First, we prove the following weak continuity lemma, which is a generalization of the classical result given in [50, Theorem 4.1]. Here, the coefficients may depend on the deformed configuration.
Lemma 4.1
Let \(\phi ^k \rightharpoonup \phi \in W^{1,p}(\Omega ; \mathbb {R}^n)\) and \(p>n\). Moreover, let \(V_i \in C^0(\overline{\Omega } \times \mathbb {R}^{n}; \mathbb {S}^{n-1})\), \(i=1,2\) and we denote
$$\begin{aligned} \mathbf{{v}}_i^k(\cdot ):=V_i\big (\cdot , \phi ^k(\cdot )\big ) \text { and } \mathbf{{v}}_i:=V_i\big (\cdot , \phi (\cdot )\big )\,,\quad i=1,2\;. \end{aligned}$$
Then
Moreover, for every symmetric positive definite \(M_i\), \(i=1,2\) with \(M_1^{-\frac{1}{2}} \in C^0(\overline{\Omega } \times \mathbb {R}^{n}; \mathbb {R}^{n \times n})\) and \(M_2^{\frac{1}{2}} \in C^0(\overline{\Omega } \times \mathbb {R}^{n}; \mathbb {R}^{n \times n})\) and the corresponding compositions
$$\begin{aligned} M_i^k(\cdot ):=M_i\big (\cdot , \phi ^k(\cdot )\big ) \text { and } \overline{M}_i:=M_i\big (\cdot , \phi (\cdot )\big ) \end{aligned}$$
we have
Proof
To prove (4.1) let \(\zeta \in L^{\frac{p}{p-n}}(\Omega )\). We show that
$$\begin{aligned} I^k&:=\int _\Omega \zeta \det \big ( \text {P}(\mathbf{{v}}_2^k) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1^k) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_1^k \big )\,{{\text {d}}}x \\ \rightarrow I&:=\int _\Omega \zeta \det \big ( \text {P}(\mathbf{{v}}_2) \mathcal {D}\phi \, \text {P}(\mathbf{{v}}_1) + \mathbf{{v}}_2 \otimes \mathbf{{v}}_1 \big )\,{{\text {d}}}x. \end{aligned}$$
Moreover, we denote
$$\begin{aligned} \overline{I}^k:=\int _\Omega \zeta \det \big ( \text {P}(\mathbf{{v}}_2) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1) + \mathbf{{v}}_2 \otimes \mathbf{{v}}_1 \big )\,{{\text {d}}}x\;. \end{aligned}$$
Using the inequality (cf. [27, Theorem 4.7])
$$\begin{aligned} |\det A - \det B | \le C |A-B|\max (|A|, |B|)^{n-1} \end{aligned}$$
and Hölder’s inequality it follows that
$$\begin{aligned}&\left| I^k - \overline{I}^k \right| \\&\quad \le C \int _\Omega |\zeta | \left| \text {P}(\mathbf{{v}}_2^k) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1^k) - \text {P}(\mathbf{{v}}_2) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_1^k - \mathbf{{v}}_2 \otimes \mathbf{{v}}_1 \right| \\&\qquad \cdot \max \bigg (\big |\text {P}(\mathbf{{v}}_2^k) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1^k) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_1^k\big |, \big |\text {P}(\mathbf{{v}}_2) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1) + \mathbf{{v}}_2 \otimes \mathbf{{v}}_1 \big |\bigg )^{n-1} {{\text {d}}}x\\&\quad \le C \,\Vert \zeta \Vert _{L^\frac{p}{p-n}} \left\| \big |\mathcal {D}\phi ^k\big |^{n-1} + 1 \right\| _{L^{\frac{p}{n-1}}} \\&\qquad \cdot \left\| \text {P}(\mathbf{{v}}_2^k) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1^k) - \text {P}(\mathbf{{v}}_2) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_1^k - \mathbf{{v}}_2 \otimes \mathbf{{v}}_1 \right\| _{L^p}\\&\quad \le C \, \Vert \zeta \Vert _{L^\frac{p}{p-n}} \Big ( \Vert \mathcal {D}\phi ^k\Vert _{L^p}^{n-1} + 1 \Big ) \\&\qquad \cdot \Big [ \Vert \mathcal {D}\phi ^k\Vert _{L^p} \Big ( \Vert \text {P}(\mathbf{{v}}_2^k)\Vert _{L^\infty }\Vert \text {P}(\mathbf{{v}}_1^k)-\text {P}(\mathbf{{v}}_1)\Vert _{L^\infty } \\&\qquad + \Vert \text {P}(\mathbf{{v}}_1)\Vert _{L^\infty }\Vert \text {P}(\mathbf{{v}}_2^k)-\text {P}(\mathbf{{v}}_2)\Vert _{L^\infty }\Big ) + \Big ( \Vert \mathbf{{v}}_1^k- \mathbf{{v}}_1\Vert _{L^\infty } + \Vert \mathbf{{v}}_2^k- \mathbf{{v}}_2\Vert _{L^\infty } \Big ) \Big ]\,. \end{aligned}$$
Here, we have used that
$$\begin{aligned} \big |\text {P}(\mathbf{{v}}_2) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1) + \mathbf{{v}}_2 \otimes \mathbf{{v}}_1 \big |^{n-1} \le ( \big |\mathcal {D}\phi ^k\big | + 1 )^{n-1} \le C ( \big |\mathcal {D}\phi ^k\big |^{n-1} + 1 )\,. \end{aligned}$$
By the Rellich–Kondrakov embedding theorem ([1], Theorem 6.3 III), there exist subsequences of \(\mathbf{{v}}_i^k\), \(i=1,2\), which for simplicity of notation are again denoted by \(\mathbf{{v}}_i^k\), \(i=1,2\), that converge uniformly to \(\mathbf{{v}}_i\), \(i=1,2\), respectively. Taking into account the Lipschitz continuity estimate
$$\begin{aligned} |\text {P}(e)-\text {P}(f)|=|(e-f)\otimes e + f \otimes (e-f)| \le 2 \sqrt{n}|e - f| \end{aligned}$$
and that \(\mathbf{{v}}_i^k \rightarrow \mathbf{{v}}_i\), \(i=1,2\) in \(L^\infty \) we obtain \(| I^k - \overline{I}^k | \rightarrow 0\) for \(k\rightarrow \infty \).
