Lipschitz Solutions for the Gradient Flow of Polyconvex Functionals

TL;DR

This paper constructs specific smooth polyconvex functions to demonstrate that the gradient flow of such functionals can be highly ill-posed, with solutions converging weakly to non-solutions.

Contribution

It provides explicit examples of smooth polyconvex functions satisfying certain conditions that lead to ill-posed gradient flow problems.

Findings

Gradient flow problems can be ill-posed for certain polyconvex functionals.

Existence of Lipschitz weak solutions that do not converge to actual solutions.

Ill-posedness persists even with smooth initial-boundary data.

Abstract

In this sequel to a previous paper, we construct certain smooth strongly polyconvex functions $F$ on $\mathbb M^{2\times 2}$ such that $\sigma=DF$ satisfies the Condition (OC) in that paper. As a result, we show that the initial-boundary value problem for the gradient flow of such polyconvex energy functionals is highly ill-posed even for some smooth initial-boundary data in the sense that the problem possesses a weakly* convergent sequence of Lipschitz weak solutions whose limit is not a weak solution.

Equations

$$\mathcal{E}(u)={\int }_{\Omega }F(Du(x))dx,$$

$${\int }_{\Omega }(F(A+D\varphi (x))-F(A))dx\ge \frac{\nu }{2}{\int }_{\Omega }|D\varphi (x){|}^{2}dx$$

$$⟨DF(A+p\otimes \alpha )-DF(A),p\otimes \alpha ⟩\ge \nu |p{|}^2|\alpha {|}^2$$

$$\sum _{i,j=1}^m\sum _{k,l=1}^n\frac{{\partial }^{2}F(A)}{\partial a_{ik}\partial a_{jl}}{p}_i{p}_j{\alpha }_k{\alpha }_l\ge \nu |p{|}^2|\alpha {|}^2.$$

$$\operatorname{div}DF(Du)=0\quad\mbox{in $\Omega.$}$$

$$\begin{cases}u_{t}=\operatorname{div}DF(Du)\;\;\;\mbox{in $\Omega_{T}=\Omega\times(0,T)$,}\\ \hfill u(x,t)=u_{0}(x)\;(x\in\partial\Omega,\,0<t<T),\;\;u(x,0)=u_{0}(x)&(x\in\Omega).\end{cases}$$

$${\mathcal{I}}_{j+1}^h(u)=\mathcal{E}(u)+\frac{1}{2h}\Vert u-u_j^h{\Vert }_{L^2(\Omega )}^2,h>0,j=0,1,\dots ,$$

$$\begin{cases}u_{t}-\operatorname{div}DF(Du)=f\quad\mbox{in $\Omega_{T}$,}\\ u(x,t)=0\;(x\in\partial\Omega,\,0<t<T),\;\;u(x,0)=0\;(x\in\Omega),\end{cases}$$

$$\sum _{j=1}^N{p}_j\otimes {\alpha }_j=0,\sum _{j=1}^N{q}_j\otimes {\alpha }_j\otimes {\alpha }_j=0.$$

$$\mathcal{L}([A,(b^i)])=\left[\begin{array}{c}A\\ BJ\end{array}\right]\forall B=(b_k^i)\in {\mathbb{M}}^{2\times 2}.$$

$$DF(A)=ϵA+G_A(\tilde{A})+G_s(\tilde{A})\mathrm{cof}A,$$

$$K_F=\mathcal{L}({\mathbb{K}}_F)=\left\{\left[\begin{array}{c}A\\ DF(A)J\end{array}\right]:A\in {\mathbb{M}}^{2\times 2}\right\}.$$

$$ϵA_j+G_A({\tilde{A}}_j)+G_s({\tilde{A}}_j)\mathrm{cof}{A}_j=-B_{j}J(j=1,\dots ,N).$$

$$c_j-c_i>⟨Q_i,A_j-A_i⟩+d_i(\det A_j-\det A_i)\forall i\ne j.$$

$$c_i-c_j+d_i\det (A_i-A_j)+⟨A_i-A_j,B_{i}J⟩<-ϵ⟨A_i,A_i-A_j⟩\forall i\ne j.$$

$$c_i-c_j+d_i\det (A_i-A_j)+⟨A_i-A_j,B_{i}J⟩<0\forall i\ne j,$$

$$C_j=\left(\begin{array}{c}{p}_j\\ ({\alpha }_j\cdot \delta )q_j\end{array}\right)\otimes {\alpha }_j(j=1,\dots ,5),$$

$${\alpha }_1=(-1,0),{\alpha }_2=(0,-1),{\alpha }_3=(1,z_3),{\alpha }_4=(1,z_4),{\alpha }_5=(y_5,1),$$

$$\{\begin{array}{l}{p}_1=p_3+p_4+y_5{p}_5,p_2=z_3{p}_3+z_4{p}_4+p_5,\\ q_1=(z_4/z_3-1)q_4+(y_5/z_3-y_5^2)q_5,\\ q_2=(z_3{z}_4-z_4^2)q_4+(y_5{z}_3-1)q_5,q_3=-(z_4/z_3)q_4-(y_5/z_3)q_5.\end{array}$$

$$\{\begin{array}{l}{y}_5=2,z_3=1,z_4=4,\\ p_3=(1,0),p_4=(1,1),p_5=(0,1),\\ {\kappa }_1=2,{\kappa }_2=3,{\kappa }_3=4,{\kappa }_4=3,{\kappa }_5=2.\end{array}$$

$$q_4=(-19,0),q_5=(-63,-82).$$

$$p_1=(2,3),p_2=(5,5),q_1=(69,164),q_2=(165,-82),q_3=(202,164),$$

$$C_1^0=\left(\begin{array}{cc}-2& 0\\ -3& 0\\ 69& 0\\ 164& 0\end{array}\right),C_2^0=\left(\begin{array}{cc}0& -5\\ 0& -5\\ 0& 165\\ 0& -82\end{array}\right),C_3^0=\left(\begin{array}{cc}1& 1\\ 0& 0\\ 404& 404\\ 328& 328\end{array}\right),$$

$$C_4^0=\left(\begin{array}{cc}1& 4\\ 1& 4\\ -95& -380\\ 0& 0\end{array}\right),C_5^0=\left(\begin{array}{cc}0& 0\\ 2& 1\\ -378& -189\\ -492& -246\end{array}\right).$$

