Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture

Abstract

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 upon 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.

Appendices

A Commutator Estimate

In this appendix we prove the following basic commutator estimate:

Lemma 17

Given $\Pi \in C^{2}({{\mathbb {R}}}^2)$ and $m:\Omega \rightarrow {{\mathbb {R}}}^2$ with ${\left| m\right| }\leqq R$ a.e. for some $0<R<\infty$, we have

(54)

Proof of Lemma 17

The proof follows computations presented in [14] and recently in a context closer to ours also in [18]. We write out

$$\begin{aligned}&\left[\Pi (m)\right]_{\varepsilon }(x) -\Pi (m_{\varepsilon }(x))\nonumber \\&= \int \left(\Pi (m(x-z))-\Pi (m_{\varepsilon }(x))\right)\rho _{\varepsilon }(z)\,\mathrm{d}z\nonumber \\&=\int D\Pi (m(x-z))\cdot \left(m(x-z)-m_{\varepsilon }(x)\right)\rho _{\varepsilon }(z)\,\mathrm{d}z\nonumber \\&\quad +\int \left(\Pi (m(x-z))-\Pi (m_{\varepsilon }(x)) - D\Pi (m(x-z))\cdot \left(m(x-z)-m_{\varepsilon }(x)\right)\right)\rho _{\varepsilon }(z)\,\mathrm{d}z\nonumber \\&=\int \left(D\Pi (m(x-z))-D\Pi (m_{\varepsilon }(x))\right)\cdot \left(m(x-z)-m_{\varepsilon }(x)\right)\rho _{\varepsilon }(z)\,\mathrm{d}z\nonumber \\&\quad +\int \left(\Pi (m(x-z))-\Pi (m_{\varepsilon }(x)) - D\Pi (m(x-z))\cdot \left(m(x-z)-m_{\varepsilon }(x)\right)\right)\rho _{\varepsilon }(z)\,\mathrm{d}z. \end{aligned}$$

(55)

By Taylor expansion we have

$$\begin{aligned}&|\Pi (m(x-z))-\Pi (m_{\varepsilon }(x)) - D\Pi (m(x-z))\cdot \left(m(x-z)-m_{\varepsilon }(x)\right)| \\&\quad \quad \lesssim {\Vert D^2\Pi \Vert }_{L^\infty }|m(x-z)-m_{\varepsilon }(x)|^2, \\&\left|\left(D\Pi (m(x-z))-D\Pi (m_{\varepsilon }(x))\right)\cdot \left(m(x-z)-m_{\varepsilon }(x)\right) \right|\\&\quad \quad \lesssim {\Vert D^2\Pi \Vert }_{L^\infty } \left|m(x-z)-m_{\varepsilon }(x)\right|^2, \end{aligned}$$

and plugging this into (55), we get

$$\begin{aligned} |\left[\Pi (m)\right]_{\varepsilon }(x)-\Pi (m_{\varepsilon }(x))|\lesssim {\Vert D^2\Pi \Vert }_{L^\infty }\int _{B_{\varepsilon }(0)}|m(x-z)-m_{\varepsilon }(x)|^2\rho _{\varepsilon }(z)\,\mathrm{d}z.\nonumber \\ \end{aligned}$$

(56)

Moreover, by Jensen’s inequality, we have

Plugging this estimate into (56) gives (54). $\quad \square$

Computations Needed in the Proof of Lemma 15

Lemma 18

Let $\varphi \in C^3({{\overline{B}}}_1)$ such that $\Delta \varphi =0$ in $B_1$ and $\Phi ^{\varphi }$ the corresponding harmonic entropy given by

$$\begin{aligned} \Phi ^{\varphi }(z)=\varphi (z)z+((iz)\cdot \nabla \varphi (z))iz\qquad \forall z\in {{\overline{B}}}_1. \end{aligned}$$

(57)

For any smooth map $w:\Omega \rightarrow {{\overline{B}}}_1$ we have

$$\begin{aligned} {{\,\mathrm{div}\,}}\Phi ^{\varphi }(w)&=A(w){{\,\mathrm{div}\,}}w + {{\,\mathrm{div}\,}}(({\left| w\right| }^2-1)B(w)) \\&\quad + \partial _2 B_1(w){{\,\mathrm{div}\,}}\Sigma _1(w) -\partial _1 B_1(w) {{\,\mathrm{div}\,}}\Sigma _2(w), \end{aligned}$$

where $A=A^\varphi :{{\overline{B}}}_1\rightarrow {{\mathbb {R}}}$ and $B=B^\varphi :{{\overline{B}}}_1\rightarrow {{\mathbb {R}}}^2$ are given by

$$\begin{aligned} A^\varphi (z)&=\varphi (z) -z_1\partial _1\varphi (z) - z_2\partial _2\varphi (z) \nonumber \\&\quad + z_1z_2 \Big [ \partial _{12}\varphi (z)- z_2\partial _{111}\varphi (z) + z_1\partial _{211}\varphi (z)\Big ] \nonumber \\&\quad +\frac{1}{2} (z_1^2-z_2^2)\Big [ \partial _{11}\varphi (z) + z_2\partial _{112}\varphi (z) + z_1\partial _{111}\varphi (z) \Big ], \end{aligned}$$

