About the Incorporation of Topological Prescriptions in CNNs for Medical Image Semantic Segmentation

Abstract

Incorporating prior knowledge into a segmentation task, whether it be under the form of geometrical constraints (area/volume penalisation, convexity enforcement, etc.) or of topological constraints (to preserve the contextual relations between objects, to monitor the number of connected components), proves to increase accuracy in medical image segmentation. In particular, it allows to compensate for the issue of weak boundary definition, of imbalanced classes, and to be more in line with anatomical consistency even though the data do not explicitly exhibit those features. This observation underpins the introduced contribution that aims, in a hybrid setting, to leverage the best of both worlds that variational methods and supervised deep learning approaches embody: (a) versatility and adaptability in the mathematical formulation of the problem to encode geometrical/topological constraints, (b) interpretability of the results for the former formalism, while (c) more efficient and effective processing models, (d) ability to become more proficient at learning intricate features and executing more computationally intensive tasks, for the latter one. To be more precise, a unified variational framework involving topological prescriptions in the training of convolutional neural networks through the design of a suitable penalty in the loss function is provided. These topological constraints are implicitly enforced by viewing the segmentation procedure as a registration task between the processed image and its associated ground truth under incompressibility conditions, thus making them homeomorphic. A very preliminary version (Lambert et al., in Calatroni, Donatelli, Morigi, Prato, Santacesaria (eds) Scale space and variational methods in computer vision, Springer, Berlin, 2023, pp. 363–375) of this work has been published in the proceedings of the Ninth International Conference on Scale Space and Variational Methods in Computer Vision, 2023. It contained neither all the theoretical results, nor the detailed related proofs, nor did it include the numerical analysis of the designed algorithm. Besides these more involved developments in the present version, a more complete, systematic and thorough analysis of the numerical experiments is also conducted, addressing several issues: (i) limited amount of labelled data in the training phase, (ii) low contrast or imbalanced classes exhibited by the data, and (iii) explainability of the results.

Appendices

Proof of Theorem 1

The proof follows the arguments of the classical direct method of the calculus of variations. Clearly, $\mathcal {F}(\varphi ) \ge \mu \,\Vert \nabla \varphi \Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4-4\mu \,{\text{ meas }}(\Omega )$, $\text{ meas }$ denoting the Lebesgue measure in $\mathbb {R}^2$. The quantity $\mathcal {F}(\varphi )$ is thus bounded below by $-4\mu \,{\text{ meas }}(\Omega )$ and as for $\varphi ={\text{ Id }}$, $\mathcal {F}(\varphi )=\frac{\nu }{2}\,\Vert s(\theta )-y\Vert ^2_{L^2(\Omega )}$ is finite (due to the embedding $BV(\Omega )\subset L^2(\Omega )$, or simply by the fact that $L^{\infty }(\Omega )\subset L^2(\Omega )$), the infimum is finite. Let then $(\varphi _k)_k \in \mathcal {W}$ be a minimising sequence, i.e. a sequence such that $\displaystyle {\lim _{k \rightarrow +\infty }}\,\mathcal {F}(\varphi _k)=\displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )$. There exists $K\in \mathbb {N}$ such that $\forall k \in \mathbb {N}$,

$$\begin{aligned} \left( k \ge K\,\,\Rightarrow \,\,\mathcal {F}(\varphi _k)\le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+1\right) . \end{aligned}$$

From now on, we assume that $k \ge K$. According to the coercivity inequality, one gets that $(\varphi _k)_k$ is uniformly bounded with respect to k in $W^{1,4}(\Omega ,\mathbb {R}^2)$, using the generalised Poincaré inequality [11, pp. 106–107] and the fact that $\varphi _k={\text{ Id }}$ on $\partial \Omega$. Besides, $\det \,\nabla \varphi _k=1$ a.e. on $\Omega$ so that $\det \,\nabla \varphi _k$ is uniformly bounded in $L^{\infty }(\Omega )$. Thus there exists a subsequence—still denoted by $(\varphi _k)_k$—and $\bar{\varphi } \in W^{1,4}(\Omega ,\mathbb {R}^2)$ such that

$$\begin{aligned} \varphi _k \underset{k\rightarrow +\infty }{\rightharpoonup } \bar{\varphi } \text { in } W^{1,4}(\Omega ,\mathbb {R}^2). \end{aligned}$$

Furthermore, $\varphi _k \underset{k\rightarrow +\infty }{\rightarrow }\ \bar{\varphi }$ in $\mathcal {C}^{0,w}(\bar{\Omega },\mathbb {R}^2)$ due to the compact embedding $W^{1,4}(\Omega ,\mathbb {R}^2) \subset \mathcal {C}^{0,w}(\bar{\Omega },\mathbb {R}^2)$ [8, Sobolev embedding theorem, Theorem 12.12] for any $w<\frac{1}{2}$. Moreover, there exist a subsequence (common with the previous one, which is always possible)—still denoted by $(\det \,\nabla \varphi _k)_k$—and $\delta \in L^{\infty }(\Omega )$ such that

$$\begin{aligned} \det \,\nabla \varphi _k \underset{k\rightarrow +\infty }{\overset{*}{\rightharpoonup }} \delta \text { in } L^{\infty }(\Omega ), \end{aligned}$$

meaning that $\forall \phi \in L^1(\Omega )$,

$$\begin{aligned} \displaystyle {\int _{\Omega }}\,\det \,\nabla \varphi _k\,\phi \,\textrm{d}x \underset{k\rightarrow +\infty }{\rightarrow } \displaystyle {\int _{\Omega }}\,\delta \,\phi \,\textrm{d}x. \end{aligned}$$

As $\Omega$ is bounded, $L^2(\Omega ) \subset L^1(\Omega )$ so that $(\det \,\nabla \varphi _k)_k$ weakly converges to $\delta$ in $L^2(\Omega )$. By applying [8, Theorem 8.20], we deduce (by uniqueness of the weak limit in $L^2(\Omega ))$ that $\det \,\nabla \bar{\varphi }=\delta$ and $\det \,\nabla \varphi _k \underset{k\rightarrow +\infty }{\overset{*}{\rightharpoonup }} \det \,\nabla \bar{\varphi }$. Then $\det \,\nabla \bar{\varphi }=1$ a.e.. By continuity of the trace operator [5, Theorem III.9], we get that $\bar{\varphi }\in {\text{ Id }}+W_0^{1,4}(\Omega ,\mathbb {R}^2)$.