$$A_1^0=\left(\begin{array}{cc}-4& 0\\ -6& 0\end{array}\right),A_2^0=\left(\begin{array}{cc}-2& -15\\ -3& -15\end{array}\right),A_3^0=\left(\begin{array}{cc}2& -1\\ -3& -5\end{array}\right),$$

$$A_4^0=\left(\begin{array}{cc}2& 8\\ 0& 7\end{array}\right),A_5^0=\left(\begin{array}{cc}0& 0\\ 2& 1\end{array}\right).$$

$$B_1^0=\left(\begin{array}{cc}138& 0\\ 328& 0\end{array}\right),B_2^0=\left(\begin{array}{cc}69& 495\\ 164& -246\end{array}\right),B_3^0=\left(\begin{array}{cc}1685& 1781\\ 1476& 1230\end{array}\right),$$

$$B_4^0=\left(\begin{array}{cc}188& -571\\ 492& 246\end{array}\right),B_5^0=\left(\begin{array}{cc}-378& -189\\ -492& -246\end{array}\right).$$

$$(c_1^0,\dots ,c_5^0,d_1^0,\dots ,d_5^0)=(0,-3650,-3318,5044,580,58,-7.5,772,57,376).$$

$$\{\begin{array}{l}{P}^0=0,{\delta }^0=(1,1),\\ {\alpha }_1^0=(-1,0),{\alpha }_2^0=(0,-1),{\alpha }_3^0=(1,1),{\alpha }_4^0=(1,4),{\alpha }_5^0=(2,1),\\ p_1^0=(2,3),p_2^0=(5,5),p_3^0=(1,0),p_4^0=(1,1),p_5^0=(0,1),\\ q_1^0=(69,164),q_2^0=(165,-82),q_3^0=(202,164),\\ q_4^0=(-19,0),q_5^0=(-63,-82),\\ {\kappa }_1^0=2,{\kappa }_2^0=3,{\kappa }_3^0=4,{\kappa }_4^0=3,{\kappa }_5^0=2.\end{array}$$

Full text

1. Introduction

Let $m,n\geq 2$ be integers and $\mathbb{M}^{m\times n}$ be the space of $m\times n$ real matrices. Given a function $F\colon\mathbb{M}^{m\times n}\to\mathbb{R}$, we consider the energy functional

[TABLE]

where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with Lipschitz boundary $\partial\Omega$ and $u\colon\Omega\to\mathbb{R}^{m}.$ Here, if $u=(u^{1},\dots,u^{m}),$ $Du=(\frac{\partial u^{i}}{\partial x_{k}})$ denotes the Jacobian matrix of $u.$ It is well-known that weak lower semicontinuity of functional $\mathcal{E}$ in a Sobolev space is equivalent to Morrey’s quasiconvexity of function $F;$ see, e.g., Acerbi & Fusco [1], Ball [3], Dacorogna [7] and Morrey [18]. Also closely related to regularity of the energy minimizers of $\mathcal{E}$ is a stronger quasiconvexity condition (see, e.g., Evans [11]); we say that $F$ is strongly quasiconvex if, for some constant $\nu>0$,

[TABLE]

holds for all $A\in\mathbb{M}^{m\times n}$ and $\phi\in C_{c}^{\infty}(\Omega;\mathbb{R}^{m}).$ (If $\nu=0$, this condition becomes the usual Morrey’s quasiconvexity.) Since $m,n\geq 2,$ it is well-known that a quasiconvex function may not be convex.

If $F$ is $C^{1}$, then condition (1.2) implies

[TABLE]

for all $A\in\mathbb{M}^{m\times n},p\in\mathbb{R}^{m},$ and $\alpha\in\mathbb{R}^{n}.$ Note that, for $C^{2}$ functions $F$, condition (1.3) is equivalent to the uniform strong Legendre-Hadamard condition:

[TABLE]

Furthermore, for a $C^{1}$ function $F$ satisfying certain natural growth conditions, any minimizer of energy $\mathcal{E}$ in a Dirichlet class is a weak solution of the Euler-Lagrange system:

[TABLE]

If $F$ is a $C^{2}$ strongly quasiconvex function, then it is well-known that every Lipschitz minimizer of $\mathcal{E}$ is of $C^{1,\alpha}(\Omega_{0};\mathbb{R}^{m})$ for all $0<\alpha<1$ on some open set $\Omega_{0}$ with $|\Omega\setminus\Omega_{0}|=0;$ see Evans [11] and Fusco & Hutchinson [13]. In a sharp contrast to such partial regularity of energy minimizers, Müller and Šverák [20] proved that, for certain smooth (at least $C^{2}$) strongly quasiconvex functions $F$ on $\mathbb{M}^{2\times 2},$ the Euler-Lagrange system (1.5) has Lipschitz weak solutions that are nowhere $C^{1}$ in $\Omega;$ such a result has been extended by Székelyhidi [21] to certain strongly polyconvex functions $F(A)=\frac{\nu}{2}|A|^{2}+G(A,\det A)$ on $\mathbb{M}^{2\times 2},$ where $G(A,s)$ is smooth and convex on $(A,s)\in\mathbb{M}^{2\times 2}\times\mathbb{R}.$ We refer to [4, 6, 17] for further related partial regularity results.

In this paper, we study a parabolic companion of system (1.5). More specifically, given $T>0$ and initial function $u_{0}\colon\bar{\Omega}\to\mathbb{R}^{m}$, we study the initial-boundary value problem for the $L^{2}$ gradient flow of energy $\mathcal{E}$ given by

[TABLE]

For convex functions $F$, standard monotonicity and semigroup methods of parabolic theory would apply to problem (1.6) (see Brézis [5]); in this case, one easily sees that (1.6) possesses at most one weak solution.