Next, we replace \(\mathbf{{v}}_i\), \(i=1,2\) in \(\bar{I}^k\) by a piecewise constant approximation on a grid superimposed to the computational domain \(\Omega \). Explicitly, we consider the finitely many nonempty intersection \(\omega _\delta ^z = \delta (z+ [0,1]^n) \cap \Omega \) of cubical cells with \(\Omega \) for \(z \in \mathbb {Z}^n\) and define
$$\begin{aligned} \bar{I}^k_\delta := \sum _{z\in \mathbb {Z}^n} \int _{\omega _\delta ^z} \zeta \det \big ( \text {P}(\mathbf{{v}}_2(z_\delta )) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1(z_\delta )) + \mathbf{{v}}_2(z_\delta ) \otimes \mathbf{{v}}_1(z_\delta ) \big ){{\text {d}}}x\,, \end{aligned}$$
where \(z_\delta \) is any point in \(\bar{\Omega }\cap \omega _\delta ^z\) if this set is nonempty. Using analogous estimates as above, we obtain
$$\begin{aligned}&\left| \overline{I}_\delta ^k - \overline{I}^k \right| \\&\quad \le C \Vert \zeta \Vert _{L^\frac{p}{p-n}} \Big ( \Vert \mathcal {D}\phi ^k\Vert _{L^p}^{n-1} + 1 \Big )\\&\qquad \cdot \Big [ \Vert \mathcal {D}\phi ^k\Vert _{L^p} \Big ( \Vert \text {P}(\mathbf{{v}}_{2,\delta })\Vert _{L^\infty }\Vert \text {P}(\mathbf{{v}}_{1,\delta })-\text {P}(\mathbf{{v}}_1)\Vert _{L^\infty }\\&\qquad +\Vert \text {P}(\mathbf{{v}}_1)\Vert _{L^\infty }\Vert \text {P}(\mathbf{{v}}_{2,\delta })-\text {P}(\mathbf{{v}}_2)\Vert _{L^\infty }\Big ) + \Big ( \Vert \mathbf{{v}}_{2,\delta }- \mathbf{{v}}_2\Vert _{L^\infty } + \Vert \mathbf{{v}}_{1,\delta }- \mathbf{{v}}_1\Vert _{L^\infty } \Big ) \Big ]\,, \end{aligned}$$
where \(\mathbf{{v}}_{1,\delta }\) and \(\mathbf{{v}}_{2,\delta }\) are piecewise constant functions in \(L^\infty \) with \(\mathbf{{v}}_{1,\delta }|_{\omega _\delta ^z} = \mathbf{{v}}_1(z_\delta )\) and \(\mathbf{{v}}_{2,\delta }|_{\omega _\delta ^z} = \mathbf{{v}}_2(z_\delta )\), respectively.
Using the uniform continuity of \(\mathbf{{v}}_2\) and \(\mathbf{{v}}_1\) on \(\overline{\Omega }\), we obtain that \(\left| \overline{I}^k_\delta - \overline{I}^k \right| \le \beta (\delta )\) for a monotonically increasing continuous function \(\beta : \mathbb {R}^+_0 \rightarrow \mathbb {R}\) with \(\beta (0) =0\). In particular, the convergence is uniform with respect to k. The same argument applies for the difference of I and
$$\begin{aligned} \bar{I}_\delta := \sum _{z\in \mathbb {Z}^n} \int _{\omega _\delta ^z} \zeta \det \big ( \text {P}(\mathbf{{v}}_2(z_\delta )) \mathcal {D}\phi \text {P}(\mathbf{{v}}_1(z_\delta )) + \mathbf{{v}}_2(z_\delta ) \otimes \mathbf{{v}}_1(z_\delta ) \big ){{\text {d}}}x \end{aligned}$$
and we get \(\left| \bar{I}_\delta - I \right| <C\beta (\delta )\). Using (2.3) it follows that
$$\begin{aligned} Q(\mathbf{{v}}_{2}(z_\delta ))^T \Big (\text {P}(\mathbf{{v}}_{2}(z_\delta )) A \text {P}(\mathbf{{v}}_{1}(z_\delta )) + \mathbf{{v}}_{2}(z_\delta ) \otimes \mathbf{{v}}_{1}(z_\delta )\Big )Q(\mathbf{{v}}_{1}(z_\delta )) = \left( \begin{array}{c|c} \tilde{A} &{} 0 \\ \hline 0 &{} 1 \end{array} \right) . \end{aligned}$$
Thus \( \det ( \text {P}(\mathbf{{v}}_2(z_\delta )) A \text {P}(\mathbf{{v}}_1(z_\delta )) + \mathbf{{v}}_2(z_\delta ) \otimes \mathbf{{v}}_1(z_\delta )) = \det (\tilde{A})\) represents an \((n-1)\times (n-1)\) minor of the linear mapping corresponding to the matrix A with respect to different orthogonal basis in preimage space (associated with \(\text {P}(\mathbf{{v}}_1(z_\delta ))\) and \(\mathbf{{v}}_1(z_\delta )\)) and the image space (associated with \(\text {P}(\mathbf{{v}}_2(z_\delta ))\) and \(\mathbf{{v}}_2(z_\delta )\)). Indeed, denoting \(Q_i:=Q(\mathbf{{v}}_i(z_\delta ))\) we have