Now, for all $k\ge K$,

$$\begin{aligned}&\displaystyle {\int _{\Omega }}\,\Vert \left( \nabla \varphi _k\right) ^{-1}\Vert ^4\,\det \,\nabla \varphi _k\,\textrm{d}x\\&\quad =\displaystyle {\int _{\Omega }}\,\frac{1}{\left( \det {\nabla \varphi _k}\right) ^3}\,\Vert \nabla \varphi _k\Vert ^4\,\textrm{d}x\le C, \end{aligned}$$

C being a positive constant depending only on $\Omega$. The assumptions of Ball’s theorems [3, Theorems 1 and 2] thus hold, yielding that $\varphi _k$ is a homeomorphism of $\bar{\Omega }$ onto $\bar{\Omega }$ and $\varphi _k^{-1}\in W^{1,4}(\Omega ,\mathbb {R}^2)$. The classical change of variable formula holds.

$$\begin{aligned} \displaystyle {\int _{A}}\,f(u(x))\,\det {\,\nabla u(x)}\,\textrm{d}x=\displaystyle {\int _{u(A)}}\,f(v)\,dv \end{aligned}$$

holds for any measurable $A\subset \bar{\Omega }$ and any measurable function $f:\mathbb {R}^2 \rightarrow \mathbb {R}$, provided only that one of the integrals exists in the previous relation. The matrix of weak derivatives of $\varphi _k^{-1}$ is given by $\nabla \varphi _k^{-1}=\left( \nabla \varphi _k\right) ^{-1} \circ \varphi _k^{-1}$ almost everywhere in $\Omega$.

The same reasoning applies to $\bar{\varphi }$ so that it inherits the same smoothness properties: it is a homeomorphism of $\bar{\Omega }$ onto $\bar{\Omega }$ and $\bar{\varphi }^{-1}\in W^{1,4}(\Omega ,\mathbb {R}^2)$.

The stored energy function W is continuous and convex. Let $\psi _k$ be such that $\psi _k \underset{k\rightarrow +\infty }{\longrightarrow }\ \bar{\psi }$ in $W^{1,4}(\Omega ,\mathbb {R}^2)$. Let then $\psi _{\sigma _1(k)}$ be such that $I(\psi _{\sigma _1(k)})=\int _{\Omega }\,W(\nabla \psi _{\sigma _1(k)})\,\textrm{d}x \underset{k\rightarrow +\infty }{\longrightarrow }\ \displaystyle {\liminf _{k\rightarrow +\infty } }I(\psi _k)$, which exists by definition of the $\liminf$. From this subsequence, one can extract a second subsequence $\psi _{\sigma _1 \circ \sigma _2(k)}$ such that $\nabla \psi _{\sigma _1 \circ \sigma _2(k)} \underset{k\rightarrow +\infty }{\longrightarrow }\nabla \bar{\psi }$ almost everywhere in $\Omega$. Applying Fatou’s lemma yields

$$\begin{aligned} I(\bar{\psi })&\le \underset{k\rightarrow +\infty }{\lim \inf } I(\psi _{\sigma _1 \circ \sigma _2(k)})=\underset{k\rightarrow +\infty }{\lim }\ I(\psi _{\sigma _1 \circ \sigma _2(k)})\\&=\underset{k\rightarrow +\infty }{\lim } I(\psi _{\sigma _1(k)})=\underset{k\rightarrow +\infty }{\lim \inf } I(\psi _{k}), \end{aligned}$$

or rewritten in terms of W,

$$\begin{aligned} \int _\Omega W(\nabla \bar{\psi })\,\textrm{d}x \le \underset{k\rightarrow +\infty }{\lim \inf }\ \int _\Omega W(\nabla \psi _k)\,\textrm{d}x, \end{aligned}$$

showing that I is (convex), strongly lower semi-continuous, so, in accordance with [5, Corollary III.8], lower semi-continuous for the weak topology. It remains to study $\underset{k\rightarrow +\infty }{\lim \inf }\ \,\Vert s(\theta )-y\circ \varphi _k\Vert _{L^2(\Omega )}^2$—in fact, as will be seen later, we handle $\underset{k\rightarrow +\infty }{\lim } \,\Vert s(\theta )-y\circ \varphi _k\Vert _{L^2(\Omega )}^2$—, inspired by prior work by Wirth [33]. We first prove that $\varphi _k \circ \bar{\varphi }^{-1} \,\underset{k\rightarrow +\infty }{\longrightarrow }\ \,{\text{ Id }}$ in $\mathcal {C}^{0,\alpha }(\bar{\Omega },\mathbb {R}^2)$ with $\alpha < \frac{1}{2}$. Recall that (see [8, Definition 12.5]) with $0<\alpha \le 1, \mathcal {C}^{0,\alpha }(\bar{\Omega },\mathbb {R}^2)$ is the set of functions $u\in \mathcal {C}^{0}(\bar{\Omega },\mathbb {R}^2)$ such that

$$\begin{aligned} \left[ u\right] _{\alpha ,\bar{\Omega }}:=\displaystyle {\sup _{(x,y)\in \bar{\Omega } \times \bar{\Omega } \begin{array}{c} x \end{array}\ne y}}\,\dfrac{|u(x)-u(y)|}{|x-y|^{\alpha }}<+\infty . \end{aligned}$$

It is equipped with the norm

$$\begin{aligned} \Vert u\Vert _{\mathcal {C}^{0,\alpha }(\bar{\Omega },\mathbb {R}^2)}:=\Vert u \Vert _{\mathcal {C}^{0}(\bar{\Omega },\mathbb {R}^2)}+\left[ u\right] _{\alpha ,\bar{\Omega }}, \end{aligned}$$

where the notation $|\cdot |$ is interpreted as the Euclidean norm. Additionally, $W^{1,4}(\Omega ,\mathbb {R}^2) \subset \mathcal {C}^{0,\lambda } (\bar{\Omega },\mathbb {R}^2)$ [8, Sobolev embedding theorem, Theorem 12.11] for every $\lambda \in [0,\frac{1}{2}]$ and the embedding is compact for every $0\le \lambda <\frac{1}{2}$ [8, Rellich-Kondrachov theorem, Theorem 12.12]. Let us take $(\alpha ,\lambda )$ positive real numbers such that $\left\{ \begin{array}{ccc} \lambda &{}<&{}\frac{1}{2} \\ \frac{\alpha }{\lambda }&{}<&{}\frac{1}{2} \end{array} \right.$, which is always possible. Straightforward computations thus give

