arXiv subjects

Andrew Lorent

Publications and source records attributed to Andrew Lorent.

At least 19 recordsLinked to original sources

arXiv2026

On Aviles-Giga limit states with $L^p$ entropy productions

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colon\Omega \subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(\Omega),\quad |m|=1\text{ a.e. in }\Omega\,, \end{align*} whose entropy productions $\mathrm{div}\,\Phi(m)$ are Radon measures in $\Omega$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $\Phi\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,\Phi(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions.

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arXiv2025

Another regularizing property of the 2D eikonal equation

A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\colon\Omega\subset\mathbb R^2\to\mathbb R^2$ such that $|m|=1$ a.e. and $\mathrm{div}\,m=0$ in $\mathcal D'(\Omega)$. It is known that, if $m$ has some low regularity, e.g., continuous or $W^{1/3,3}$, then $m$ is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption $m\in B^{1/3}_{p,\infty}$ for $p>3$. In this note we establish that regularizing effect in the borderline case $p=6$, above which the Besov regularity assumption implies continuity. If the domain is a disk and $m$ satisfies tangent boundary conditions, we also prove this for $p$ slightly below $6$.

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arXiv2024

On regularity and rigidity of $2\times 2$ differential inclusions into non-elliptic curves

We study differential inclusions $Du\in \Pi$ in an open set $\Omega\subset\mathbb R^2$, where $\Pi\subset \mathbb R^{2\times 2}$ is a compact connected $C^2$ curve without rank-one connections, but non-elliptic: tangent lines to $\Pi$ may have rank-one connections, so that classical regularity and rigidity results do not apply. For a wide class of such curves $\Pi$, we show that $Du$ is locally Lipschitz outside a discrete set, and is rigidly characterized around each singularity. Moreover, in the partially elliptic case where at least one tangent line to $\Pi$ has no rank-one connections, or under some topological restrictions on the tangent bundle of $\Pi$, there are no singularities. This goes well beyond previously known particular cases related to Burgers' equation and to the Aviles-Giga functional. The key is the identification and appropriate use of a general underlying structure: an infinite family of conservation laws, called entropy productions in reference to the theory of scalar conservation laws.

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arXiv2022

Quantitative rigidity of differential inclusions in two dimensions

For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ which has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate \[ \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\,dx \leq C \int_{B_1} \mathrm{dist}^2(Du, K)\, dx, \qquad\forall u\in H^1(B_1;\mathbb R^2). \] This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$, of the celebrated quantitative rigidity estimate of Friesecke, James and M\"uller 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 PDE satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold true in $\mathbb R^{n\times n}$ for $n\geq 3$.

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arXiv2022

On optimal regularity estimates for finite-entropy solutions of scalar conservation laws

We consider finite-entropy solutions of scalar conservation laws $u_t +a(u)_x =0$, that is, bounded weak solutions whose entropy productions are locally finite Radon measures. Under the assumptions that the flux function $a$ is strictly convex (with possibly degenerate convexity) and $a''$ forms a doubling measure, we obtain a characterization of finite-entropy solutions in terms of an optimal regularity estimate involving a cost function first used by Golse and Perthame.

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arXiv2022

On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds

Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in $\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional \[I_{\epsilon}(m):=\int_{\Omega} \left(\epsilon \left|\nabla m\right|^2 + \frac{1}{\epsilon}\left(1-\|m\|^2\right)^2\right) \, dx,\] for $\Omega\subset\mathbb R^2$ and $m\colon\Omega\to\mathbb R^2$ satisfying $\nabla\cdot m=0$. Using, as in the euclidean case $\|\cdot\|=|\cdot|$, the concept of entropies for the limit equation $\|m\|=1$, $\nabla\cdot m=0$, 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 $\|\cdot\|=|\cdot|$, and the last two points are sensitive to the anisotropy of the norm $\|\cdot\|$.

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arXiv2021

Factorization for entropy production of the Eikonal equation and regularity

The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose $\Gamma$-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 $\mathcal{H}^1$-rectifiable sets in $2$D, is arguably the key missing part for a proof of the full $\Gamma$-convergence of the Aviles-Giga functional. In the first part of this work, for $p\in \left(1,\frac{4}{3}\right]$ we establish an $L^p$ version of the main theorem of Ghiraldin and Lamy [Comm. Pure Appl. Math. 73 (2020), no. 2, 317-349]. Specifically we show that if $m$ is a solution to the Eikonal equation, then $m\in B^{\frac{1}{3}}_{3p,\infty,loc}$ is equivalent to all entropy productions of $m$ being in $L^p_{loc}$. 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^{\frac{1}{3}}_{3,\infty,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 an earlier result of the authors [Annales de l'Institut Henri Poincar\'{e}. Analyse Non Lin\'{e}aire 35 (2018), no. 2, 481-516].