$$\begin{aligned}&\int _{\omega _\delta ^z} \zeta (x) \det \big ( \text {P}(\mathbf{{v}}_2(z_\delta )) \mathcal {D}\phi ^k(x) \text {P}(\mathbf{{v}}_1(z_\delta )) + \mathbf{{v}}_2(z_\delta ) \otimes \mathbf{{v}}_1(z_\delta ) \big ){{\text {d}}}x\,\\&\quad =\int _{\omega _\delta ^z} \zeta (x) \det \big ( Q_2^T \big (\text {P}(\mathbf{{v}}_2(z_\delta )) \mathcal {D}\phi ^k(x) \text {P}(\mathbf{{v}}_1(z_\delta )) + \mathbf{{v}}_2(z_\delta ) \otimes \mathbf{{v}}_1(z_\delta ) \big ) Q_1 \big ){{\text {d}}}x\,\\&\quad =\int _{\omega _\delta ^z} \zeta (x) \det \big ( Q_2^T \text {P}(\mathbf{{v}}_2(z_\delta )) Q_2 Q_2^T \mathcal {D}\phi ^k(x) Q_1 Q_1^T \text {P}(\mathbf{{v}}_1(z_\delta )) Q_1 + e_n \otimes e_n \big ){{\text {d}}}x\,\\&\quad =\int _{\omega _\delta ^z} \zeta (x) \det \big ( \text {P}(e_n) Q_2^T \mathcal {D}\phi ^k(x) Q_1 \text {P}(e_n) + e_n \otimes e_n \big ){{\text {d}}}x\,\\&\quad =\int _{Q_1^T\omega _\delta ^z} \zeta (Q_1y) \det \big ( \text {P}(e_n) \mathcal {D}\big ( Q_2^T \circ \phi ^k \circ Q_1\big ) (y) \text {P}(e_n) + e_n \otimes e_n \big ){{\text {d}}}y\,\\&\quad =\int _{Q_1^T\omega _\delta ^z} \zeta (Q_1y) {{\text {Cof}}}_{nn}\big ( \mathcal {D}\big ( Q_2^T \circ \phi ^k \circ Q_1\big ) (y) \big ){{\text {d}}}y\,, \end{aligned}$$
where we have used the orthogonal change of variables \(y = Q_1^T x\) and \({{\text {Cof}}}_{nn}\) denotes the minor obtained by erasing the last column and the last row. This change of orthogonal coordinates is fixed on each cell \(\omega _\delta ^z\). Since for each \(\delta \) the domain \(\Omega \) is covered by finitely many cells \(\omega _\delta ^z\), using the above computation and standard weak continuity results [20, Theorem 8.20] for determinants of minors of the Jacobian we obtain that \(\bar{I}^k_\delta \rightarrow \bar{I}_\delta \) for \(k\rightarrow \infty \). Finally, for given \(\epsilon \) we first choose \(\delta \) small enough to ensure that \(\left| \bar{I}_\delta - I \right| + \left| \bar{I}^k_\delta - \bar{I}^k \right| \le \tfrac{\epsilon }{2}\). Then, we choose k large enough to ensure that \(\left| I^k - \bar{I}^k \right| + \left| \bar{I}^k_\delta - \bar{I}_\delta \right| \le \tfrac{\epsilon }{2}\). This proves that a subsequence of \(I^k\) converges to I for \(k\rightarrow \infty \). Since the limit does not depend on the subsequence, we finally obtain weak convergence for the whole sequence.
To prove (4.2), consider the three sequences of matrix functions
$$\begin{aligned}&\text {P}(\mathbf{{v}}_2^k) (M_2^k)^{\frac{1}{2}} \text {P}(\mathbf{{v}}_2^k) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_2^k,\; \text {P}(\mathbf{{v}}_2^k) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1^k) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_1^k,\nonumber \\&\quad \text { and } \text {P}(\mathbf{{v}}_1^k) (M_1^k)^{-\frac{1}{2}} \text {P}(\mathbf{{v}}_1^k) + \mathbf{{v}}_1^k \otimes \mathbf{{v}}_1^k. \end{aligned}$$
(4.3)
The determinant of the second expression above converges weakly as \(k \rightarrow \infty \) by the first part of the lemma, while the determinants of the first and third can be assumed to converge uniformly. Moreover, the matrices in (4.3) have the block structure shown in (2.3), so multiplying the three together and taking into account that \(\text {P}\) is a projection (depending on the argument) recovers the matrix
$$\begin{aligned} \text {P}(\mathbf{{v}}_2^k) (M_2^k)^{\frac{1}{2}} \text {P}(\mathbf{{v}}_2^k) \mathcal {D}\phi ^k \text {P}(\mathbf{{v}}_1^k) (M_1^k)^{-\frac{1}{2}} \text {P}(\mathbf{{v}}_1^k) + \mathbf{{v}}_2^k \otimes \mathbf{{v}}_1^k \end{aligned}$$
appearing in the statement. Multiplicativity of the determinant and the fact that a product of strongly converging and one weakly converging sequence converges weakly then finishes the proof. \(\square \)
We are now in a position to prove existence of a minimizing deformation for the hypersurface matching energy E in a suitable set of admissible deformations. Of particular difficulty is that derivatives of \(\mathbf{{d}}_2\) are not defined in the whole of \(\Omega \) and that in the functional these derivatives are evaluated at deformed positions. We handle this by ensuring that the involved deformations are such that terms involving these derivatives are not evaluated near the singularities. We obtain the following theorem:
Theorem 4.2
(Existence of minimizing deformations) Let \(\mathcal {M}_1,\mathcal {M}_2\) be \(C^{2,1}\) compact embedded hypersurfaces in \(\mathbb {R}^n\) such that a \(C^1\) diffeomorphism \(\varphi : \mathcal {M}_1 \rightarrow \mathcal {M}_2\) exists between them.