There is no general theory on solvability of problem (1.6) under the sole condition (1.3). However, for certain quasiconvex functions $F,$ (1.6) may be studied by a time-discretization method based on recursively minimizing the energies

[TABLE]

where $u^{h}_{j}$ is a minimizer of $\mathcal{I}_{j}^{h}$ with given boundary data $u_{0}.$ We refer to Ambrosio, Gigli & Savaré [2] for general study of such a method. In general, when $h\to 0$, the time-discretization scheme only produces the so-called generalized minimizing movements (see [2]) or the Young measure solutions to (1.6) (see, e.g., Kinderlehrer & Pedregal [15] and Demoulini [8]). Existence of the true weak solutions to (1.6) remains essentially open even for general polyconvex functions $F;$ see Evans, Savin & Gangbo [12] for a special case of $F=\Phi(\det A)$. The same existence issue has also remained open for elastodynamics problems as studied in [9, 10], but see [14] for a recent result on existence of the true weak solutions for one-dimensional nonconvex elastodynamics.

A parabolic problem similar to problem (1.6) has been studied through the stationary elliptic system (1.5) by Müller, Rieger and Šverák in [19], where they proved that for certain smooth quasiconvex functions $F$, for all $\epsilon>0$ and $\alpha\in(0,1)$, the elliptic system (1.5) in $\Omega\subset\mathbb{R}^{2}$ possesses a family of Lipschitz but nowhere $C^{1}$ weak solutions $u(\cdot,t)$ for each $t\in(0,T)$ satisfying $u(\cdot,0)=0,\,u(x,t)=0\,(x\in\partial\Omega)$ and $\|u_{t}\|_{C^{\alpha}(\Omega_{T})}<\epsilon.$ Such a function $u=u(x,t)$ is thus a Lipschitz weak solution to the problem

[TABLE]

with right-hand side $f\equiv u_{t}\in C^{\alpha}(\Omega_{T};\mathbb{R}^{2})$ satisfying $\|f\|_{C^{\alpha}}<\epsilon.$ However, their method cannot be adapted to the gradient flow problem (1.6), where $f=0$.

The main purpose of the present paper is establish that even for smooth strongly polyconvex functions $F$ and smooth initial-boundary functions $u_{0}$ the initial-boundary value problem (1.6) may be highly ill-posed. Our main result is the following nonuniqueness and instability theorem.

Theorem 1.1**.**

There exist smooth strongly polyconvex functions $F$ on $\mathbb{M}^{2\times 2}$ and smooth functions $u_{0}$ such that problem $(\ref{ibvp-1})$ possesses a sequence of Lipschitz weak solutions that converges weakly to a function which is not a Lipschitz weak solution itself.*

We point out that in the theorem one can choose $u_{0}(x)=Ax$ for some $A\in\mathbb{M}^{2\times 2}.$ In this case, quasiconvexity condition (1.2) and the standard energy identity for $C^{2}$ solutions imply that the only $C^{2}$ solution to problem (1.6) is $u(x,t)\equiv Ax;$ thus, the Lipschitz weak solutions consisting of the sequence stated in the theorem will eventually be all distinct and non-$C^{2}$ on $\bar{\Omega}_{T},$ yielding infinitely many nonclassical, nonstationary Lipschitz weak solutions to problem (1.6).

Theorem 1.1 follows from the general results of Yan [22, Theorem 1.3 and Corollary 1.4] once we have constructed the smooth strongly polyconvex functions $F\colon\mathbb{M}^{2\times 2}\to\mathbb{R}$ with $\sigma=DF$ satisfying the Condition (OC) given by [22, Definition 3.2]. Our construction (see Proposition 2.1) relies on finding a special $\tau_{5}$-configuration supported by a smooth strongly polyconvex function. The search for such special $\tau_{5}$-configuration and polyconvex function is accomplished similarly as in Székelyhidi [21, Lemma 3]; however, in our case, we need to solve a more restrictive linear programming problem and rely on more specific results (see Lemmas 4.1 and 4.5) obtained by some necessary MATLAB computations.

2. Special $\tau_{5}$-configuration supported by a polyconvex function

We refer to Yan [22, Sections 2 and 3] for certain definitions and notations. In particular, for $N\geq 3,$ let $M^{\prime}_{N}$ be the set of $T_{N}$-configurations $(X_{1},\dots,X_{N})$ in $\mathbb{M}^{4\times 2}$ whose determining rank-one matrices are given by $C_{j}=\begin{pmatrix}p_{j}\\ (\alpha_{j}\cdot\delta)q_{j}\end{pmatrix}\otimes\alpha_{j},$ where $p_{j},q_{j},\alpha_{j},\delta\in\mathbb{R}^{2}$, $\alpha_{j}\neq 0$, at least three of $\alpha_{j}$’s are mutually non-collinear, and

[TABLE]

Let $J=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ and let $\mathcal{L}\colon\mathbb{M}^{2\times 2}\times(\mathbb{R}^{2})^{2}\to\mathbb{M}^{4\times 2}$ be defined by

[TABLE]

Define $\mathcal{M}^{\prime}_{N}=\mathcal{L}^{-1}(M_{N}^{\prime})$ to be the set of special $\tau_{N}$-configurations in $\mathbb{M}^{2\times 2}\times(\mathbb{R}^{2})^{2}$ (see [22, Definition 3.3]).

Consider a function $F\colon\mathbb{M}^{2\times 2}\to\mathbb{R}$ given by $F(A)=\frac{\epsilon}{2}|A|^{2}+G(A,\det A),$ where $G=G(A,s)$ is smooth on $\mathbb{M}^{2\times 2}\times\mathbb{R}.$ Then

[TABLE]

where $\tilde{A}=(A,\det A)$ and $\operatorname{cof}A=\begin{pmatrix}a_{22}&-a_{21}\\ -a_{12}&a_{11}\end{pmatrix}$ for $A=\begin{pmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix}.$ Let $\mathbb{K}_{F}=\{[A,(DF(A))]:A\in\mathbb{M}^{2\times 2}\}$ be the graph of $DF$ and define

[TABLE]

Let $K=K_{F}$ and $K_{N}=K\times\dots\times K$ ($N$ copies). Suppose $(X_{1},\dots,X_{N})\in M^{\prime}_{N}\cap K_{N}$ with $X_{j}=\begin{bmatrix}A_{j}\\ B_{j}\end{bmatrix}$. Then

[TABLE]

As in [21], if we are given $c_{j},d_{j}\in\mathbb{R}$ and $Q_{j}\in\mathbb{M}^{2\times 2}$ for $j=1,\dots,N$, then it is well known that there exists a smooth convex function $G\colon\mathbb{M}^{2\times 2}\times\mathbb{R}\to\mathbb{R}$ satisfying $G(\tilde{A}_{j})=c_{j},\,G_{A}(\tilde{A}_{j})=Q_{j}$ and $G_{s}(\tilde{A}_{j})=d_{j}$ provided the following condition holds:

[TABLE]

If $G_{A}(\tilde{A}_{j})=Q_{j}$ and $G_{s}(\tilde{A}_{j})=d_{j}$ also satisfy (2.4), then (2.5) becomes

[TABLE]

In particular, if

[TABLE]

then we can choose $\epsilon>0$ sufficiently small so that (2.6) is satisfied.