$$\begin{aligned}&\Vert \varphi _k\circ \bar{\varphi }^{-1}-\bar{\varphi } \circ \bar{\varphi }^{-1} \Vert _{\mathcal {C}^{0,\alpha }(\bar{\Omega },\mathbb {R}^2)} \\&\quad \le \, \underset{x\in \bar{\Omega }}{\sup } |\varphi _k(x)-\bar{\varphi }(x)| +\underset{(x,y)\in \bar{\Omega }\times \bar{\Omega }\begin{array}{c} x \end{array}\ne y}{\sup }\\&\quad \frac{|\varphi _k\circ \bar{\varphi }^{-1}(x)-\varphi _k\circ \bar{\varphi }^{-1}(y) -\bar{\varphi }\circ \bar{\varphi }^{-1}(x)+\bar{\varphi }\circ \bar{\varphi }^{-1} (y)|}{|\bar{\varphi }^{-1}(x)-\bar{\varphi }^{-1}(y)|^\lambda }\\&\quad \frac{|\bar{\varphi }^{-1}(x)-\bar{\varphi }^{-1}(y)|^\lambda }{|x-y|^\alpha },\\&\quad \le \underset{x\in \bar{\Omega }}{\sup } |\varphi _k(x)-\bar{\varphi }(x)|\\&\quad + \underset{(x,y)\in \bar{\Omega }\times \bar{\Omega }\begin{array}{c} x \end{array}\ne y}{\sup } \frac{|\varphi _k(x)-\varphi _k(y)-\bar{\varphi }(x)+\bar{\varphi }(y)|}{|x-y|^\lambda }\\&\quad \underset{(x,y)\in \bar{\Omega }\times \bar{\Omega }\begin{array}{c} x \end{array}\ne y}{\sup } \frac{|\bar{\varphi }^{-1}(x)-\bar{\varphi }^{-1}(y)|^\lambda }{|x-y|^\alpha },\\&\le \underset{x\in \bar{\Omega }}{\sup } |\varphi _k(x)-\bar{\varphi }(x)|\\&\quad + \underset{(x,y)\in \bar{\Omega }\times \bar{\Omega }\begin{array}{c} x \end{array}\ne y}{\sup } \frac{|\varphi _k(x)-\varphi _k(y)-\bar{\varphi }(x)+\bar{\varphi }(y)|}{|x-y|^\lambda }\\&\quad \underset{(x,y)\in \bar{\Omega }\times \bar{\Omega }\begin{array}{c} x \end{array}\ne y}{\sup }\left| \frac{\bar{\varphi }^{-1}(x)-\bar{\varphi }^{-1}(y)}{|x-y|^{\frac{\alpha }{\lambda }}}\right| ^{\lambda } ,\\&\le (1+\Vert \bar{\varphi }^{-1}\Vert _{\mathcal {C}^{0,\frac{\alpha }{\lambda }} (\bar{\Omega },\mathbb {R}^2)}^\lambda )\Vert \varphi _k\\&\quad -\bar{\varphi }\Vert _{\mathcal {C}^{0,\lambda } (\bar{\Omega },\mathbb {R}^2)}\underset{k\rightarrow +\infty }{\longrightarrow }0. \end{aligned}$$

From the above, for almost every $x\in \Omega , \nabla \varphi _k^{-1}=(\nabla \varphi _k)^{-1}(\varphi _k^{-1})$. Additionally, $\det {\left( \nabla \varphi _k\right) ^{-1}}=1$ almost everywhere. Let $\mathcal {N}_k\subset \Omega$ be such that ${\text{ meas }}(\mathcal {N}_k)=0$ and for all $x\in \Omega \backslash \mathcal {N}_k, \det {\left( \nabla \varphi _k\right) ^{-1}}=1$. Let now $\mathcal {N}_k'$ be such that $\mathcal {N}_k'=\varphi _k(\mathcal {N}_k)$. From [26, Theorem 1 and Corollary 1], ${\text{ meas }}(\mathcal {N}_k')=0$. For every $y\notin \mathcal {N}_k', \det {\left( \left( \nabla \varphi _k\right) ^{-1}(\varphi _k^{-1}(y))\right) } =\det {\left( \nabla \varphi _k\right) ^{-1}(x)}$ with $x \in \Omega \backslash \mathcal {N}_k$, yielding $\det {\nabla \varphi _k^{-1}}=1$ a.e..

Note first of all that $y \circ \varphi _k \in L^2(\Omega )$ (the same holds for $y\circ \bar{\varphi }$ by applying a similar reasoning). Indeed, according to the change of variable formula, valid from Ball’s results (see in particular [3, Proof of Theorem 3]),

$$\begin{aligned} \int _{\Omega }\,|y \circ \varphi _k|^2\,\textrm{d}x&= \int _{\Omega }\,|y|^2\,\det {\nabla \varphi _k^{-1}}\,\textrm{d}x,\\&\le L^2\,\int _{\Omega }\,\det {\nabla \varphi _k^{-1}}\,\textrm{d}x, \end{aligned}$$

the rightmost quantity being uniformly bounded in k, having indeed $y\in L^{\infty }(\Omega )$ (more precisely, $y\in BV(\Omega ,\left\{ 1,\ldots ,L\right\}$) and $\det {\,\nabla \varphi _k^{-1}}=1$ almost everywhere.

Let now $\mathcal {O}_i=\left\{ x\in \Omega \,|\,y(x)=i\right\}$ with i any index in $\left\{ 1,\ldots ,L\right\}$. We first observe that $\Omega =\displaystyle {\cup _{i=1}^{L}}\,\mathcal {O}_i\cup \mathcal {N}$ with $\mathcal {N}$ such that $\text{ meas }(\mathcal {N})=0$.

Let us now consider any measurable set $\mathcal {S}\subset \Omega$ with Lebesgue measure $\text{ meas }(\mathcal {S}) \le \eta$. One has, according to the change of variable formula, valid from Ball’s results, and Hölder’s inequality with parameter $s\ge 2$

$$\begin{aligned} \int _{\bar{\varphi }^{-1}(\mathcal {S})}\,\textrm{d}x&=\int _{\mathcal {S}}\,\,\det {\nabla \bar{\varphi }^{-1}}\,\textrm{d}x=\int _{\mathcal {S}}\,\,1 \times \,\dfrac{1}{\det {\nabla \bar{\varphi }\circ \left( \bar{\varphi }\right) ^{-1}}}\,\textrm{d}x,\nonumber \\&\le \left( \int _{\mathcal {S}}\,\textrm{d}x\right) ^{\frac{s-1}{s}}\,\left( \int _{\mathcal {S}}\, \left( \frac{1}{\det {\nabla \bar{\varphi }\circ \left( \bar{\varphi }\right) ^{-1}}} \right) ^s\,\textrm{d}x\right) ^{\frac{1}{s}},\nonumber \\&\le \,\left( \text{ meas }(\mathcal {S})\right) ^{\frac{s-1}{s}}\,\left( \int _{\Omega }\, \left( \frac{1}{\det {\nabla \bar{\varphi } }}\right) ^{s-1}\,\textrm{d}x\right) ^{\frac{1}{s}},\nonumber \\&\le \left( \text{ meas }(\Omega )\right) ^{\frac{1}{s}}\,\left( \text{ meas }(\mathcal {S}) \right) ^{\frac{s-1}{s}}\le \left( \text{ meas }(\Omega )\right) ^{\frac{1}{s}} \,\eta ^{\frac{s-1}{s}}, \end{aligned}$$