Assume further that
$$\begin{aligned} 0< \sigma < \min ( {\text {dist}}( \mathcal {M}_1, {\text {sing}}\,\mathbf{{d}}_1 ), {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) ), \end{aligned}$$
(4.4)
where \({\text {sing}}\,\mathbf{{d}}_i\) is the set of points where \(\mathbf{{d}}_i\) is not differentiable, and that \({\mathcal {C}}:\mathbb {R}^{n \times n} \rightarrow \text {SPD}(n)\) is continuous. Then there exists a constant \(0 < \nu _0 := \nu _0(\Omega , \mathcal {M}_1, \mathcal {M}_2, \sigma , p, \alpha _p)\) such that for \(0 < \nu \le \nu _0\), the functional \(E_\nu \) has at least one minimizer \(\phi \) among deformations in the space \(W^{1,p}_0(\Omega ; \mathbb {R}^{n})+{\text {id}}\). Moreover, \(\phi \) is a homeomorphism of \(\Omega \) into \(\Omega \), and \(\phi ^{-1}\in W^{1,\theta }(\Omega ; \mathbb {R}^{n})\), where \(\theta \) is given by \(\theta =q(1+s)/(q+s)\).
Proof
We proceed in several steps.
Step 1 Coercivity First, we point out the coercivity enjoyed by our functional. Using the Poincaré and Morrey inequalities ([41], Theorem 12.30 and 11.34), and the Dirichlet boundary conditions we have
$$\begin{aligned} \Vert \phi \Vert _{C^{0,\alpha }(\Omega )} \le C \Vert \phi \Vert _{W^{1,p}(\Omega )} \le C ( 1 + \Vert \mathcal {D}\phi \Vert _{L^p(\Omega )}) \le C (1 + E_\nu [\phi ]^{\frac{1}{p}}), \end{aligned}$$
(4.5)
for any \(\phi \in W^{1,p}_0(\Omega )+{\text {id}}\) and \(\alpha =1-n/p\).
Step 2 Lower semicontinuity along sequences of constrained deformations For the remainder of the proof, a deformation \(\phi \in W^{1,p}_0(\Omega ; \mathbb {R}^n) + {\text {id}}\), \(p>n\) is termed \(\rho \)-admissible for \(\rho > 0 \), if
-
\(E_{\text {vol}}[\phi ]< +\infty \),
-
\(\det \mathcal {D}\phi (x) > 0\) for a.e. \(x \in \Omega \), and
-
for all \(x \in {\text {supp}}\,\big ( \eta _\sigma \circ \mathbf{{d}}_1 \big )\) and every \(y \in {\text {sing}}\,(\mathbf{{d}}_2)\), we have \(\left| \phi (x) - y \right| \ge \rho \).
Notice that since \(p>n\), \(\phi \) has a unique continuous representative, so the third property is well defined.
First, notice that with the assumption (4.4) we have
$$\begin{aligned} {\text {supp}}\,(\eta _\sigma \circ \mathbf{{d}}_i) = \{|\mathbf{{d}}_i|\le \sigma \} \subset \Omega \setminus \overline{{\text {sing}}\,(\mathbf{{d}}_i)}\,,\quad i=1,2. \end{aligned}$$
(4.6)
Let \(\phi ^k\) be a sequence of \(\rho \)-admissible deformations with \(E_{\text {vol}}[\phi ^k]\le C\). By (4.5) and using the Banach–Alaoglu and Rellich–Kondrakov theorems, a subsequence (again denoted by \((\phi ^k)\)) converges to a deformation \(\phi \), both in the \(W^{1,p}\)-weak and uniform topologies.
Now, we have ([20, Theorem 8.20])
$$\begin{aligned} (\det \mathcal {D}\phi ^k, {\text {Cof}}\,\mathcal {D}\phi ^k) \rightharpoonup (\det \mathcal {D}\phi , {\text {Cof}}\,\mathcal {D}\phi ) \text { in } L^{\frac{p}{n}}(\Omega ) \times \left( L^{\frac{p}{n-1}}(\Omega )\right) ^{n^2}. \end{aligned}$$
(4.7)
Additionally, since (4.7) holds and because \(E_\nu [\phi _k]\) is bounded, \(\int _\Omega (\det \mathcal {D}\phi _k)^{-s} d x\) is bounded by the definition of \(\hat{W}\) and \(\det \mathcal {D}\phi _k \ge 0\) a.e. Together with (4.7), we have
$$\begin{aligned} \det \mathcal {D}\phi (x) > 0 \text { a.e.,} \end{aligned}$$
(4.8)
so that \(\phi \) is again \(\rho \)-admissible.
Notice also that by a.e. positivity of the determinants, (4.7) and a standard lower semicontinuity result for convex integrands (see, e.g., [20, Theorem 3.23]) implies
$$\begin{aligned} E_{\text {vol}}[\phi ] \le \liminf _{k \rightarrow \infty } E_{\text {vol}}[\phi ^k], \end{aligned}$$
and uniform convergence of \(\phi ^k\) immediately leads to
$$\begin{aligned} E_{\text {match}}[\phi ] = \lim _{k \rightarrow \infty } E_{\text {match}}[\phi ^k]. \end{aligned}$$
We claim that under the assumptions of this theorem, we also have that
$$\begin{aligned} E_{\text {mem}}\left[ \,\phi \,\right] \le \liminf _{k \rightarrow \infty } E_{\text {mem}}[\phi ^k] \end{aligned}$$
(4.9)
and
$$\begin{aligned} E_{\text {bend}}\left[ \,\phi \,\right] \le \liminf _{k \rightarrow \infty } E_{\text {bend}}[\phi ^k]. \end{aligned}$$
(4.10)
To see this, notice that \(\phi ^k, \phi \) being \(\rho \)-admissible ensures that the normal vectors satisfy
$$\begin{aligned} \mathbf{{n}}_1,\, \mathbf{{n}}_2 \circ \phi ^k,\, \mathbf{{n}}_2 \circ \phi \in C^{0}( \{|\mathbf{{d}}_1|\le \sigma \}; \mathbb {R}^{n}). \end{aligned}$$
Consequently, the first part of Lemma 4.1 (with \(V_i=\mathbf{{n}}_i\)) implies
$$\begin{aligned} \begin{aligned}&\chi _{\{|\mathbf{{d}}_1|\le \sigma \}}\Big ( \mathcal {D}_{\text {tg}}\phi ^k, \det \big (\mathcal {D}_{\text {tg}}\phi ^k + (\mathbf{{n}}_2\circ \phi ^k) \otimes \mathbf{{n}}_1\big )\Big ) \\&\rightharpoonup \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}\Big ( \mathcal {D}_{\text {tg}}\phi , \det \big (\mathcal {D}_{\text {tg}}\phi + (\mathbf{{n}}_2\circ \phi ) \otimes \mathbf{{n}}_1\big )\Big )\text { in } \left( L^p(\Omega ) \right) ^{n^2} \times L^{\frac{p}{n}}(\Omega ), \end{aligned} \end{aligned}$$
(4.11)
with \(\chi _{\{|\mathbf{{d}}_1|\le \sigma \}}\) denoting the indicator function. Combining (4.11) with the polyconvexity of W, defining the function \(E_{\text {mem}}\), both introduced in (3.1) we find the assertion (4.9).