(58)

$$\begin{aligned} B^\varphi (z)&=\left( \begin{array}{c} \partial _1\varphi (z) +\frac{1}{2} z_2\partial _{12}\varphi (z)-\frac{1}{2} z_1\partial _{22}\varphi (z)\\ \partial _2\varphi (z) -\frac{1}{2} z_2\partial _{11}\varphi (z)+\frac{1}{2} z_1\partial _{12}\varphi (z) \end{array} \right) . \end{aligned}$$

(59)

Proof of Lemma 18

We have

$$\begin{aligned} \Phi ^{\varphi }(w)\overset{(57)}{=}\left(\begin{array}{c} w_1 \varphi -w_2 \left(-w_2 \partial _1 \varphi +w_1 \partial _2 \varphi \right)\\ w_2 \varphi +w_1 \left(-w_2 \partial _1 \varphi +w_1 \partial _2 \varphi \right) \end{array}\right), \end{aligned}$$

so

$$\begin{aligned}&{{\,\mathrm{div}\,}}\Phi ^{\varphi }(w)\nonumber \\&\quad = w_1 \partial _1 \varphi \partial _1 w_1+ w_1 \partial _2 \varphi \partial _1 w_2+\varphi \partial _1 w_1\nonumber \\&\quad \quad -\left(-w_2 \partial _1 \varphi + w_1 \partial _2 \varphi \right) \partial _1 w_2\nonumber \\&\quad \quad -w_2\left( -\partial _1 \varphi \partial _1 w_2- w_2 \partial _{11} \varphi \partial _1 w_1- w_2 \partial _{12} \varphi \partial _1 w_2\right.\nonumber \\&\quad \quad \quad \quad \quad \left.+\partial _2 \varphi \partial _1 w_1+ w_1 \partial _{12} \varphi \partial _1 w_1+ w_1 \partial _{22} \varphi \partial _1 w_2\right)\nonumber \\&\quad \quad + w_2 \partial _1 \varphi \partial _2 w_1+ w_2 \partial _2 \varphi \partial _2 w_2+\varphi \partial _2 w_2\nonumber \\&\quad \quad +\left(-w_2 \partial _1 \varphi + w_1 \partial _2 \varphi \right) \partial _2 w_1\nonumber \\&\quad \quad + w_1\left( -\partial _1 \varphi \partial _2 w_2- w_2 \partial _{11} \varphi \partial _2 w_1- w_2 \partial _{12} \varphi \partial _2 w_2\right.\nonumber \\&\quad \quad \quad \quad \quad \left.+\partial _2 \varphi \partial _2 w_1+ w_1 \partial _{12} \varphi \partial _2 w_1+ w_1 \partial _{22} \varphi \partial _2 w_2\right)\nonumber \\&\quad = \left(\varphi +w_1 \partial _1 \varphi - w_2 \partial _2 \varphi +w_2^2 \partial _{11} \varphi - w_1 w_2 \partial _{12} \varphi \right) \partial _1 w_1\nonumber \\&\quad \quad + \left(\varphi +w_2 \partial _2 \varphi - w_1 \partial _1 \varphi +w_1^2 \partial _{22} \varphi - w_1 w_2 \partial _{12} \varphi \right) \partial _2 w_2\nonumber \\&\quad \quad +\left(w_1 \partial _2 \varphi + w_2 \partial _1 \varphi -w_1 \partial _2 \varphi +w_2^2 \partial _{12} \varphi - w_1 w_2 \partial _{22} \varphi +w_2 \partial _1 \varphi \right) \partial _1 w_2\nonumber \\&\quad \quad +\left(w_2 \partial _1 \varphi - w_2 \partial _1 \varphi +w_1 \partial _2 \varphi -w_1 w_2 \partial _{11} \varphi + w_1^2 \partial _{12} \varphi + w_1 \partial _2 \varphi \right) \partial _2 w_1\nonumber \\&\quad = \left(\varphi +w_1 \partial _1 \varphi - w_2 \partial _2 \varphi +w_2^2 \partial _{11} \varphi - w_1 w_2 \partial _{12} \varphi \right) \partial _1 w_1\nonumber \\&\quad \quad + \left(\varphi +w_2 \partial _2 \varphi - w_1 \partial _1 \varphi +w_1^2 \partial _{22} \varphi - w_1 w_2 \partial _{12} \varphi \right) \partial _2 w_2\nonumber \\&\quad \quad +\left(2\partial _1 \varphi +w_2 \partial _{12} \varphi - w_1 \partial _{22} \varphi \right) w_2 \partial _1 w_2\nonumber \\&\quad \quad +\left(2\partial _2 \varphi -w_2 \partial _{11} \varphi + w_1 \partial _{12} \varphi \right) w_1 \partial _2 w_1. \end{aligned}$$

