main result of [LP18].
For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without bo... more For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset{{\mathbb{R}}}^{2\times 2}$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold tr...
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
Archive for Rational Mechanics and Analysis, Jul 11, 2019
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional who... more The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose Γ-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on H 1 -rectifiable sets in 2D, is arguably the key missing part for a proof of the full Γ-convergence of the Aviles-Giga functional. In the first part of this work, for p ∈ 1, 4 3 we establish an L p version of the main theorem of [GL20]. Specifically we show that if m is a solution to the Eikonal equation, then m ∈ B 1 3 3p,∞,loc is equivalent to all entropy productions of m being in L p loc . Given the main result of [GL20], this result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in L p loc are rigid) -the rigidity/flexibility threshold of the Eikonal equation is exactly the space B 1 3 3,∞,loc . In the second part of this paper, under the assumption that all entropy productions are in L p loc , we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the L p setting -this is a strong extension of the main result of [LP18].
For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without bo... more For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset{{\mathbb{R}}}^{2\times 2}$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold tr...
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
Archive for Rational Mechanics and Analysis, Jun 17, 2020
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 beyond Šverák'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.
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.
Archive for Rational Mechanics and Analysis, Jul 11, 2019
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
Given any strictly convex norm • on R 2 that is C 1 in R 2 \ {0}, we study the generalized Aviles... more Given any strictly convex norm • on R 2 that is C 1 in R 2 \ {0}, we study the generalized Aviles-Giga functional for Ω ⊂ R 2 and m : Ω → R 2 satisfying ∇ • m = 0. Using, as in the euclidean case • = | • |, the concept of entropies for the limit equation m = 1, ∇ • m = 0, we obtain the following. First, we prove compactness in L p of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in BV , we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case • = | • |, and the last two points are sensitive to the anisotropy of the norm • .
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional who... more The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose Γ-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on H 1 -rectifiable sets in 2D, is arguably the key missing part for a proof of the full Γ-convergence of the Aviles-Giga functional. In the first part of this work, for p ∈ 1, 4 3 we establish an L p version of the main theorem of [GL20]. Specifically we show that if m is a solution to the Eikonal equation, then m ∈ B 1 3 3p,∞,loc is equivalent to all entropy productions of m being in L p loc . Given the main result of [GL20], this result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in L p loc are rigid) -the rigidity/flexibility threshold of the Eikonal equation is exactly the space B 1 3 3,∞,loc . In the second part of this paper, under the assumption that all entropy productions are in L p loc , we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the L p setting -this is a strong extension of the main result of [LP18].
For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without bo... more For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset{{\mathbb{R}}}^{2\times 2}$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold tr...
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
Archive for Rational Mechanics and Analysis, Jul 11, 2019
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional who... more The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose Γ-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on H 1 -rectifiable sets in 2D, is arguably the key missing part for a proof of the full Γ-convergence of the Aviles-Giga functional. In the first part of this work, for p ∈ 1, 4 3 we establish an L p version of the main theorem of [GL20]. Specifically we show that if m is a solution to the Eikonal equation, then m ∈ B 1 3 3p,∞,loc is equivalent to all entropy productions of m being in L p loc . Given the main result of [GL20], this result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in L p loc are rigid) -the rigidity/flexibility threshold of the Eikonal equation is exactly the space B 1 3 3,∞,loc . In the second part of this paper, under the assumption that all entropy productions are in L p loc , we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the L p setting -this is a strong extension of the main result of [LP18].
For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without bo... more For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset{{\mathbb{R}}}^{2\times 2}$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold tr...
Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the... more Compensated compactness is an important method used to solve nonlinear PDEs, in particular in the study of hyperbolic conservation laws. One of the simplest formulations of a compensated compactness problem is to ask for conditions on a compact set K ⊂ M m×n such that lim j→∞ dist(Du j , K) L p = 0 and sup (1) Let M 1 , M 2 , . . . , M q denote the set of all minors of M m×n . A sufficient condition for (1) is that any probability measure µ supported on K satisfying is a Dirac measure. We call measures that satisfy (2) Null Lagrangian Measures and following [M ü 99], we denote the set of Null Lagrangian Measures supported on K by M pc (K). For general m, n, a necessary and sufficient condition for triviality of M pc (K) was an open question even in the case where K is a linear subspace of M m×n . We answer this question and provide a necessary and sufficient condition for any linear subspace K ⊂ M m×n . The ideas also allow us to show that for any d ∈ {1, 2, 3}, d-dimensional subspaces K ⊂ M m×n support non-trivial Null Lagrangian Measures if and only if K has Rank-1 connections. This is known to be false for d ≥ 4 from [Bh-Fi-Ja-Ko 94]. Further using the ideas developed we are able to answer a question of Kirchheim, M üller and for some function a and its primitive F. The set K 1 arises in the study of entropy solutions to the 2 × 2 system of conservation laws In [Ki-M ü-Sv 03], the authors asked what are the conditions on the function a such that M pc (K 1 ∩ U) consists of Dirac measures, where U is an open neighborhood of an arbitrary matrix in K 1 . Given α = (α 1 , α 2 ) ∈ IR 2 , if a ′ (α 2 ) > 0 then we construct non-trivial measures in M pc (K 1 ∩ B δ (P 1 (α))) for any δ > 0. On the other hand if a ′ (α 2 ) < 0 then for sufficiently small δ > 0, we show that M pc (K 1 ∩ B δ (P 1 (α))) consists of Dirac measures.
Archive for Rational Mechanics and Analysis, Jun 17, 2020
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 beyond Šverák'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.
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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