Furthermore, by our assumptions on \(\mathcal {M}_i\) (see Sect. 2), we have that
$$\begin{aligned} \chi _{\{|\mathbf{{d}}_i|\le \sigma \}}{\mathcal {S}}_i = \chi _{\{|\mathbf{{d}}_i|\le \sigma \}}\mathcal {D}^2 \mathbf{{d}}_i \in C^0( \overline{\Omega }; \mathbb {R}^{n \times n}). \end{aligned}$$
Since \({\mathcal {C}}\) produces uniformly positive matrices, we have \(\chi _{\{|\mathbf{{d}}_1|\le \sigma \}}({\mathcal {C}}({\mathcal {S}}^{ext}_1))^{-1}\in C^0(\overline{\Omega }; \mathbb {R}^{n \times n})\). We can then use a continuity result for square roots of nonnegative definite matrix-valued functions defined on \(\Omega \) [15, Theorem 1.1] to see that
$$\begin{aligned} \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}({\mathcal {C}}({\mathcal {S}}^{ext}_1))^{-\frac{1}{2}}\in & {} C^0(\overline{\Omega }; \mathbb {R}^{n \times n}),\\ \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}( {\mathcal {C}}({\mathcal {S}}^{ext}_2) \circ \phi ^k )^{\frac{1}{2}}\in & {} C^0(\overline{\Omega }; \mathbb {R}^{n \times n}),\\ \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}({\mathcal {C}}({\mathcal {S}}^{ext}_2) \circ \phi )^{\frac{1}{2}}\in & {} C^0(\overline{\Omega }; \mathbb {R}^{n \times n}). \end{aligned}$$
The second part of Lemma 4.1 implies the weak convergence
from which (4.10) follows by using the polyconvexity of W.
Step 3 Existence of minimizers restricted to admissible deformations Since we have already seen that the set of \(\rho \)-admissible deformations is weakly closed and weakly compact, and that every term of E is weakly lower semicontinuous on this set, we just need to check that for all fixed \(\nu > 0\), the set of \(\rho \)-admissible deformations, with adequate \(\rho \), is not empty.
For some given \(\sigma \) satisfying \({\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \sigma > 0\) let \(\rho \) satisfy
$$\begin{aligned} 0< \rho < {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \sigma \;. \end{aligned}$$
(4.12)
We construct a deformation \(\hat{\varphi }\), which is \(\rho \)-admissible and satisfies \(E_\nu [\hat{\varphi }] < \infty \). By assumption, there exists a diffeomorphism \(\varphi :\mathcal {M}_1\rightarrow \mathcal {M}_2\). Thus, we construct an extension of this diffeomorphism to \(\{|\mathbf{{d}}_1|\le \sigma \}\) along the normal directions using
$$\begin{aligned} \hat{\varphi }( x+s\mathbf{{n}}_1(x) ):=\varphi (x)+s \mathbf{{n}}_2(\varphi (x)), \text { for }x \in \mathcal {M}_1, -\sigma \le s \le \sigma . \end{aligned}$$
(4.13)
We can then extend \(\hat{\varphi }\) to the inside and outside components \(\Omega _i, \Omega _o\) of \(\Omega \setminus \{|\mathbf{{d}}_1|\le \sigma \}\) by solving the minimization problems for \(E_{{\text {vol}}}\) with Dirichlet boundary conditions given by (4.13) on \(\partial \Omega _i\) and \(\partial \Omega _o \setminus \partial \Omega \), and by \(\hat{\varphi }(x)=x\) on \(\partial \Omega \). For the resulting \(\hat{\varphi }\), we have
$$\begin{aligned} E_{\text {match}}[\hat{\varphi }]=0,\, E_{\text {vol}}[\hat{\varphi }]<\infty ,\, E_{\text {mem}}[\hat{\varphi }]<\infty ,\, E_{\text {bend}}[\hat{\varphi }]<\infty , \end{aligned}$$
where the first two statements follow by construction, and the last two by virtue of \(\varphi \) being a diffeomorphism and the choice of \(\sigma \). Moreover, we note that since \(\hat{\varphi }\) has finite energy and the growth conditions assumed for \(\hat{W}\) [see (3.7)], the condition \(\det \mathcal {D}\hat{\varphi }(x) > 0\) for a.e. x is also satisfied [4].
Step 4 A priori estimate to remove the constraint Next, we show that for any \(\rho \) satisfying (4.12) there exists a parameter \(\nu _0 > 0\) such that for all \(0< \nu < \nu _0\) the constrained minimizers of \(E_\nu \) subject to (4.4) solves the unconstrained optimization problem, consisting in minimizing \(E_\nu \) on \(W_0^{1,p}+{\text {id}}\).
To this end, we verify that every \(\phi \) that satisfies
$$\begin{aligned} E_\nu [ \phi ] \le E_\nu [ \hat{\varphi }] \end{aligned}$$
(4.14)
is \(\rho \)-admissible. It is immediate from (4.14) that \(E_{\text {vol}}(\phi ) < + \infty \), and from the definition of \(\hat{W}\) in (3.7) it follows with the same arguments as in (4.8) that \(\det \phi > 0\) a.e.