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arXiv2020

Rigidity of a non-elliptic differential inclusion related to the Aviles-Giga conjecture

In this paper we prove sharp regularity for a differential inclusion into a set $K\subset\mathbb{R}^{2\times 2}$ 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.

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arXiv2019

On the Rank-$1$ convex hull of a set arising from a hyperbolic system of Lagrangian elasticity

We address the questions (P1), (P2) asked in Kirchheim-M\"{u}ller-\v{S}ver\'{a}k (2003) concerning the structure of the Rank-$1$ convex hull of a submanifold $\mathcal{K}_1\subset M^{3\times 2}$ that is related to weak solutions of the two by two system of Lagrangian equations of elasticity studied by DiPerna (1985) with one entropy augmented. This system serves as a model problem for higher order systems for which there are only finitely many entropies. The Rank-$1$ convex hull is of interest in the study of solutions via convex integration: the Rank-$1$ convex hull needs to be sufficiently non-trivial for convex integration to be possible. Such non-triviality is typically shown by embedding a $\mathbb{T}_4$ (Tartar square) into the set. We show that in the strictly hyperbolic, genuinely nonlinear case considered by DiPerna (1985), no $\mathbb{T}_4$ configuration can be embedded into $\mathcal{K}_1$.

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arXiv2018

Null Lagrangian Measures in subspaces, compensated compactness and conservation laws

Compensated compactness is an important method used to solve nonlinear PDEs. A simple formulation of a compensated compactness problem is to ask for conditions on a set $\mathcal{K}\subset M^{m\times n}$ such that $$ \lim_{n\rightarrow \infty} \mathrm{dist}(Du_n,\mathcal{K})\overset{L^p}{\rightarrow} 0\; \Rightarrow \{Du_{n}\}_{n}\text{ is precompact.} $$ Let $M_1,M_2,\dots, M_q$ denote the set of minors of $M^{m\times n}$. A sufficient condition for this is that any measure $μ$ supported on $\mathcal{K}$ satisfying $$ \int M_k(X) dμ(X)=M_k\left(\int X dμ(X)\right)\text{ for }k=1,2,\dots, q $$ is a Dirac measure. We call measures that satisfy the above equation "Null Lagrangian Measures" and we denote the set of Null Lagrangian Measures supported on $\mathcal{K}$ by $\mathcal{M}^{pc}(\mathcal{K})$. For general $m,n$, a necessary and sufficient condition for triviality of $\mathcal{M}^{pc}(\mathcal{K})$ was an open question even in the case where $\mathcal{K}$ is a linear subspace of $M^{m\times n}$. We answer this question and provide a necessary and sufficient condition for any linear subspace $\mathcal{K}\subset M^{m\times n}$. The ideas also allow us to show that for any $d\in \left\{1,2,3\right\}$, $d$-dimensional subspaces $\mathcal{K}\subset M^{m\times n}$ support non-trivial Null Lagrangian Measures if and only if $\mathcal{K}$ has Rank-$1$ connections. This is known to be false for $d\ge 4$. Using the ideas developed we are able to answer (up to first order) a question of Kirchheim, Müller and Sverak on the Null Lagrangian measures arising in the study of a (one) entropy solution of a $2\times 2$ system of conservation laws that arises in elasticity.