(9)

since $\det {\nabla \bar{\varphi }}=1$ a.e.. Again, due to the smoothness properties of $\bar{\varphi }$, $\Omega =\bar{\varphi }^{-1}(\Omega )=\cup _{i=1}^{L}\,\bar{\varphi }^{-1}(\mathcal {O}_i) \cup \bar{\varphi }^{-1}(\mathcal {N})$, which, combined with inequality (9), shows that to achieve our goal, it suffices to prove that $y \circ \varphi _k \underset{k\rightarrow +\infty }{\longrightarrow }\ y \circ \bar{\varphi }$ in $L^2(\bar{\varphi }^{-1}\left( \mathcal {O}_i\right) )$. One has:

$$\begin{aligned}&\Vert y\circ \varphi _k - y\circ \bar{\varphi }\Vert ^2_{L^2(\bar{\varphi }^{-1}\left( \mathcal {O}_i\right) )}\\&\quad = \displaystyle {\int _{\bar{\varphi }^{-1}\left( \mathcal {O}_i\right) }}\,\left| y\circ \varphi _k - y\circ \bar{\varphi }\right| ^2\,\textrm{d}x\\&\quad =\displaystyle {\int _{\bar{\varphi }^{-1} \left( \mathcal {O}_i\right) }}\,\left| y\circ \varphi _k - i\right| ^2\,\textrm{d}x,\\&\quad =\displaystyle {\int _{\varphi _k \circ \bar{\varphi }^{-1}\left( \mathcal {O}_i\right) }}\,\left| y - i\right| ^2\,\det {\nabla \,\varphi _k^{-1}}\,\textrm{d}x,\\&\quad \le \,\displaystyle {\int _{\varphi _k \circ \bar{\varphi }^{-1} \left( \mathcal {O}_i\right) \Delta \mathcal {O}_i}}\,\left| y - i \right| ^2\,\det {\nabla \,\varphi _k^{-1}}\,\textrm{d}x\\&\qquad +\displaystyle {\int _{\mathcal {O}_i}} \,\left| y - i\right| ^2\,\det {\nabla \,\varphi _k^{-1}}\,\textrm{d}x,\\&\quad \le (L-1)^2\,\displaystyle {\int _{\varphi _k \circ \bar{\varphi }^{-1} \left( \mathcal {O}_i\right) \Delta \mathcal {O}_i}}\,\det {\nabla \,\varphi _k^{-1}}\,\textrm{d}x, \end{aligned}$$

where $\Delta$ stands for the symmetric difference. According to the change of variable, valid from Ball’s results, and Hölder’s inequality with parameter $s\ge 2$,

$$\begin{aligned}&\Vert y\circ \varphi _k - y\circ \bar{\varphi }\Vert ^2_{L^2(\bar{\varphi }^{-1}\left( \mathcal {O}_i\right) )}\\&\quad \le (L-1)^2\,\left( \displaystyle {\int _{\varphi _k \circ \bar{\varphi }^{-1}\left( \mathcal {O}_i\right) \Delta \mathcal {O}_i}}\,\textrm{d}x\right) ^{\frac{s-1}{s}}\\&\qquad \times \left( \displaystyle {\int _{\varphi _k \circ \bar{\varphi }^{-1}\left( \mathcal {O}_i\right) \Delta \mathcal {O}_i}}\, \left( \dfrac{1}{\det \,\nabla \varphi _k}\right) ^{s-1}\,\textrm{d}x\right) ^{\frac{1}{s}},\\&\quad \le (L-1)^2\, \left( {\text{ meas }}\left( \varphi _k \circ \bar{\varphi }^{-1}\left( \mathcal {O}_i\right) \Delta \mathcal {O}_i\right) \right) ^{\frac{s-1}{s}}\\&\quad \times \left( \displaystyle {\int _{\Omega }}\, \left( \dfrac{1}{\det \,\nabla \varphi _k}\right) ^{s-1}\,\textrm{d}x\right) ^{\frac{1}{s}}. \end{aligned}$$

The rightmost term is uniformly bounded with respect to k since $\det \,\nabla \varphi _k=1$ a.e.. Besides, as $\varphi _k \circ \bar{\varphi }^{-1} \,\underset{k\rightarrow +\infty }{\longrightarrow } \,{\text{ Id }}$ in $\mathcal {C}^{0,\lambda }(\bar{\Omega },\mathbb {R}^2)$, it can be shown (see [33, Footnote 1., p.455]) that ${\text{ meas }}\,\left( \varphi _k \circ \bar{\varphi }^{-1}\,\left( \mathcal {O}_i\right) \Delta \mathcal {O}_i\right) \underset{k\rightarrow +\infty }{\longrightarrow }0$, yielding $y \circ \varphi _k \underset{k\rightarrow +\infty }{\longrightarrow }\ y \circ \bar{\varphi }$ in $L^2(\bar{\varphi }^{-1}\left( \mathcal {O}_i\right) )$.

Proof of Theorem 2

The proof is an adaptation of the one of [15, Theorem 2, Chapter 5]. We follow the steps of [15, Theorem 2, Chapter 5] and highlight the main differences [15, Theorem 2, Chapter 5] without dwelling on common elements. In particular, we refer the reader to [15, Theorem 2, Chapter 5] for the proof of (ii).

Let us fix $\varepsilon >0$. Let m be a positive integer. Let us define the open sets

$$\begin{aligned} \Omega _k&=\left\{ x\in \Omega \,|\,\text{ dist }(x,\partial \Omega )>\frac{1}{m+k}\right\} \\&\cap B(0,k+m),\,\,k=1,\ldots , \end{aligned}$$

B(x, r) denoting the open ball with centre x and radius r, and m being chosen large enough so that $\Vert Dy\Vert (\Omega -\Omega _1)<\varepsilon$.

Set $\Omega _0 \equiv \emptyset$ and define $V_k=\Omega _{k+1} {\setminus } \overline{\Omega _{k-1}}, k=1,\ldots$. Let $\left\{ \zeta _k\right\} _{k=1}^{\infty }$ be a sequence of smooth functions such that

$$\begin{aligned} \left\{ \begin{array}{ccc} \zeta _k\in \mathcal {C}_c^{\infty }(V_k)\,\,&{}\,\,0\le \zeta _k\le 1\,\,&{}(k=1,\ldots )\\ \displaystyle {\sum _{k=1}^{\infty }}\,\zeta _k=1\,\,&{}&{}\,\,{\text{ on }} \,\,\Omega . \end{array}\right. \end{aligned}$$

Let $\rho \in \mathcal {C}_c^{\infty }(\mathbb {R}^2)$ be a mollifier with ${\text{ supp }}\,(\rho )\subset B(0,1), \rho \ge 0$ and $\int _{\mathbb {R}^2}\,\rho (x)\,\textrm{d}x=1$, and define next $\rho _{\varepsilon }(x)=\frac{1}{\varepsilon ^2}\,\rho \left( \frac{x}{\varepsilon }\right)$. For each k, select $\varepsilon _k>0$ so small that

$$\begin{aligned} \left\{ \begin{array}{c} {\text{ supp }}\left( \rho _{\varepsilon _k} *\left( y\zeta _k\right) \right) \subset V_k,\\ \left( \displaystyle {\int _{\Omega }}\,|\rho _{\varepsilon _k} *\left( y\zeta _k\right) -y\zeta _k|^p\,\textrm{d}x\right) ^{\frac{1}{p}}\,< \dfrac{\varepsilon }{2^k},\\ \displaystyle {\int _{\Omega }}\,|\rho _{\varepsilon _k} *\left( yD\zeta _k\right) -yD\zeta _k|\,\textrm{d}x\,< \dfrac{\varepsilon }{2^k}. \end{array}\right. \end{aligned}$$