We prove now that for all deformations \(\phi \) satisfying (4.14) also satisfy
$$\begin{aligned} \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \le {\text {dist}}(\mathcal {M}_2,{\text {sing}}\,\mathbf{{d}}_2) - \rho \;. \end{aligned}$$
(4.15)
This is sufficient because from (4.15) it follows for all x satisfying \(|\mathbf{{d}}_1(x)| \le \sigma \) by the triangle inequality that
$$\begin{aligned} \begin{aligned} \rho&\le {\text {dist}}(\mathcal {M}_2,{\text {sing}}\,\mathbf{{d}}_2) - \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}\\&= {\text {dist}}(\mathcal {M}_2,{\text {sing}}\,\mathbf{{d}}_2) - {\text {dist}}(\phi (x),\mathcal {M}_2) \\&\le {\text {dist}}(\phi (x),{\text {sing}}\,\mathbf{{d}}_2), \end{aligned} \end{aligned}$$
which is the third property of a \(\rho \)-admissible deformation \(\phi \).
To prove (4.15), we use the triangle inequality and estimate
$$\begin{aligned} \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \le \sigma + \Vert \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}. \end{aligned}$$
(4.16)
By the monotonicity of \(\eta _\sigma \) and the fact that the signed distance functions \(\mathbf{{d}}_i\) are Lipschitz continuous with constant 1 we have, for each \(\hat{\sigma } \in (0, \sigma )\) that
$$\begin{aligned} \begin{aligned}&\Vert \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \\&\quad \le \left( 1 + \Vert \phi \Vert _{C^{0,\alpha }(\{\sigma - \hat{\sigma } \le |\mathbf{{d}}_1| \le \sigma \})}\right) \hat{\sigma }^\alpha \\&\qquad + \frac{ \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )}}{\eta _\sigma (\sigma - \hat{\sigma } )}. \end{aligned} \end{aligned}$$
(4.17)
Estimates (4.5) and (4.14) imply in turn
$$\begin{aligned} \Vert \phi \Vert _{C^{0,\alpha }(\{\sigma - \hat{\sigma } \le |\mathbf{{d}}_1| \le \sigma \})} \le C \Vert \phi \Vert _{W^{1,p}(\Omega )} \le C (1+E_\nu [\hat{\varphi }]^{\frac{1}{p}}). \end{aligned}$$
(4.18)
Finally, combining (4.16), (4.17), and (4.18) we obtain
$$\begin{aligned} \begin{aligned}&\Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \\&\quad \le \sigma + \left( 1 + C ( 1 + E_\nu [\hat{\varphi }]^{\frac{1}{p}} ) \right) \hat{\sigma }^\alpha \\&\qquad + \frac{1}{\eta _\sigma (\sigma - \hat{\sigma } )} \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )}. \end{aligned} \end{aligned}$$
(4.19)
Now we can apply Ehrling’s lemma [54, Theorem 7.30] for the embeddings \( W^{1,p}(\Omega ) \subset \subset L^\infty (\Omega ) \subset L^2(\Omega )\) to control the last term in (4.19). Taking into account, the Poincaré inequality and Dirichlet boundary conditions, we obtain for any \(\epsilon > 0\) a constant \(C(\epsilon ) > 0\) such that
$$\begin{aligned} \begin{aligned}&\Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )} \\&\quad \le \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \Omega )} \\&\quad \le C(\epsilon )\Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^2( \Omega )}\\&\qquad + \epsilon \, C \left( \Vert \nabla ( \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) )\Vert _{L^p( \Omega )}+1\right) . \end{aligned} \end{aligned}$$
(4.20)
Now, for the first term in the right-hand side of (4.20) we can estimate
$$\begin{aligned} \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^2( \Omega )} = \nu ^{\frac{1}{2}} E_{\text {match}}[\phi ]^{\frac{1}{2}} \le \nu ^{\frac{1}{2}} E_\nu [\hat{\varphi }]^{\frac{1}{2}}. \end{aligned}$$
(4.21)
For the second term, denoting \({\text {diam}}\,\Omega = \sup _{x,y \in \Omega }|x-y|\),
$$\begin{aligned} \begin{aligned}&\Vert \nabla ( \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) )\Vert _{L^p( \Omega )} \\&\quad \le \Vert \nabla ( \eta _\sigma \circ \mathbf{{d}}_1 ) ( \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 ) \Vert _{L^p(\Omega )} \\&\qquad + \Vert ( \eta _\sigma \circ \mathbf{{d}}_1 ) \nabla ( \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 ) \Vert _{L^p(\Omega )} + 1 \\&\quad \le C \nu ^{\frac{1}{p}} \left( \Vert \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 \Vert ^{\frac{p-2}{p}}_{L^\infty ( \Omega )} E_{\text {match}}[\phi ]^{\frac{1}{p}}\right) \\&\qquad + C \Big ( \Vert \mathcal {D}\phi \Vert _{L^p(\Omega )} + \Vert \nabla \mathbf{{d}}_1 \Vert _{L^{p}(\Omega )} + 1 \Big ) \\&\quad \le C \nu ^{\frac{1}{p}} \left( ( \Vert \phi \Vert _{C^{0,\alpha }(\Omega )} + 2 {\text {diam}}\,\Omega )^{\frac{p-2}{p}}E_\nu [\hat{\varphi }]^{\frac{1}{p}} \right) + C \left( E_\nu [\hat{\varphi }]^{\frac{1}{p}} + 1 \right) \\&\quad \le C \nu ^{\frac{1}{p}} \left( ( 1 + E_\nu [\hat{\varphi }]^{\frac{1}{p}} )^{\frac{p-2}{p}} E_\nu [\hat{\varphi }]^{\frac{1}{p}}\right) + C \left( E_\nu [\hat{\varphi }]^{\frac{1}{p}} + 1\right) , \end{aligned} \end{aligned}$$
(4.22)
where we have applied the product rule, the definition of \(E_{\text {match}}\), \(\eta _\sigma \in C^\infty _0\), \(\eta _\sigma \le C\), that \(| \nabla \mathbf{{d}}_i | = 1\) a.e., \(i=1,2\), the chain rule, and (4.14). The use of the chain rule is justified by [46, Theorem 2.2], since \(\mathbf{{d}}_2\) has Lipschitz constant 1.