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arXiv2016

Regularity of the Eikonal equation with two vanishing entropies

The Aviles-Giga functional $I_ε(u)=\int_Ω \frac{\left|1-\left|\nabla u\right|^2\right|^2}ε+ε\left|\nabla^2 u\right|^2 \, dx$ is a well known second order functional that models phenomena from blistering to liquid crystals. The zero energy states of the Aviles-Giga functional have been characterized by Jabin, Otto, Perthame. Among other results they showed that if $\lim_{n\rightarrow \infty} I_{ε_n}(u_n)=0$ for some sequence $u_n\in W^{2,2}_0(Ω)$ and $u=\lim_{n\rightarrow \infty} u_n$ then $\nabla u$ is Lipschitz continuous outside a locally finite set. This is essentially a corollary to their theorem that if $u$ is a solution to the Eikonal equation $\left|\nabla u\right|=1$ a.e. and if for every "entropy" $Φ$ function $u$ satisfies $\nabla\cdot\left[Φ(\nabla u^{\perp})\right]=0$ distributionally in $Ω$ then $\nabla u$ is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result by showing that if $Ω$ is bounded and simply connected, $u$ satisfies the Eikonal equation and if \begin{equation} \label{eqi88} \nabla\cdot\left(Σ_{e_1 e_2}(\nabla u^{\perp})\right)=0\text{and}\nabla\cdot\left(Σ_{ε_1 ε_2}(\nabla u^{\perp})\right)=0\text{distributionally in}Ω, \end{equation} where $Σ_{e_1 e_2}$ and $Σ_{ε_1 ε_2}$ are the entropies introduced by Ambrosio, DeLellis, Mantegazza, Jin, Kohn, then $\nabla u$ is locally Lipschitz continuous outside a locally finite set.

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arXiv2014

On functions whose symmetric part of gradient agree and a generalization of Reshetnyak's compactness theorem

We consider the following question: Given a connected open domain $Ω\subset R^n$, suppose $u,v:Ω\rightarrow R^n$ with $\det(\nabla u)>0$, $\det(\nabla v)>0$ a.e. are such that $\nabla u^T(x)\nabla u(x)=\nabla v(x)^T \nabla v(x)$ a.e. does this imply a global relation of the form $\nabla v(x)= R\nabla u(x)$ a.e. in $Ω$ where $R\in SO(n)$? If $u,v$ are $C^1$ it is an exercise to see this true, if $u,v\in W^{1,1}$ we show this is false. We prove this question has a positive answer if $v\in W^{1,1}$ and $u\in W^{1,n}$ is a mapping of $L^p$ integrable dilatation for $p>n-1$. These conditions are sharp in two dimensions and this result represents a generalization of the corollary to Liouville's theorem that states that the differential inclusion $\nabla u\in SO(n)$ can only be satisfied by an affine mapping. Liouville's corollary for rotations has been generalized by Reshetnyak who proved convergence of gradients to a fixed rotation for any weakly converging sequence $v_k\in W^{1,1}$ for which $$ \int_Ω \mathrm{dist}(\nabla v_k,SO(n)) dz\rightarrow 0 \text{as} k\rightarrow \infty. $$ Let $S(\cdot)$ denote the (multiplicative) symmetric part of a matrix. In Theorem 3 we prove an analogous for any pair of weakly converging sequences $v_k\in W^{1,p}$ and $u_k\in W^{1,\frac{p(n-1)}{p-1}}$ (where $p\in \left[1,n\right]$ and the sequence $(u_k)$ has its dilatation pointwise bounded above by an $L^r$ integrable function, $r>n-1$) that satisfy $\int_Ω \left|S(\nabla u_k)-S(\nabla v_k)\right|^p dz\rightarrow 0$ as $k\rightarrow \infty$ and for which the sign of the $\det(\nabla v_k)$ tends to 1 in $L^1$. This result contains Reshetnyak's theorem as the special case $(u_k)\equiv Id$, p=1.

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arXiv2014

Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of studies proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization.

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arXiv2014

A generalized Stoilow decomposition for pairs of mappings of integrable dilatation

We prove a rigidity result for pairs of mappings of integrable dilatation whose gradients pointwise deform the unit ball to similar ellipses. Our result implies as corollaries a version of the generalized Stoilow decomposition provided by Theorem 5.5.1 of a recent monograph of Astala-Iwaniec-Martin and the two dimensional rigidity result of our previous paper for mappings whose symmetric part of gradient agrees. Specifically let $u,v\in W^{1,2}(Ω,\mathbb{R}^2)$ where $\det(Du)>0$, $\det(Dv)>0$ a.e. and $u$ is a mapping of integrable dilatation. Suppose for a.e. $z\in Ω$ we have $Du(z)^T Du(z)=λDv(z)^T Dv(z)$ for some $λ>0$. Then there exists a meromorphic function $ψ$ and a homeomorphism $w\in W^{1,1}(Ω:\mathbb{R}^2)$ such that $Du(z)=\mathcal{P}(ψ(w(z)))Dv(z)$ where $\mathcal{P}(a+ib)=(\begin{smallmatrix} a & -b \\ b & a \end{smallmatrix})$. We show by example that this result is sharp in the sense that there can be no continuous relation between the gradients of $Du$ and $Dv$ on a dense open connected subset of $Ω$ unless one of the mappings is of integrable dilatation.