Define $y_{\varepsilon }=\displaystyle {\sum _{k=1}^{\infty }}\,\rho _{\varepsilon _k}*\left( y\zeta _k\right)$. By construction of the sets $\left\{ \Omega _k\right\} _k$, there exists $n_0$ such that $\text{ ess }\,\text{ supp }\,(y)\subset \Omega '\subset \subset \Omega _{n_0}$. The quantity $\displaystyle {\sum _{k=1}^{\infty }}\,\rho _{\varepsilon _k}*\left( y\zeta _k\right)$ can thus be restricted to the indices $\displaystyle {\sum _{k=1}^{n_0}}\,\rho _{\varepsilon _k}*\left( y\zeta _k\right)$, yielding $y_{\varepsilon }=\displaystyle {\sum _{k=1}^{n_0}}\,\rho _{\varepsilon _k}*\left( y\zeta _k\right) \in \mathcal {C}^{\infty }(\Omega )$ with ${\text{ supp }}(y_{\varepsilon }) \subset \Omega _{n_0+1}$, set that depends on $\varepsilon$. Note that taking $\tilde{\varepsilon }\le \varepsilon$ would generate another partitioning $\tilde{\Omega }_{k}$ that may be obtained by relabelling the partitioning ${\Omega }_{k}$.

Since $y=\displaystyle {\sum _{k=1}^{n_0}}y\zeta _k$, using Minkowski inequality,

$$\begin{aligned} \Vert y_{\varepsilon }-y\Vert _{L^p(\Omega )}&=\left( \displaystyle {\int _{\Omega }} \,\left| \displaystyle {\sum _{k=1}^{n_0}}\,\rho _{\varepsilon _k}*\left( y\zeta _k\right) -y\zeta _k\right| ^p\,\textrm{d}x\right) ^{\frac{1}{p}},\\&\le \displaystyle {\sum _{k=1}^{n_0}}\,\left( \displaystyle {\int _{\Omega }}\, \left| \rho _{\varepsilon _k}*\left( y\zeta _k\right) -y\zeta _k\right| ^p\,\textrm{d}x\right) ^{\frac{1}{p}}<\varepsilon . \end{aligned}$$

Consequently, $y_{\varepsilon } \, \underset{}{\longrightarrow }\,y$ in $L^p(\Omega )$ as $\varepsilon$ tends to 0 (and in particular, in $L^1(\Omega )$). Thus, as $\Vert Dy\Vert (\Omega ) \le \displaystyle {\liminf _{\varepsilon \rightarrow 0}}\,\Vert Dy_{\varepsilon }(\Omega )\Vert$, it suffices to prove that $\displaystyle {\limsup _{\varepsilon \rightarrow 0}}\,\Vert Dy_{\varepsilon }(\Omega )\Vert \le \Vert Dy\Vert (\Omega )$, which is led to the reader following [15, Theorem 2, Chapter 5].

Now, if $x\in \Omega _1 \subset \subset \Omega _2$ then

$$\begin{aligned} y_{\varepsilon }(x)&=\rho _{\varepsilon _1}*\left( y\zeta _1\right) (x),\\&=\displaystyle {\int _{V_1=\Omega _2}}\,\rho _{\varepsilon _1}(x-s)\,y(s)\zeta _1(s)\,\textrm{d}s, \end{aligned}$$

yielding $|y_{\varepsilon }(x)| \le (L-1)\,\int _{}\rho _{\varepsilon _1}(x-s)\,\textrm{d}s=L-1$.

If $x \in \Omega _l{\setminus } \overline{\Omega _{l-1}}$ with $l>1$, then

$$\begin{aligned} y_{\varepsilon }(x)&=\rho _{\varepsilon _{l-1}}*\left( y\zeta _{l-1}\right) (x)+\rho _{\varepsilon _{l}}*\left( y\zeta _{l}\right) (x),\\&=\displaystyle {\int _{V_{l-1}}}\,\rho _{\varepsilon _{l-1}}(x-s)\,y(s) \zeta _{l-1}(s)\,\textrm{d}s\\&\quad +\displaystyle {\int _{V_{l}}}\, \rho _{\varepsilon _{l}}(x-s)\,y(s)\zeta _{l}(s)\,\textrm{d}s,\\&\le 2(L-1), \end{aligned}$$

showing that $y_{\varepsilon }$ is uniformly bounded in the $L^{\infty }$-norm (the bound depending only on L which is fixed).

Proof of Theorem 3

First observe that with $\varphi _l$ defined above,

$$\begin{aligned}&\displaystyle {\inf _{(\varphi ,V,W)\in \overline{\mathcal {W}} \times L^4(\Omega ,M_2 (\mathbb {R})) \begin{array}{c} \times \end{array}\overline{\overline{\mathcal {W}}}}}\, E_{l,j}(\varphi ,V,W)\\&\quad \le E_{l,j}(\varphi _l,\nabla \varphi _l,\nabla \varphi _l) \\&\quad \le \mathcal {F}(\varphi _l)+\dfrac{C}{l}\le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\frac{C+1}{l}. \end{aligned}$$

$E_{l,j}(\varphi ,V,W)\ge 0$ and taking $\varphi ={\text{ Id }}$, $V=I_2$ and $W=I_2$ shows that the functional is proper since $s(\theta )\in L^2(\Omega )$ and $y_l\in \mathcal {C}^{\infty }(\Omega )$ (bounded uniformly with respect to l), and subsequently, that the infimum is finite.

Let us denote by $\left( \varphi _{l,j,k},V_{l,j,k},W_{l,j,k}\right)$ a minimising sequence of the decoupled problem $(DP)_{l,j}$, that is, a sequence such that

$$\begin{aligned}&\displaystyle {\lim _{k\rightarrow +\infty }}\,E_{l,j}(\varphi _{l,j,k},V_{l,j,k},W_{l,j,k})\\&\quad =\displaystyle {\inf _{(\varphi ,V,W)\in \overline{\mathcal {W}} \times L^4(\Omega ,M_2(\mathbb {R})) \begin{array}{c} \times \end{array}\overline{\overline{\mathcal {W}}}}}\, E_{l,j}(\varphi ,V,W). \end{aligned}$$

In particular,

$$\begin{aligned}&\forall \varepsilon >0,\,\,\exists N_{\varepsilon ,l,j}\in \mathbb {N},\,\,\forall k \in \mathbb {N},\\&\quad \left( k\ge N_{\varepsilon ,l,j}\,\,\Longrightarrow \,\,E_{l,j}(\varphi _{l,j,k},V_{l,j,k},W_{l,j,k})\right. \\&\quad \le \displaystyle \left. \inf _{(\varphi ,V,W)\in \overline{\mathcal {W}} \times L^4(\Omega ,M_2(\mathbb {R})) \begin{array}{c} \times \end{array}\overline{\overline{\mathcal {W}}}}\,E_{l,j}(\varphi ,V,W)+\varepsilon \right) . \end{aligned}$$