Proposition 2.1**.**

Let $N=5.$ Then there exists $(X_{1},\dots,X_{5})\in M^{\prime}_{5}$ with $X_{j}=\begin{bmatrix}A_{j}\\ B_{j}\end{bmatrix}$ such that $(\ref{ineq-3})$ holds for some constants $c_{1},\dots,c_{5}$ and $d_{1},\dots,d_{5}.$

Proof.

Suppose $(X_{1},\dots,X_{5})\in M^{\prime}_{5}$ is determined by $P,\,C_{i}$ and $\kappa_{i}$ such that $(\ref{ineq-3})$ holds for some constants $c_{1},\dots,c_{5}$ and $d_{1},\dots,d_{5}.$ Then, by adjusting the constants, we may assume $P=0,\,c_{1}=0.$ Let

[TABLE]

where $\alpha_{j}\neq 0$ and at least three $\alpha_{j}$’s are mutually noncollinear. We attempt to find such a $T_{5}$-configuration with $\delta=(1,1)$ and

[TABLE]

where $z_{3}\neq 0.$ From condition (2.1), we solve $p_{1},\,p_{2}$ as linear combinations of $p_{3},p_{4},p_{5}$ and solve $q_{1},q_{2},q_{3}$ as linear combinations of $q_{4},q_{5}$ to obtain

[TABLE]

Then $\delta_{ij}=\det(A_{i}-A_{j})$ becomes a function of $y_{5},z_{3},z_{4},\kappa_{i},p_{3},p_{4}$ and $p_{5}$, while $\mu_{ij}=\langle A_{i}-A_{j},B_{i}J\rangle$ becomes a linear function of $q_{4},q_{5}$ with coefficients depending on 14 free-variables $y_{5},z_{3},z_{4},\kappa_{i},p_{3},p_{4}$ and $p_{5}$. With a fixed choice of $(y_{5},z_{3},z_{4},\kappa_{i},p_{3},p_{4},p_{5})\in\mathbb{R}^{14},$ condition (2.7) becomes a system of 20 linear inequalities on 13 variables: $c_{2},\dots,c_{5},\,d_{1},\dots,d_{5},\,q_{4},\,q_{5}.$ The matrix representing the LHS of (2.7) is a $20\times 13$ matrix $A$ depending on $(y_{5},\,z_{3},\,z_{4},\,\kappa_{i},\,p_{3},\,p_{4},\,p_{5}).$ Using the linear programming by MATLAB, after numerous trials on different choices of $(y_{5},\,z_{3},\,z_{4},\,\kappa_{i},\,p_{3},\,p_{4},\,p_{5})$ for the matrix $A$, the linear inequality $AX<0$ finally becomes feasible with

[TABLE]

With this choice of parameters, a feasible solution $X$ for $AX<0$ gives

[TABLE]

Hence by (2.8) we have

[TABLE]

and therefore the rank-one matrices $C^{0}_{j}=\begin{pmatrix}p_{j}\\ (\alpha_{j}\cdot\delta)q_{j}\end{pmatrix}\otimes\alpha_{j}$ are given by

[TABLE]

[TABLE]

The corresponding $T_{5}$-configuration $(X^{0}_{1},\dots,X^{0}_{5})\in M^{\prime}_{5}$ is illustrated in Fig. 1, where $X^{0}_{i}=\begin{pmatrix}A^{0}_{i}\\ B^{0}_{i}\end{pmatrix}$ with

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For these $A^{0}_{j}$ and $B^{0}_{j}$, it is straightforward (with the aid of MATLAB) to verify that condition (2.7) is satisfied with the constants:

[TABLE]

This completes the proof.

∎

Remark 2.1**.**

Our construction is substantially different from that of [21, Lemma 3] because the rank-one matrices $C_{j}^{0}$ satisfy more restrictive conditions. The special and more restrictive $T_{5}$-configuration constructed above provides a correct $T_{5}$-configuration for [21, Example 1] as the example of $T_{5}$-configuration there was incorrectly printed.

Proposition 2.2**.**

Let $(X_{1}^{0},\dots,X_{5}^{0})\in M^{\prime}_{5}$ be the special $T_{5}$-configuration constructed above. Then for all sufficiently small $\epsilon>0$ there exists a smooth convex function $G\colon\mathbb{M}^{2\times 2}\times\mathbb{R}\to\mathbb{R}$ such that the strongly polyconvex function $F_{0}(A)=\frac{\epsilon}{2}|A|^{2}+G(A,\det A)$ satisfies that $X_{j}^{0}\in K_{F_{0}}$ for $j=1,\dots,5.$

Proof.

Let $c_{1}^{0},\dots,c_{5}^{0}$ and $d_{1}^{0},\dots,d_{5}^{0}$ be the constants determined above. Then (2.6) is satisfied for all sufficiently small $\epsilon>0.$ For such an $\epsilon>0$, let $Q^{0}_{j}=-\epsilon A^{0}_{j}-B^{0}_{j}J-d^{0}_{j}\operatorname{cof}A^{0}_{j}.$ Following the proof of [21, Lemma 3], there exists a smooth convex function $G\colon\mathbb{M}^{2\times 2}\times\mathbb{R}\to\mathbb{R}$ such that $G(\tilde{A}^{0}_{j})=c_{j}^{0}$, $G_{A}(\tilde{A}_{j}^{0})=Q_{j}^{0}$ and $G_{s}(\tilde{A}_{j}^{0})=d_{j}^{0}$ for $j=1,\dots,5.$ Let $F_{0}(A)=\frac{\epsilon}{2}|A|^{2}+G(A,\det A).$ Then $F_{0}$ is smooth and strongly polyconvex; clearly, for each $j=1,\dots,5$, one has $DF_{0}(A^{0}_{j})=\epsilon A^{0}_{j}+G_{A}(\tilde{A}^{0}_{j})+G_{s}(\tilde{A}^{0}_{j})\operatorname{cof}A^{0}_{j}=-B_{j}^{0}J,$ which proves $X_{j}^{0}\in K_{F_{0}}.$ ∎