(60)

Noting that

$$\begin{aligned} \partial _1 \left(\frac{\left|w\right|^2}{2}\right)-w_1 \partial _1 w_1=w_2 \partial _1 w_2\quad \text { and }\quad \partial _2 \left(\frac{\left|w\right|^2}{2}\right)-w_{2} \partial _2 w_2=w_1 \partial _2 w_1, \end{aligned}$$

and plugging this into (60),

$$\begin{aligned}&{{\,\mathrm{div}\,}}\Phi ^{\varphi }(w)\nonumber \\&\quad = \left(\varphi +w_1 \partial _1 \varphi - w_2 \partial _2 \varphi +w_2^2 \partial _{11} \varphi - w_1 w_2 \partial _{12} \varphi \right.\nonumber \\&\quad \quad \quad \quad \left. -2 w_1 \partial _1 \varphi -w_1 w_2 \partial _{12} \varphi +w_1^2 \partial _{22} \varphi \right) \partial _1 w_1\nonumber \\&\quad \quad + \left(\varphi +w_2 \partial _2 \varphi - w_1 \partial _1 \varphi +w_1^2 \partial _{22} \varphi - w_1 w_2 \partial _{12} \varphi \right.\nonumber \\&\quad \quad \quad \quad \left. -2 w_2 \partial _2 \varphi -w_1 w_2 \partial _{12} \varphi +w_2^2 \partial _{11} \varphi \right) \partial _2 w_2\nonumber \\&\quad \quad + \left(\partial _1 \varphi +\frac{1}{2} w_2 \partial _{12} \varphi - \frac{1}{2} w_1 \partial _{22} \varphi \right)\partial _1 \left(\left|w\right|^2\right)\nonumber \\&\quad \quad + \left(\partial _2 \varphi -\frac{1}{2} w_2 \partial _{11} \varphi + \frac{1}{2} w_1 \partial _{12} \varphi \right)\partial _2 \left(\left|w\right|^2\right)\nonumber \\&\quad =\left(\varphi -w_1 \partial _1 \varphi -w_2 \partial _2 \varphi +w_1^2 \partial _{22} \varphi +w_2^2 \partial _{11} \varphi -2 w_1 w_2\partial _{12} \varphi \right) {{\,\mathrm{div}\,}}w\nonumber \\&\quad \quad + \left(\partial _1 \varphi +\frac{1}{2} w_2 \partial _{12} \varphi - \frac{1}{2} w_1 \partial _{22} \varphi \right)\partial _1 \left(\left|w\right|^2\right)\nonumber \\&\quad \quad + \left(\partial _2 \varphi -\frac{1}{2} w_2 \partial _{11} \varphi + \frac{1}{2} w_1 \partial _{12} \varphi \right)\partial _2 \left(\left|w\right|^2\right) \nonumber \\&\quad = C(w){{\,\mathrm{div}\,}}w + B(w)\cdot \nabla ({\left| w\right| }^2), \end{aligned}$$

(61)

where $B=B^\varphi$ is as in (59) and

$$\begin{aligned} C(z)=\varphi -z_1 \partial _1 \varphi -z_2 \partial _2 \varphi +z_1^2 \partial _{22} \varphi +z_2^2 \partial _{11} \varphi -2 z_1 z_2\partial _{12} \varphi . \end{aligned}$$

(62)

We rewrite (61) as

$$\begin{aligned} {{\,\mathrm{div}\,}}\Phi ^{\varphi }(w)&=C(w) {{\,\mathrm{div}\,}}w+ {{\,\mathrm{div}\,}}\left(\left(\left|w\right|^2-1\right)B(w)\right)\nonumber \\&\quad +\left(\partial _1 B_1 \partial _1 w_1+ \partial _2 B_1 \partial _1 w_2+ \partial _1 B_2 \partial _2 w_1+ \partial _2 B_2 \partial _2 w_2 \right)\left( 1-\left|w\right|^2\right). \end{aligned}$$

(63)

Note that

$$\begin{aligned} \partial _1 B_1&=\partial _{11} \varphi +\frac{1}{2} z_2 \partial _{112} \varphi - \frac{1}{2} \partial _{22}\varphi -\frac{1}{2} z_{1}\partial _{221} \varphi , \end{aligned}$$