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arXiv2013

Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close

For $A\in M^{2\times 2}$ let $S(A)=\sqrt{A^T A}$, i.e. the symmetric part of the polar decomposition of $A$. We consider the relation between two quasiregular mappings whose symmetric part of gradient are close. Our main result is the following. Suppose $v,u\in W^{1,2}(B_1(0):\mathbb{R}^2)$ are $Q$-quasiregular mappings with $\int_{B_1(0)} \det(Du)^{-p} dz\leq C_p$ for some $p\in (0,1)$ and $\int_{B_1(0)} |Du|^2 dz\leq 1$. There exists constant $M>1$ such that if $$ \int_{B_1(0)} |S(Du)-S(Dv)|^2 dz=ε$$ then $$ \int_{B_{\frac{1}{2}}(0)} |Dv-R Du| dz\leq c C_p^{\frac{1}{p}}ε^{\frac{p^3}{M Q^5\log(10 C_p Q)}}\text{ for some }R\in SO(2). $$ Taking $u=Id$ we obtain a special case of the quantitative rigidity result of Friesecke, James and Muller. Our main result can be considered as a first step in a new line of generalization of F-J-M Theorem in which $Id$ is replaced by a mapping of non-trivial degree.

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arXiv2013

On indecomposable sets with applications

In this note we show the characteristic function of every indecomposable set $F$ in the plane is $BV$ equivalent to the characteristic function a closed set $\mathbb{F}$, i.e. $||\mathbb{1}_{F}-\mathbb{1}_{\mathbb{F}}||_{BV(\mathbb{R}^2)}=0$. We show by example this is false in dimension three and above. As a corollary to this result we show that for every $ε>0$ a set of finite perimeter $S$ can be approximated by a closed subset $\mathbb{S}_ε$ with finitely many indecomposable components and with the property that $H^1(\partial^M \mathbb{S}_ε\backslash \partial^M S)=0$ and $||\mathbb{1}_{S}-\mathbb{1}_{\mathbb{S}_ε}||_{BV(\mathbb{R}^2)}<ε$. We apply this corollary to give a short proof that locally quasiminimizing sets in the plane are $BV_l$ extension domains.

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arXiv2012

A quantitative characterisation of functions with low Aviles Giga energy on convex domains

Given a connected Lipschitz domain U we let L(U) be the subset of functions in 2nd order Sobolev space whose gradient (in the sense of trace) is equal to the inward pointing unit normal to U. The the Aviles Giga functional over L(U) serves as a model in connection with problems in liquid crystals and thin film blisters, it is also the most natural higher order generalisation of the Modica Mortola functional. Jabin, Otto, Perthame characterised a class of functions which includes all limits of sequences whose Aviles Giga energy goes to zero. A corollary to their work is that if there exists such a sequence for a bounded domain U, then U must be a ball and the limiting function must be the distance from the boundary. We prove a quantitative generalisation of this corollary for the class of bounded convex sets. As a consequence of this we show that if U has C^2 boundary and is close to a ball, then for all small enough \ep the minimiser of I_{\ep} is close to the distance function from the boundary.

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arXiv2011

A simple proof of the characterization of functions of low Aviles Giga energy on a ball via regularity

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 $Ω\subset R^2$ the functional is $I_ε(u)=1/2\int_Ω ε^{-1}|1-|Du|^2|^2+ε|D^2 u|^2$ where $u$ belongs to the subset of functions in $W^{2,2}_{0}(Ω)$ whose gradient (in the sense of trace) satisfies $Du(x)\cdot η_x=1$ where $η_x$ is the inward pointing unit normal to $\partial Ω$ at $x$. In Jabin, Otto, Perthame characterized a class of functions which includes all limits of sequences $u_n\in W^{2,2}_0(Ω)$ with $I_{ε_n}(u_n)\to 0$ as $ε_n\to 0$. 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) $u:=\lim_{n\to \infty} u_n =\mathrm{dist}(\cdot,\partialΩ)$. Recently we provided a quantitative generalization of this corollary over the space of convex domains using `compensated compactness' inspired calculations originating from the proof of coercivity of $I_ε$ by DeSimone, Muller, Kohn, Otto. 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 $Du$.

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