3. The perturbations of $(X^{0}_{1},\dots,X^{0}_{5})$ and $F_{0}$

Let $(X^{0}_{1},\dots,X^{0}_{5})\in M^{\prime}_{5}$ and strongly polyconvex function $F_{0}$ be determined as in Propositions 2.1 and 2.2 above. The parameters of $(X^{0}_{1},\dots,X^{0}_{5})$ are listed as follows for later usage:

[TABLE]

We aim to construct the required strongly polyconvex functions $F$ as a perturbation of function $F_{0},$ with the set $K_{F}$ containing $(X^{0}_{1},\dots,X^{0}_{5})$ and all suitable perturbations of $(X^{0}_{1},\dots,X^{0}_{5}).$

First of all, to perturb the function $F_{0}$, let $B_{1}(0)$ be the unit ball in $\mathbb{M}^{2\times 2}$ and $\zeta\in C^{\infty}_{c}(B_{1}(0))$ be such that $0\leq\zeta(A)\leq 1$ and $\zeta(0)=1.$ Given $r>0$ and tensor $H=(H^{pqij})$ with $H^{pqij}=H^{ijpq}\in\mathbb{R}$ for all $i,j,p,q\in\{1,2\}$ (thus $H$ can be viewed as a $4\times 4$ symmetric matrix), define the function

[TABLE]

for $A=(a_{ij})\in\mathbb{M}^{2\times 2}.$ Then $V_{H,r}\in C^{\infty}_{c}(\mathbb{M}^{2\times 2})$ has support in $B_{r}(0),$ and

[TABLE]

where $C$ is a constant independent of $H,r.$

Let $r_{0}=\min_{i\neq j}|A_{i}^{0}-A_{j}^{0}|>0$. The functions $F$ will be constructed as a perturbation of $F_{0}$ of the form:

[TABLE]

with $\tilde{H}_{1},\dots,\tilde{H}_{5}$ to be chosen later. Note that

[TABLE]

thus, the specific $T_{5}$-configuration $(X_{1}^{0},\dots,X_{5}^{0})$ also lies on $K_{F}.$ Furthermore, $F$ is strongly polyconvex if $\sum_{j=1}^{5}|D^{2}V_{\tilde{H}_{j},r_{0}}(A)|<\epsilon/2,$ which will be satisfied if

[TABLE]

We now perturb the $T_{5}$-configuration $(X_{1}^{0},\dots,X_{5}^{0})$ around the parameters in (3.1) by the following 28 parameters (notice that we do not perturb $\delta_{0}$):

[TABLE]

Let $Y=(z_{1},y_{2},z_{3},z_{4},y_{5},p_{3},p_{4},p_{5},q_{4},q_{5},\kappa_{1},\dots,\kappa_{5})\in\mathbb{R}^{20},$ and define $p_{1},p_{2}$ and $q_{1},q_{2},q_{3}$ according to (2.1) by

[TABLE]

For $j=1,\dots,5,$ define $C_{j}=C_{j}(Y)=\begin{pmatrix}p_{j}\\ (\alpha_{j}\cdot\delta^{0})q_{j}\end{pmatrix}\otimes\alpha_{j}$, and for $\nu=1,\dots,5,$ define

[TABLE]

Henceforth, the indices $\nu,j$ appearing in $Z^{\nu}_{j}$, $C_{j}$ and $\kappa_{j}$ are always taken modulo 5. For all $\nu$ and $j$, define

[TABLE]

Then $(X^{\nu}_{1}(Y,Q),\dots,X^{\nu}_{5}(Y,Q))\in M^{\prime}_{5}$ for all $\nu$ and $(Y,Q)$. Moreover,

[TABLE]

where $Y^{0}=(0,0,1,4,2,p_{3}^{0},p_{4}^{0},p_{5}^{0},q_{4}^{0},q_{5}^{0},2,3,4,3,2)\in\mathbb{R}^{20}$ and $P_{\nu}^{0}$’s are the vertices of the pentagon as shown in Figure 1.

The following result is immediate.

Lemma 3.1**.**

Let $C_{j}=C_{j}(Y)$ be defined as above. Define

[TABLE]

Then $X^{\nu}_{j}(Y,Q)=X^{\nu+i-1}_{j-i+1}(Y,P^{\nu}_{i}(Y,Q))$ for all $\nu,j,i\mod 5,$ with $j\geq i.$

In what follows, we fix the identifications: $\mathbb{M}^{2\times 2}\cong\mathbb{R}^{4}$ and $\mathbb{M}^{4\times 2}\cong\mathbb{R}^{8}$ as follows, in the same way as used in MATLAB:

[TABLE]

Lemma 3.2**.**

For each $\nu=1,\dots,5$, the $8\times 20$ matrix $\dfrac{\partial Z^{\nu}_{1}}{\partial Y}$ has rank less than or equal to 5.

Proof.