Let us take in particular $\varepsilon =\frac{1}{\gamma _j}$. Then there exists $N_{l,j}\in \mathbb {N}$ such that $\forall k \in \mathbb {N}$,

$$\begin{aligned}&\left( k\ge N_{l,j}\right. \,\,\Longrightarrow \, \,E_{l,j}(\varphi _{l,j,k},V_{l,j,k},W_{l,j,k})\nonumber \\&\quad \le \displaystyle {\inf _{(\varphi ,V,W)\in \overline{\mathcal {W}} \times L^4(\Omega ,M_2(\mathbb {R})) \begin{array}{c} \times \end{array}\overline{\overline{\mathcal {W}}}}} \,E_{l,j}(\varphi ,V,W)+\dfrac{1}{\gamma _j}, \nonumber \\&\quad \left. \le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\frac{C+1}{l}+\dfrac{1}{\gamma _0}\right) . \end{aligned}$$

(10)

We now take $k:=N_{l,j}$ and for the sake of readability, we denote by $\varphi _{l,j}:=\varphi _{l,j,N_{l,j}}$ and similarly for $V_{l,j}$ and $W_{l,j}$. Using the inequality $(a-b)^2\ge \frac{1}{2}a^2-b^2$, a first coercivity-type inequality can be obtained, of the form

$$\begin{aligned}&\,\mu \,\Vert V_{l,j}\Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\mu \alpha }{4}\,\Vert \det {V_{l,j}}\Vert _{L^2(\Omega )}^2-\dfrac{\mu \alpha }{2}\,{\text{ meas }}(\Omega )\\&\quad \le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\frac{C+1}{l}+\dfrac{1}{\gamma _0}, \end{aligned}$$

showing that

$$\begin{aligned} \left| \begin{array}{l} (V_{l,j})\,\,\,\text{ is } \text{ uniformly } \text{ bounded } \text{ in } L^4(\Omega ,M_2(\mathbb {R})) \text{ and } \text{ so } \text{ in } \\ L^2(\Omega ,M_2(\mathbb {R})),\\ {\text{ and }}\, (\det {V_{l,j}})\,\,\,{\text{ is } \text{ uniformly } \text{ bounded } \text{ in } L^2(\Omega )}. \end{array}\right. \end{aligned}$$

Thus there exist a common subsequence, obtained by applying a diagonal extraction procedure, denoted by $(V_{l,\Psi (j)})$ (respectively, $(\det {V_{l,\Psi (j)}})$ and $\bar{V}_l \in L^4(\Omega ,M_2(\mathbb {R}))$ and $\bar{\delta }_l\in L^2(\Omega )$ such that

$$\begin{aligned} V_{l,\Psi (j)} \,\,\underset{j \rightarrow +\infty }{\rightharpoonup }\,\,\bar{V}_l\,\,\,\,{\text{ in } L^4(\Omega ,M_2(\mathbb {R}))},\\ \det {V_{l,\Psi (j)}} \,\,\underset{j \rightarrow +\infty }{\rightharpoonup }\,\,\bar{\delta }_l\,\,\,\,{\text{ in } L^2(\Omega )}. \end{aligned}$$

In addition,

$$\begin{aligned}&\dfrac{\gamma _{\Psi (j)}}{2}\,\Vert V_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}^2\\&\quad \le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\frac{C+1}{l}+\dfrac{1}{\gamma _0}, \end{aligned}$$

yielding

$$\begin{aligned}&\left| \Vert \nabla \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}-\Vert V_{l,\Psi (j)}\Vert \right| _{L^2(\Omega ,M_2(\mathbb {R}))}\\&\quad \le \Vert V_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}\\&\quad \le \left( \dfrac{2}{\gamma _{0}}\,\left( \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\frac{C+1}{l}+\dfrac{1}{\gamma _0}\right) \right) ^{\frac{1}{2}}. \end{aligned}$$

The sequence $(\varphi _{l,\Psi (j)})$ is thus uniformly bounded in $W^{1,2}(\Omega ,\mathbb {R}^2)$ according to the generalised Poincaré inequality, and there exist a subsequence obtained by applying a diagonal extraction procedure and still denoted by $(\varphi _{l,\Psi (j)})$ and $\bar{\varphi }_l\in W^{1,2}(\Omega ,\mathbb {R}^2)$ such that

$$\begin{aligned} \varphi _{l,\Psi (j)} \,\,\underset{j \rightarrow +\infty }{\rightharpoonup }\,\,\bar{\varphi }_l\,\,\,\,{\text{ in } W^{1,2}(\Omega ,\mathbb {R}^2)}. \end{aligned}$$

The same kind of argument applies to $(W_{l,\Psi (j)})$ yielding

$$\begin{aligned} W_{l,\Psi (j)} \,\,\underset{j \rightarrow +\infty }{\rightharpoonup }\,\,\bar{W}_l\,\,\,\,{\text{ in } L^{2}(\Omega ,M_2(\mathbb {R}))}. \end{aligned}$$

Additionally, $(\det {W_{l,\Psi (j)}})$ is uniformly bounded in $L^{\infty }$-norm and so there exists $\bar{\bar{\delta }}_l\in L^{\infty }(\Omega )$ such that

$$\begin{aligned} \det {W_{l,\Psi (j)}} \,\,\overset{*}{\underset{j \rightarrow +\infty }{\rightharpoonup }}\,\,\bar{\bar{\delta }}_l\,\,\,\,{\text{ in } L^{\infty }(\Omega )}, \end{aligned}$$

and finally, $\bar{\bar{\delta }}_l=1$ a.e.. 

Let us now set $x_{l,\Psi (j)}=V_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)}$.

On the one hand, since $\Vert V_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}\le \frac{2}{\gamma _{\Psi (j)}}\,\left[ \inf _{\varphi \in \mathcal {W}}\,\mathcal {F}(\varphi )+\frac{C+1}{l}+\frac{1}{\gamma _0}\right]$, it implies that $x_{l,\Psi (j)} \underset{j \rightarrow +\infty }{\longrightarrow }\ 0$ in $L^2(\Omega ,M_2(\mathbb {R}))$. Consequently, $V_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)} \underset{j \rightarrow +\infty }{\rightharpoonup }\ 0$ in $L^2(\Omega ,M_2(\mathbb {R}))$, yielding $\nabla \varphi _{l,\Psi (j)} \underset{j \rightarrow +\infty }{\rightharpoonup }\ \bar{V}_l$ in $L^2(\Omega ,M_2(\mathbb {R}))$. Indeed, $\forall \phi \in L^2(\Omega ,M_2(\mathbb {R}))$, $\int _{\Omega }\,x_{l,\Psi (j)}\,:\,\phi \,\textrm{d}x \underset{j \rightarrow +\infty }{\rightarrow }\ 0$, that is, $\int _{\Omega }\,V_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)}\,:\,\phi \,\textrm{d}x \underset{j \rightarrow +\infty }{\rightarrow }\ 0$. But $V_{l,\Psi (j)}\underset{j \rightarrow +\infty }{\rightharpoonup }\ \bar{V}_l$ in $L^4(\Omega ,M_2(\mathbb {R}))$ so in $L^2(\Omega ,M_2(\mathbb {R}))$, which shows that, $\forall \phi \in L^2(\Omega ,M_2(\mathbb {R}))$,