Together, (4.20), (4.21), and (4.22) imply
$$\begin{aligned} \begin{aligned}&\Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )} \\&\quad \le \nu ^{\frac{1}{p}} \left( C(\epsilon ) \nu ^{\frac{1}{2}-\frac{1}{p}} E_\nu [\hat{\varphi }]^{\frac{1}{2}} + \epsilon \, C ( 1 + E_\nu [\hat{\varphi }]^{\frac{1}{p}} )^{\frac{p-2}{p}} E_\nu [\hat{\varphi }]^{\frac{1}{p}}\right) \\&\qquad + \epsilon \, C \left( E_\nu [\hat{\varphi }]^{\frac{1}{p}} + 1\right) . \end{aligned} \end{aligned}$$
(4.23)
In light of (4.19) and (4.23), and since \(E_\nu [\hat{\varphi }]\) is independent of \(\nu \), we can now choose first \(\hat{\sigma }\), then \(\epsilon \) and finally \(\nu \) small enough to obtain
$$\begin{aligned} \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}&\le \sigma + ( {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \sigma - \rho ) \\&\le {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \rho \;. \end{aligned}$$
Step 5 Injectivity The injectivity and regularity of the inverse follow by the growth conditions satisfied by \(E_\text {vol}\) and classical results of Ball [4, Theorems 2 and 3]. Note that Theorem 3 in [4] is stated in the mechanical application context in dimension \(n=3\), but it holds also in \(\mathbb {R}^n\) following the same proof and using the condition \(s>(n-1)q/(q-n)\). \(\square \)
We have particularized the statement of Theorem 4.2 to the case of Dirichlet boundary conditions to ensure global invertibility. In fact, we also have existence of minimizing deformations for the case of Neumann boundary conditions.
Corollary 4.3
(Natural boundary conditions) Under the assumptions of Theorem 4.2, there exists a constant
$$\begin{aligned} 0 < \nu _N = \nu _N(\Omega , \mathcal {M}_1, \mathcal {M}_2, \sigma , p, \alpha _p) \end{aligned}$$
such that for \(0 < \nu \le \nu _N\), the functional \(E_\nu \) possesses at least one minimizer among deformations in the space \(W^{1,p}(\Omega ; \mathbb {R}^{n})\).
Proof
The proof follows the same arguments used for Theorem 4.2, so we only point out the necessary modifications. We need a replacement for the coercivity estimate (4.5) and claim
$$\begin{aligned} \begin{aligned} \Vert \phi \Vert _{W^{1,p}(\Omega )}&\le C(1 + \nu ^{\frac{1}{2}} E_{\text {match}}[\phi ]^{\frac{1}{2}} + \Vert \mathcal {D}\phi \Vert _{L^p(\Omega )}) \\&\le C (1 + \nu ^{\frac{1}{2}} E_\nu [\phi ]^{\frac{1}{2}}+ E_\nu [\phi ]^{\frac{1}{p}} ). \end{aligned} \end{aligned}$$
(4.24)
To verify this let us consider \(\omega :=\{|\mathbf{{d}}_1| \le \sigma /2 \}\). An adequate Poincaré inequality (see, e.g., [41, Theorem 12.23]) implies that
$$\begin{aligned} \Vert \phi \Vert _{W^{1,p}(\Omega )} \le C \left( \Vert \mathcal {D}\phi \Vert _{L^p(\Omega )}+\left| \int _\omega \phi \,{{\text {d}}}x \right| \,\right) , \end{aligned}$$
and we estimate the second term in the right-hand side by
$$\begin{aligned} \begin{aligned} \left| \int _\omega \phi \,{{\text {d}}}x \right|&\le \int _\omega | \phi | \,{{\text {d}}}x \le \int _\omega | \mathbf{{d}}_2 \circ \phi | \,{{\text {d}}}x + |\omega |\sup _{x \in \mathcal {M}_2}|x| \\&\quad \le \int _\omega | \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 | \,{{\text {d}}}x + \int _\omega |\mathbf{{d}}_1| \,{{\text {d}}}x + |\omega |\sup _{x \in \mathcal {M}_2}|x| \\&\quad \le \eta _\sigma \left( \frac{\sigma }{2}\right) ^{-1} |\omega |^{-\frac{1}{2}} \left( \nu E_{\text {match}}[\phi ]\right) ^{\frac{1}{2}} + \int _\omega |\mathbf{{d}}_1| \,{{\text {d}}}x + |\omega |\sup _{x \in \mathcal {M}_2}|x|, \end{aligned} \end{aligned}$$
where Hölder’s inequality has been used to compare \(L^1\) and \(L^2\) norms. Therefore, (4.24) follows.
The proof of the estimate for \(\Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}\) (to ensure that deformations stay away from the singularities of \(\mathbf{{d}}_2\)) is still valid with minor modifications, since \(\nu \) appears in (4.24) multiplicatively. \(\square \)
We conclude this section with the following proposition, which explores the penalization limit in which the parameter \(\nu \) tends to zero.
Proposition 4.4
Let \(\{\nu _k\}_{k \in \mathbb {N}}\), be a sequence of penalty matching parameters such that \(\nu _k \rightarrow 0\) as \(k \rightarrow \infty \), and \(\phi ^k\) be solutions of the Dirichlet minimization problem for \(E_{\nu _k}\). Then, up to a choice of subsequence, the \(\phi ^k\) converge strongly in \(W^{1,p}\) to a minimizer of
$$\begin{aligned} E_{\mathrm{mem}}+E_{\mathrm{bend}}+E_{\mathrm{vol}} \end{aligned}$$
in \(W_0^{1,p}(\Omega ;\mathbb {R}^n)+{\mathbb {1}}\) under the constraint \(\phi (\mathcal {M}_1^{c}) = \mathcal {M}_2^{c}\) for all \(c \in (-\sigma , \sigma )\).