Write $Z^{\nu}_{1}(Y)=\gamma_{\nu}\otimes\alpha_{\nu},$ where $\gamma_{\nu}=\kappa_{\nu}\begin{pmatrix}p_{\nu}\\ (\alpha_{\nu}\cdot\delta^{0})q_{\nu}\end{pmatrix}=(f_{i}(Y))\in\mathbb{R}^{4}.$ Set $\alpha_{\nu}=(a,b)\in\mathbb{R}^{2},$ where one of $a,b$ is constant and the other is a variable in $Y$. Thus, with the identification (3.8), as a $8\times 20$ matrix,

[TABLE]

Since one of $\partial a/\partial Y$ or $\partial b/\partial Y$ vanishes, we easily have $\operatorname{rank}\frac{\partial Z^{\nu}_{1}}{\partial Y}\leq 5.$ ∎

4. The construction of polyconvex functions $F$

Suppose that $F$ is a function defined by (3.3) with $\tilde{H}_{j}$ to be chosen. To study the set $K_{F}$, we define the function $\Phi\colon\mathbb{M}^{4\times 2}\cong\mathbb{R}^{8}\to\mathbb{M}^{2\times 2}\cong\mathbb{R}^{4}$ by

[TABLE]

where $X=\begin{bmatrix}A\\ B\end{bmatrix}\in\mathbb{M}^{4\times 2}\cong\mathbb{R}^{8}.$ Then $X\in K_{F}$ if and only if $\Phi(X)=0.$

With the identification (3.8), we have $A=PX$ and $BJ=EX$, where

[TABLE]

with $I=I_{2}$ ($2\times 2$ identity matrix) and $O=O_{2}$ ($2\times 2$ zero matrix). Therefore, as a $4\times 8$ matrix, we have

[TABLE]

where the tensor $D^{2}F(A)$ is viewed as a symmetric $4\times 4$ matrix. Hence

[TABLE]

For $\nu=1,\dots,5$, we define the functions

[TABLE]

Then $\Psi^{\nu}(Y^{0},P_{\nu}^{0})=0.$ By Lemma 3.1, we have, component-wise, modulo 5,

[TABLE]

To study the equation $\Psi^{\nu}(Y,Q)=0$ near $(Y^{0},P_{\nu}^{0}),$ note that, by (4.1) and (4.4), the partial Jacobian matrix $\dfrac{\partial\Psi^{\nu}}{\partial Y}(Y,Q)$ is a $20\times 20$ matrix given by

[TABLE]

Thus, the matrix $\frac{\partial\Psi^{\nu}}{\partial Y}(Y,Q)$ depends on the Hessian tensors $D^{2}F(PX^{\nu}_{k})$ $(k=1,\dots,5)$ and is otherwise independent of $F$ and $Q$. Define

[TABLE]

Since $X^{\nu}_{j}(Y^{0},P_{\nu}^{0})=X^{0}_{\nu+j-1}$ for all $\nu,j=1,\dots,5,$ we have

[TABLE]

and thus $J_{\nu}$ is a polynomial of $H_{1}=D^{2}F(A^{0}_{1}),\dots,H_{5}=D^{2}F(A_{5}^{0})$ with coefficients being independent of $F$. We write this polynomial as

[TABLE]

Lemma 4.1**.**

The polynomial $j_{\nu}(H_{1},\dots,H_{5})$ is not identically zero for each $\nu=1,\dots,5.$

Proof.

Given $s,t$, we consider the tensors (with $I=I_{2}$, $O=O_{2}$):

[TABLE]

and let $g_{\nu}(s,t)=j_{\nu}(h_{1}(s),h_{2}(t),h_{1}(s),h_{1}(s),h_{2}(t)).$ Then, using the jacobian computations by MATLAB, we verify that

[TABLE]

Therefore, the polynomial $j_{\nu}(H_{1},\dots,H_{5})$ is not identically zero for each $\nu=1,\dots,5.$ ∎

Since polynomial $j_{\nu}(H_{1},\dots,H_{5})$ is not identically zero for each $\nu=1,\dots,5,$ we select $(H_{1}^{0},\dots,H_{5}^{0})$ with the following properties:

[TABLE]

Lemma 4.2**.**

With $\tilde{H}_{j}$ chosen to satisfy $(\ref{ImFT-1})$, define $F$ by $(\ref{def-F})$ and $\Psi^{\nu}$ by $(\ref{Psi}).$ Then there exist positive numbers $\rho,\delta$ and smooth functions

[TABLE]

for $\nu=1,\dots,5$, such that $Y_{\nu}(P_{\nu}^{0})=Y^{0}$ and, for all $Y\in B_{\delta}(Y^{0})$ and $Q\in B_{\rho}(P_{\nu}^{0})$, it follows that

[TABLE]

Proof.

Note that $\Psi^{\nu}(Y^{0},P_{\nu}^{0})=0$ and, by (4.9),

[TABLE]

Thus, the conclusion follows from the implicit function theorem. ∎

Lemma 4.3**.**

Let $z^{\nu}(Q)=Z^{\nu}_{1}(Y_{\nu}(Q))$ for $Q\in B_{\rho}(P_{\nu}^{0})\subset\mathbb{R}^{8}.$ Then all eigenvalues of matrix $M=Dz^{\nu}(Q)\in\mathbb{M}^{8\times 8}$ are real numbers $\{-1,0,\mu_{M}\},$ where $\mu_{M}=4+\operatorname{tr}(M),$ with $-1$ being an eigenvalue of multiplicity at least 4, and 0 an eigenvalue of multiplicity at least 3.

Proof.

Clearly $M=Dz^{\nu}(Q)=\frac{\partial Z^{\nu}_{1}}{\partial Y}(Y_{\nu}(Q))\frac{\partial Y_{\nu}}{\partial Q}(Q).$ Differentiate the equation $\Psi^{\nu}(Y_{\nu}(Q),Q)=0$ with respect to $Q$ to have

[TABLE]

Thus

[TABLE]

Let $R_{j}=D\Phi(X^{\nu}_{j})=D\Phi(Q+Z^{\nu}_{j}(Y_{\nu}(Q)))\in\mathbb{M}^{4\times 8}$ and

[TABLE]

where each $T_{j}\in\mathbb{M}^{20\times 4}.$ Then

[TABLE]

and thus

[TABLE]

Therefore $R_{1}M=-R_{1}$ and so $R_{1}(I_{8}+M)=0.$ By (4.3), $\operatorname{rank}R_{1}=4;$ thus $\dim\ker(R_{1})=4$ and hence $\operatorname{rank}(I_{8}+M)\leq 4,$ which gives

[TABLE]

By Lemma 3.2, $\operatorname{rank}(S_{1})\leq 5;$ thus $\operatorname{rank}(M)\leq 5$ and hence

[TABLE]

By (4.14) and (4.15), $M$ has $-1$ as an eigenvalue of multiplicity at least 4 and has 0 as an eigenvalue of multiplicity at least 3. Since the sum of all eigenvalues of $M$ is $\operatorname{tr}(M)$, it follows that all eigenvalues of $M$ consist of $\{-1,0,\mu_{M}\}.$ ∎

Lemma 4.4**.**

Let $M=Dz^{\nu}(Q)$ be defined as above. If $\mu_{M}\notin\{0,-1\},$ then $\operatorname{rank}[\operatorname{adj}(I-\mu_{M}^{-1}M)]=1$ and, for any $b\in\mathbb{R}^{8}$,

[TABLE]

where $\operatorname{adj}A$ is the adjoint of $A\in\mathbb{M}^{8\times 8}$ satisfying $A(\operatorname{adj}A)=(\det A)I_{8}.$

Proof.