$$\begin{aligned} \int _{\Omega }\,\nabla \varphi _{l,\Psi (j)}\,:\,\phi \,\textrm{d}x&= \int _{\Omega }\,\nabla \varphi _{l,\Psi (j)}-V_{l,\Psi (j)}\,:\,\phi \,\textrm{d}x\\&\quad +\int _{\Omega }\,V_{l,\Psi (j)}\,:\,\phi \,\textrm{d}x,\\&\underset{j \rightarrow +\infty }{\longrightarrow } \int _{\Omega }\,\bar{V}_{l}\,:\,\phi \,\textrm{d}x. \end{aligned}$$

Then $\nabla \varphi _{l,\Psi (j)} \underset{j \rightarrow +\infty }{\rightharpoonup }\ \bar{V}_l$ in $L^2(\Omega ,M_2(\mathbb {R}))$. On the other hand, $\nabla \varphi _{l,\Psi (j)} \underset{j \rightarrow +\infty }{\rightharpoonup }\ \nabla \bar{\varphi }_l$ in $L^2(\Omega ,M_2(\mathbb {R}))$, which, by uniqueness of the weak limit in $L^2(\Omega ,M_2(\mathbb {R}))$, enables one to conclude that $\nabla \bar{\varphi }_l=\bar{V}_l\in L^4(\Omega ,M_2(\mathbb {R}))$.

Let us now focus on $(\det {V_{l,\Psi (j)}})$. One has $\det {V_{l,\Psi (j)}}=\det {\nabla \varphi _{l,\Psi (j)}}+d_{l,\Psi (j)}$ with

$$\begin{aligned} d_{l,\Psi (j)}=&\det {x_{l,\Psi (j)}}+\left( x_{l,\Psi (j)}\right) _{1,1}\dfrac{\partial \varphi _{l,\Psi (j)}^2}{\partial x_2}+\left( x_{l,\Psi (j)}\right) _{2,2}\dfrac{\partial \varphi _{l,\Psi (j)}^1}{\partial x_1}\\&-\left( x_{l,\Psi (j)}\right) _{1,2}\dfrac{\partial \varphi _{l,\Psi (j)}^2}{\partial x_1}-\left( x_{l,\Psi (j)}\right) _{2,1}\dfrac{\partial \varphi _{l,\Psi (j)}^1}{\partial x_2}, \end{aligned}$$

$\left( x_{l,\Psi (j)}\right) _{k,n}$ being the element of the $k^{\textrm{th}}$ row and $n^{\textrm{th}}$ column of the matrix $x_{l,\Psi (j)}$ and with $\varphi _{l,\Psi (j)}=\left( \varphi _{l,\Psi (j)}^1,\varphi _{l,\Psi (j)}^2\right)$. Then it holds that

$$\begin{aligned} \Vert d_{l,\Psi (j)}\Vert _{L^1(\Omega )}&\le \dfrac{1}{2}\,\Vert x_{l,\Psi (j)}\Vert ^2_{L^2(\Omega , M_2(\mathbb {R}))}\\&\quad +\Vert x_{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}\,\Vert \nabla \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}. \end{aligned}$$

The quantity $\Vert \nabla \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega ,M_2(\mathbb {R}))}$ being bounded independently of j and l and having $x_{l,\Psi (j)}\underset{j \rightarrow +\infty }{\longrightarrow }\ 0$ in $L^2(\Omega ,M_2(\mathbb {R}))$, it follows that $d_{l,\Psi (j)}\underset{j \rightarrow +\infty }{\longrightarrow }\ 0$ in $L^1(\Omega )$.

From [8, Theorem 1.14], if $\Psi _j \,\,\underset{j \rightarrow +\infty }{\rightharpoonup }\ \bar{\Psi }$ in $W^{1,2}(\Omega ,\mathbb {R}^2)$, then $\det {\nabla \Psi _j}\,\,\underset{j \rightarrow +\infty }{\longrightarrow } \det {\nabla \bar{\Psi }}$ in the sense of distributions. On the one hand, $\forall \phi \in \mathcal {D}(\Omega )$,

$$\begin{aligned} \displaystyle {\int _{\Omega }}\,\det {V_{l,\Psi (j)}}\,\phi \,\textrm{d}x \,\,\underset{j \rightarrow +\infty }{\longrightarrow } \,\,\overline{\delta }_l\,\phi \,\textrm{d}x, \end{aligned}$$

as $(\det {V_{l,\Psi (j)}})$ weakly converges to $\overline{\delta }_l$ in $L^2(\Omega )$ as j tends to $+\infty$. On the other hand,

$$\begin{aligned} \displaystyle {\int _{\Omega }}\,\det {V_{l,\Psi (j)}}\,\phi \,\textrm{d}x =\displaystyle {\int _{\Omega }}\,\det {\nabla \varphi _{l,\Psi (j)}}\,\phi \,\textrm{d}x+\displaystyle {\int _{\Omega }}\,d_{l,\Psi (j)}\,\phi \,\textrm{d}x, \end{aligned}$$

with ${\int _{\Omega }}\,\det {\nabla \varphi _{l,\Psi (j)}}\,\phi \,\textrm{d}x \,\,\underset{j \rightarrow +\infty }{\longrightarrow } \,\,\int _{\Omega }\,\det {\nabla \bar{\varphi }_l}\,\phi \,\textrm{d}x$, as $\det {\nabla \varphi _{l,\Psi (j)}}$ converges to $\det {\nabla \bar{\varphi }_l}$ in the sense of distributions when j tends to $+\infty$ and $\left| \int _{\Omega }\,d_{l,\Psi (j)}\,\phi \,\textrm{d}x\right| \le \Vert d_{l,\Psi (j)}\Vert _{L^1(\Omega )}\Vert \phi \Vert _{\mathcal {C}^0(\bar{\Omega })}$ $\underset{j\rightarrow +\infty }{\rightarrow }0$. Consequently, $L^2(\Omega ) \ni \det {\nabla \bar{\varphi }_l}=\bar{\delta }_l\in L^2(\Omega )$ in the sense of distributions and finally $\det {\nabla \bar{\varphi }_l}=\bar{\delta }_l$ a.e..