(64)

$$\begin{aligned} \partial _2 B_2&=\partial _{22} \varphi -\frac{1}{2} z_2 \partial _{112} \varphi - \frac{1}{2} \partial _{11} \varphi +\frac{1}{2} z_{1}\partial _{221} \varphi , \nonumber \\ \text {so }\quad&\partial _1 B_1+\partial _2 B_2=\frac{1}{2}\Delta \varphi , \end{aligned}$$

(65)

and

$$\begin{aligned} \partial _2 B_1&=\frac{3}{2}\partial _{12} \varphi +\frac{1}{2}z_2 \partial _{122} \varphi -\frac{1}{2}z_1 \partial _{222} \varphi , \end{aligned}$$

(66)

$$\begin{aligned} \partial _1 B_2&=\frac{3}{2}\partial _{12} \varphi -\frac{1}{2}z_2 \partial _{111} \varphi +\frac{1}{2}z_1 \partial _{112} \varphi ,\nonumber \\ \text {so }\quad&\partial _2 B_1-\partial _1 B_2=\frac{1}{2}z_2\left(\partial _1 \Delta \varphi \right)-\frac{1}{2}z_1\left(\partial _2 \Delta \varphi \right). \end{aligned}$$

(67)

Since $\varphi$ is harmonic we have from (65) and (67) that

$$\begin{aligned} \partial _1 B_1+\partial _2 B_2=0\quad \text { and }\quad \partial _2 B_1-\partial _1 B_2=0. \end{aligned}$$

(68)

Recalling moreover the explicit expressions of ${{\,\mathrm{div}\,}}\Sigma _j(w)$ computed in (25), (26), we find that (63) can be rewritten as

$$\begin{aligned} {{\,\mathrm{div}\,}}\Phi ^{\varphi }(w)&\overset{(68), (63)}{=} C(w) {{\,\mathrm{div}\,}}w+ {{\,\mathrm{div}\,}}\left(\left(\left|w\right|^2-1\right)B(w)\right)\nonumber \\&\quad +\partial _1 B_1\left(\partial _1 w_1-\partial _2 w_2\right)\left(1-\left|w\right|^2\right)\nonumber \\&\quad +\partial _2 B_1\left(\partial _1 w_2+\partial _2 w_1\right)\left(1-\left|w\right|^2\right)\nonumber \\&\overset{(25)-(26)}{=}C(w) {{\,\mathrm{div}\,}}w+ {{\,\mathrm{div}\,}}\left(\left(\left|w\right|^2-1\right)B(w)\right)\nonumber \\&\quad -\partial _1 B_1\left({{\,\mathrm{div}\,}}\Sigma _2(w)-(w_1^2-w_2^2){{\,\mathrm{div}\,}}w\right)\nonumber \\&\quad +\partial _2 B_1\left({{\,\mathrm{div}\,}}\Sigma _1(w)+2w_1 w_2{{\,\mathrm{div}\,}}w\right)\nonumber \\&=A(w){{\,\mathrm{div}\,}}w+{{\,\mathrm{div}\,}}\left(\left(\left|w\right|^2-1\right)B\right)\nonumber \\&\quad +\partial _2 B_1 {{\,\mathrm{div}\,}}\Sigma _1(w)-\partial _1 B_1 {{\,\mathrm{div}\,}}\Sigma _2(w), \end{aligned}$$

(69)

where

$$\begin{aligned} A(w)&=C(w)+2\partial _2 B_1 w_1 w_2+\partial _1 B_1 \left(w_1^2-w_2^2\right)\\&\overset{(62), (66), (64)}{=}\varphi -w_1 \partial _1 \varphi -w_2 \partial _2 \varphi +w_1^2 \partial _{22} \varphi +w_2^2 \partial _{11} \varphi -2 w_1 w_2\partial _{12}\varphi \\&\quad +2w_1 w_2\left(\frac{3}{2}\partial _{12} \varphi +\frac{1}{2}w_2 \partial _{122} \varphi -\frac{1}{2}w_1 \partial _{222} \varphi \right)\\&\quad +\left(w_1^2-w_2^2\right)\left(\partial _{11} \varphi +\frac{1}{2} w_2 \partial _{112} \varphi - \frac{1}{2} \partial _{22}\varphi -\frac{1}{2} w_{1}\partial _{221} \varphi \right)\\&=\varphi -w_1 \partial _1 \varphi -w_2 \partial _2 \varphi \\&\quad +2w_1 w_2\left(\frac{1}{2}\partial _{12} \varphi +\frac{1}{2}w_2 \partial _{122} \varphi -\frac{1}{2}w_1 \partial _{222} \varphi \right)\\&\quad +\left(w_1^2-w_2^2\right)\left(\frac{1}{2}\partial _{11} \varphi +\frac{1}{2} w_2 \partial _{112} \varphi -\frac{1}{2} w_{1}\partial _{221} \varphi \right). \end{aligned}$$

This expression agrees with (58) because $\Delta \varphi =0$, so (69) proves Lemma 18. $\quad \square$