Since $\mu_{M}\notin\{0,-1\},$ it must be a simple eigenvalue of $M$ and thus $\operatorname{rank}(\mu_{M}I-M)=7$; so, $\operatorname{rank}[\operatorname{adj}(I-\mu_{M}^{-1}M)]=1.$ Finally, (4.16) follows from the formula: $\det(A+\alpha\otimes\beta)=\det A+(\operatorname{adj}A)\alpha\cdot\beta.$ ∎

Let $M_{\nu}^{0}=Dz^{\nu}(P_{\nu}^{0}).$ Then, by (4.12) and (4.13),

[TABLE]

where $H^{0}_{j}=D^{2}F(A_{j}^{0})$ ($j=1,\dots,5$), $W(H_{1},\dots,H_{5})$ is a $8\times 8$ matrix whose entries are polynomials of tensors $(H_{1},\dots,H_{5})$, and $j_{\nu}(H_{1},\dots,H_{5})$ is the polynomial function defined by (4.7). Again, both $W$ and $j_{\nu}$ are independent of the function $F$. Therefore, both $\mu_{M_{\nu}^{0}}(1+\mu_{M_{\nu}^{0}})$ and $|\operatorname{adj}(I-\mu_{M_{\nu}^{0}}^{-1}M_{\nu}^{0})z^{\nu}_{0}|^{2}$, where $z^{\nu}_{0}=\kappa_{\nu}^{0}C_{\nu}^{0}\in\mathbb{R}^{8}$, are rational functions of $(H^{0}_{1},\dots,H^{0}_{5})$ that are independent of the function $F$.

Lemma 4.5**.**

The rational functions of $(H_{1},\dots,H_{5})$ representing $\mu_{M_{\nu}^{0}}(1+\mu_{M_{\nu}^{0}})$ and $|\operatorname{adj}(I-\mu_{M_{\nu}^{0}}^{-1}M_{\nu}^{0})z^{\nu}_{0}|^{2}$ are not identically zero for each $\nu=1,\dots,5.$

Proof.

Let $h_{1}(s)$ and $h_{2}(t)$ be the tensors defined in the proof of Lemma 4.1. Using the determinant computations by MATLAB, one shows that the values of the two rational functions representing $\mu_{M_{\nu}^{0}}(1+\mu_{M_{\nu}^{0}})$ and $|\operatorname{adj}(I-\mu_{M_{\nu}^{0}}^{-1}M_{\nu}^{0})z^{\nu}_{0}|^{2}$ are nonzero at $[h_{1}(s),h_{2}(t),h_{1}(s),h_{1}(s),h_{2}(t)]$ with the same $(s,t)$ and $\nu$ as given in the proof of Lemma 4.1. ∎

Finally, we construct the function $F$ as follows.

Definition 4.1**.**

Let $F\colon\mathbb{M}^{2\times 2}\to\mathbb{R}$ be defined by (3.3) with the tensors $\tilde{H}_{j}=H_{j}^{0}-D^{2}F_{0}(A_{j}^{0})$ chosen to ensure that $(H_{1}^{0},\dots,H_{5}^{0})$ satisfies $(\ref{ImFT-1})$ and

[TABLE]

The existence of such $(H_{1}^{0},\dots,H_{5}^{0})$’s is guaranteed by Lemmas 4.1 and 4.5. Note that, by (3.5), $F$ is a smooth strongly polyconvex function.

Remark 4.1**.**

Since the exceptional values of $(H_{1}^{0},\dots,H_{5}^{0})$ not satisfying (4.9) or (4.18) are zeros of certain nonzero polynomials, it follows that the values of $(H_{1}^{0},\dots,H_{5}^{0})=(D^{2}F(A_{1}^{0}),\dots,D^{2}F(A_{5}^{0}))$ satisfying (4.9) and (4.18) are generic near $(D^{2}F_{0}(A_{1}^{0}),\dots,D^{2}F_{0}(A_{5}^{0})).$ Such a result has been derived in [21, Section 5] from certain stability analysis of $T_{N}$-configurations, which may not be available for the special $T_{N}$-configurations due to possible dimension deficiency.

5. The open set $\Sigma$ and verification of Condition (OC)

Let $F$ be a function defined in Definition 4.1 and $Y_{\nu},z^{\nu},M_{\nu}$ be the functions on $B_{\rho}(P_{\nu}^{0})$ defined in Lemmas 4.2 and 4.3. Let $K=K_{F}.$

By (4.18) and continuity, we select $\eta\in(0,\rho)$ sufficiently small so that

[TABLE]

for all $\nu=1,\dots,5.$ Furthermore, from $P_{\nu}^{1}(Y_{1}(P_{1}^{0}),P_{1}^{0})=P_{\nu}^{0}$ and continuity, we select $\beta\in(0,\eta)$ sufficiently small so that

[TABLE]

Lemma 5.1**.**

Let $\hat{P}^{\nu}_{j}(Q)=P^{\nu}_{j}(Y_{\nu}(Q),Q)$ for $Q\in B_{\rho}(P_{\nu}^{0})$ and all $\nu,j\mod 5.$ Then

[TABLE]

for all $Q\in B_{\beta}(P_{1}^{0})$ and $i=1,\dots,5.$ Therefore, in particular,

[TABLE]

Proof.