A similar reasoning applies to handle the component $\Vert W_{l,\Psi (j)}-\nabla \varphi _{l,\Psi (j)}\Vert ^2_{L^2(\Omega ,M_2(\mathbb {R}))}$ and leads to

$$\begin{aligned} \left| \begin{array}{ccc} \nabla \bar{\varphi }_l=\bar{V}_l=\bar{W}_l \in L^4(\Omega ,M_2(\mathbb {R})),\\ \det {\nabla \bar{\varphi }_l}=1\,\,\,{\text{ a.e. }}. \end{array}\right. \end{aligned}$$

Invoking again the generalised Poincaré inequality, it comes that $\bar{\varphi }_l\in {\text{ Id }}+W_0^{1,4}(\Omega ,\mathbb {R}^2)$ with $\det {\nabla \bar{\varphi }_l}=1$ a.e.. Subsequently, still following Ball’s results, $\bar{\varphi }_l$ is a homeomorphism of $\bar{\Omega }$ to $\bar{\Omega }$ and $\bar{\varphi }_l^{-1}\in W^{1,4}(\Omega ,\mathbb {R}^2)$.

Using the Lipschitz property of $Ey_l$ and Rellich-Kondrachov theorem ([5, Theorem IX.16]) that gives the compact embedding $W^{1,2}(\Omega ,\mathbb {R}^2) \subset L^2(\Omega )$, it can be observed that $\Vert s(\theta )-(Ey_l)\circ \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega )}^2\,\,\underset{j \rightarrow +\infty }{\longrightarrow } \,\,\Vert s(\theta )-(Ey_l)\circ \bar{\varphi }_{l}\Vert _{L^2(\Omega )}^2$, so that going back to (10), it yields

$$\begin{aligned}&\liminf _{j\rightarrow +\infty }\,\bigg [\mu \,\Vert V_{l,\Psi (j)}\Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+ \dfrac{\nu }{2}\,\Vert s(\theta )-(Ey_l)\\&\qquad \circ \varphi _{l,\Psi (j)}\Vert _{L^2(\Omega )}^2+\dfrac{\mu \alpha }{2}\,\Vert \det {V_{l,\Psi (j)}}-1\Vert _{L^2(\Omega )}^2\bigg ]\\&\quad \le \liminf _{j\rightarrow +\infty } E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\dfrac{C+1}{l}. \end{aligned}$$

Thus

$$\begin{aligned}&\mu \,\Vert \nabla \bar{\varphi }_l\Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\nu }{2}\,\Vert s(\theta )-(Ey_l)\circ \bar{\varphi }_{l}\Vert _{L^2(\Omega )}^2\\&\quad +\dfrac{\mu \alpha }{2}\,\Vert \det {\nabla \bar{\varphi }_l}-1\Vert _{L^2(\Omega )}^2 \\&\quad \le \liminf _{j\rightarrow +\infty } E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le \limsup _{j\rightarrow +\infty } E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\dfrac{C+1}{l}, \end{aligned}$$

and finally, as $\det {\nabla \bar{\varphi }_l}=1$ a.e.,

$$\begin{aligned}&\mu \,\Vert \nabla \bar{\varphi }_l\Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\nu }{2}\,\Vert s(\theta )-y_l\circ \bar{\varphi }_{l}\Vert _{L^2(\Omega )}^2\\&\quad \le \liminf _{j\rightarrow +\infty } E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le \limsup _{j\rightarrow +\infty } E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )+\dfrac{C+1}{l}. \end{aligned}$$

Observe that $\forall \varphi \in {\text{ Id }}+W_{0}^{1,4}(\Omega ,\mathbb {R}^2)$ with $\det {\nabla \varphi }=1$ a.e.,

$$\begin{aligned}&E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le E_{l,\Psi (j)}(\varphi ,\nabla \varphi ,\nabla \varphi )+\dfrac{1}{\gamma _{\Psi (j)}},\\&\quad \le \,\mu \,\Vert \nabla \varphi \Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\nu }{2}\,\Vert s(\theta ){-} y_l\circ \varphi \Vert _{L^2(\Omega )}^2{+}\dfrac{1}{\gamma _{\Psi (j)}}, \end{aligned}$$

which enables one to conclude that $\bar{\varphi }_l$ is a minimiser of $\mu \,\Vert \nabla \cdot \Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\nu }{2}\,\Vert s(\theta )-y_l\circ \cdot \Vert _{L^2(\Omega )}^2$, after passing to the $\liminf$ in the previous inequality when j tends to $+\infty$. Reasoning as before,

$$\begin{aligned}&\left| \Vert s(\theta )-y_l\circ \bar{\varphi }_l\Vert _{L^2(\Omega )}-\Vert s(\theta )-y\circ \bar{\varphi }_l\Vert _{L^2(\Omega )}\right| \\&\quad \le C\,\Vert y_l-y\Vert _{L^p(\Omega )}\le \dfrac{C}{l}, \end{aligned}$$

so that

$$\begin{aligned} \frac{\nu }{2}\,\Vert s(\theta )-y\circ \bar{\varphi }_l\Vert _{L^2(\Omega )}^2-\dfrac{C}{l}\le \frac{\nu }{2}\,\Vert s(\theta )-y_l\circ \bar{\varphi }_l\Vert _{L^2(\Omega )}^2, \end{aligned}$$

with $C:=\frac{C\nu }{2}$ and

$$\begin{aligned}&\displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )-\dfrac{C}{l}\le \mathcal {F}(\bar{\varphi }_l)-\dfrac{C}{l} \\&\quad \le \mu \,\Vert \nabla \bar{\varphi }_l\Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\nu }{2}\,\Vert s(\theta )-y_l\circ \bar{\varphi }_{l}\Vert _{L^2(\Omega )}^2. \end{aligned}$$

Passing to the limit when l tends to $+\infty$ yields

$$\begin{aligned} \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi )&\le \liminf _{l\rightarrow +\infty } \liminf _{j\rightarrow +\infty } \,E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\le \limsup _{l\rightarrow +\infty }\limsup _{j \rightarrow +\infty }\,E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le \displaystyle {\inf _{\varphi \in \mathcal {W}}}\,\mathcal {F}(\varphi ), \end{aligned}$$

and the result follows, after noticing that

$$\begin{aligned}&E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad \le E_{l,\Psi (j)}(\bar{\varphi }_l,\nabla \bar{\varphi }_l,\nabla \bar{\varphi }_l)+\dfrac{1}{\gamma _{\Psi (j)}},\\&\quad \le \,\mu \,\Vert \nabla \bar{\varphi }_l\Vert _{L^4(\Omega ,M_2(\mathbb {R}))}^4+\dfrac{\nu }{2}\,\Vert s(\theta )-y_l\circ \bar{\varphi }_l\Vert _{L^2(\Omega )}^2+\dfrac{1}{\gamma _{\Psi (j)}}, \end{aligned}$$

yielding

$$\begin{aligned}&\liminf _{j\rightarrow +\infty } \,E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)})\\&\quad =\limsup _{j\rightarrow +\infty } \,E_{l,\Psi (j)}(\varphi _{l,\Psi (j)},V_{l,\Psi (j)},W_{l,\Psi (j)}). \end{aligned}$$