Let $Q\in B_{\beta}(P_{1}^{0}).$ From $\Psi^{1}(Y_{1}(Q),Q)=0$ and (4.5), it follows that $\Psi^{i}(Y_{1}(Q),P^{1}_{i}(Y_{1}(Q),Q))=0$ for each $i.$ Hence, by (4.10) and (5.2), one has $Y_{1}(Q)=Y_{i}(P^{1}_{i}(Y_{1}(Q),Q))=Y_{i}(\hat{P}_{i}^{1}(Q)).$ Thus, by definition of $P_{j}^{\nu}(Y,Q)$,

[TABLE]

[TABLE]

[TABLE]

Finally, (5.3) follows from differentiating $\hat{P}^{i}_{7-i}(\hat{P}^{1}_{i}(Q))=Q$ with $Q$. ∎

The following result verifies the Condition (OC) for $\sigma=DF$ and thus proves Theorem 1.1 by virtue of Yan [22, Theorem 1.3 and Corollary 1.4].

Theorem 5.2**.**

Let $\hat{X}^{\nu}_{j}(Q)=Q+Z^{\nu}_{j}(Y_{\nu}(Q))$ for all $\nu,j=1,\dots,5$ and $Q\in B_{\rho}(P_{\nu}^{0}).$ Then $(\hat{X}_{1}^{\nu}(Q),\dots,\hat{X}_{5}^{\nu}(Q))\in M^{\prime}_{5}\cap K_{5}$ for all $Q\in B_{\rho}(P_{\nu}^{0}).$ Define

[TABLE]

where $\mathcal{L}\colon\mathbb{M}^{2\times 2}\times(\mathbb{R}^{2})^{2}\to\mathbb{M}^{4\times 2}$ is the linear bijection defined by $(\ref{mapL})$. Then $\Sigma$ is nonempty, open and bounded, and the function $\sigma=DF$ satisfies the Condition (OC) with open set $\Sigma.$

Proof.

Clearly, $(\hat{X}_{1}^{\nu}(Q),\dots,\hat{X}_{5}^{\nu}(Q))\in M^{\prime}_{5}\cap K_{5}$ for all $Q\in B_{\rho}(P_{\nu}^{0})$ and hence $\tilde{\Sigma}$ is well-defined, nonempty and bounded; thus, so is $\Sigma.$ Moreover, one easily sees that $\Sigma$ satisfies the condition (3.9) of Definition 3.2 in [22] with fixed $N=5.$ It remains to show that $\Sigma$ is open; this is equivalent to showing that $\tilde{\Sigma}$ is open. So, let $\bar{X}\in\tilde{\Sigma}$ and assume

[TABLE]

for some $\bar{Q}\in B_{\beta}(P_{1}^{0})$, $i\in\{1,\dots,5\}$ and $0<\bar{\lambda}<1.$ The goal is to show that $B_{\gamma}(\bar{X})\subset\tilde{\Sigma}$ for some $\gamma>0.$ (See Figure 2 for a typical case of the proof.)

Let $\bar{U}=\hat{P}^{1}_{i}(\bar{Q});$ by (5.2), $\bar{U}\in B_{\eta}(P_{i}^{0}).$ Let $z(U)=z^{i}(U)=Z^{i}_{1}(Y_{i}(U))$ be the function defined in Lemma 4.3. By Lemmas 3.1 and 5.1, we have

[TABLE]

We proceed with the proof of $B_{\gamma}(\bar{X})\subset\tilde{\Sigma}$ in two cases.

Case 1: Assume $\det(I+\bar{\lambda}Dz(\bar{U}))\neq 0.$ In this case, we consider the function $G(Q,X)=\hat{P}_{i}^{1}(Q)+\bar{\lambda}z(\hat{P}_{i}^{1}(Q))-X$ for $Q\in B_{\beta}(P^{0}_{1})$ and all $X.$ By (5.5), one has $G(\bar{Q},\bar{X})=0;$ moreover, by (5.3),

[TABLE]

Therefore, by the implicit function theorem, there exist balls $B_{\eta^{\prime}}(\bar{Q})\subset B_{\beta}(P_{1}^{0})$ and $B_{\gamma}(\bar{X})$ such that for each $X\in B_{\gamma}(\bar{X})$ there exists $Q\in B_{\eta^{\prime}}(\bar{Q})$ such that $G(Q,X)=0$; hence

[TABLE]

This proves $B_{\gamma}(\bar{X})\subset\tilde{\Sigma}.$

Case 2: Assume $\det(I+\bar{\lambda}Dz(\bar{U}))=0$. In this case, let $\bar{M}=Dz(\bar{U}).$ Since $0<\bar{\lambda}<1,$ by Lemma 4.3, one has $\bar{\lambda}=-\mu_{\bar{M}}^{-1};$ thus, by (5.1), \bar{b}=\operatorname{adj}\big{(}I-\mu_{\bar{M}}^{-1}\bar{M}\big{)}z(\bar{U})\neq 0. Consider $G(Q,X)=H(\hat{P}_{i}^{1}(Q),X),$ where

[TABLE]

for $U\in B_{\rho}(P^{0}_{i})$ and all $X.$ Note that $G(\bar{Q},\bar{X})=H(\bar{U},\bar{X})=0$ and

[TABLE]

Thus, by Lemma 4.4, $\det\frac{\partial H}{\partial U}(\bar{U},\bar{X})=[\operatorname{adj}(I-\mu_{\bar{M}}^{-1}\bar{M})z(\bar{U})]\cdot\bar{b}=|\bar{b}|^{2}\neq 0$, and hence, by (5.3),

[TABLE]

The rest of the proof follows similarly as in Case 1.

This completes the proof. ∎

Remark 5.1**.**

For $0<\lambda<1,$ let

[TABLE]

By (5.1), $\det(I+\lambda M_{\nu}(U))\neq 0$ for all $\nu$ and $U\in B_{\eta}(P_{\nu}^{0})$ if $\lambda$ is sufficiently close to 1; thus, as in the Case 1 of the proof, it follows that the set $\tilde{\Sigma}_{\lambda}$ is open for all $\lambda$ sufficiently close to 1. Therefore, by selecting a suitable sequence $\{\lambda_{k}\}\to 1$, the open sets $\{\tilde{\Sigma}_{\lambda_{k}}\}_{k=1}^{\infty}$ form an in-approximation of set $K_{F}.$ In this regard, it would be interesting to see whether the convex integration scheme of Müller & Šverák [20] (see also [16, 19, 21]) could be adapted for constructing Lipschitz but nowhere $C^{1}$ (in space) solutions for the polyconvex gradient flow problem.