Proof
First, notice that the energy E may be written as
$$\begin{aligned} \begin{aligned} E_\nu [\phi ]&= \frac{1}{\nu }\int _\Omega \eta _\sigma \circ \mathbf{{d}}_1 |\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1|^2 + \alpha _p |\mathcal {D}\phi |^p\\&\quad \, +H\Big (\det \mathcal {D}\phi , {{\text {Cof}}}\mathcal {D}\phi , \mathcal {D}_{\text {tg}}\phi , \det ( \mathcal {D}_{\text {tg}}\phi + \mathbf{{n}}_2 \circ \phi \otimes \mathbf{{n}}_1) , \\&\quad \quad \Lambda ({\mathcal {C}}({\mathcal {S}}^{ext}_1), {\mathcal {C}}({\mathcal {S}}^{ext}_2 \circ \phi )) ,\mathcal {D}\phi ),\\&\quad \quad \det \big ( \Lambda ({\mathcal {C}}({\mathcal {S}}^{ext}_1),{\mathcal {C}} ({\mathcal {S}}^{ext}_2 \circ \phi )), \mathcal {D}\phi \big ) \Big )\,{{\text {d}}}x, \end{aligned} \end{aligned}$$
(4.25)
where \(H:\mathbb {R}^+ \times \mathbb {R}^{n \times n} \times \mathbb {R}^{n \times n} \times \mathbb {R}\times \mathbb {R}^{n \times n} \times \mathbb {R}\rightarrow \mathbb {R}^+\) is smooth and convex.
Denote by \(\hat{\varphi }\) the extension of a diffeomorphism between \(\mathcal {M}_1\) and \(\mathcal {M}_2\) used in the proof of Theorem 4.2. Since \(E_{\text {match}}[\hat{\varphi }]=0\), we have that \(E_{\nu _k}[\phi ^k] \le E_1[\hat{\varphi }]\). By the coercivity estimate (4.5), the \(\phi ^k\) are then bounded in \(W^{1,p}\) and we may extract a (not relabeled) subsequence converging uniformly and weakly in \(W^{1,p}\) to some limit \(\phi \). Since \(\{E_{\nu _k}[\phi ^k]\}\) is bounded and \(\nu _k \rightarrow 0\), the uniform convergence of \(\phi ^k\) implies that
$$\begin{aligned} \int _\Omega \eta _\sigma (\mathbf{{d}}_1)|\mathbf{{d}}_2 \circ \phi ^k - \mathbf{{d}}_1|^2 \,{{\text {d}}}x \xrightarrow [k \rightarrow \infty ]{} \int _\Omega \eta _\sigma (\mathbf{{d}}_1)|\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1|^2 \,{{\text {d}}}x = 0. \end{aligned}$$
(4.26)
In consequence, \(\phi (\mathcal {M}_1^c)\subseteq \mathcal {M}_2^c\). Since \({ \left. \phi \phantom {\big |} \right| _{\mathcal {M}_1^c} }\) is the uniform limit of the maps \({ \left. \phi ^k \phantom {\big |} \right| _{\mathcal {M}_1^c} }\) which are surjective onto \(\mathcal {M}_2^c\) and \(\mathcal {M}_1^c\) is compact, we conclude that \(\phi (\mathcal {M}_1^c)=\mathcal {M}_2^c\) for all \(c \in (-\sigma , \sigma )\). Therefore, \(\phi \) is admissible for all \(\nu _k\) and \(E_{\nu _k}[\phi ^k] \le E_{1}[\phi ]\). Combined with lower semicontinuity and (4.25), the above implies
$$\begin{aligned} \begin{aligned}&\int _\Omega \alpha _p |\mathcal {D}\phi ^k|^p + H(\det (\mathcal {D}\phi ^k), \ldots ) \,{{\text {d}}}x \\&\quad \xrightarrow [k \rightarrow \infty ]{} \int _\Omega \alpha _p |\mathcal {D}\phi |^p + H(\det (\mathcal {D}\phi ), \ldots ) \,{{\text {d}}}x. \end{aligned} \end{aligned}$$
(4.27)
From this identity, the fact that H is convex and differentiable, and \(\mathcal {D}\phi ^k \rightharpoonup \mathcal {D}\phi \) in \(L^p\) it follows that
$$\begin{aligned} \begin{aligned} 0=&\limsup _{k \rightarrow \infty } \bigg ( \int _\Omega \alpha _p \left( |\mathcal {D}\phi ^k|^p - |\mathcal {D}\phi |^p \right) \\&+ \big ( H(\det (\mathcal {D}\phi ^k), \ldots ) - H(\det (\mathcal {D}\phi ), \ldots ) \big ) \,{{\text {d}}}x \bigg ) \\ \ge&\limsup _{k \rightarrow \infty } \bigg ( \int _\Omega \alpha _p \left( |\mathcal {D}\phi ^k|^p - |\mathcal {D}\phi |^p \right) \\&+ \mathcal {D}H(\det (\mathcal {D}\phi ), \ldots ) \cdot \big (\det (\mathcal {D}\phi ^k) - \det (\mathcal {D}\phi ), \ldots \big ) \,{{\text {d}}}x\bigg )\\ =&\limsup _{k \rightarrow \infty } \int _\Omega \alpha _p |\mathcal {D}\phi ^k|^p {{\text {d}}}x - \int _\Omega \alpha _p |\mathcal {D}\phi |^p \,{{\text {d}}}x\;. \end{aligned} \end{aligned}$$
Together with the weak lower semicontinuity of the \(L^p\)-norm, the above shows that
$$\begin{aligned} \int _\Omega \alpha _p |\mathcal {D}\phi |^p \,{{\text {d}}}x = \lim _{k \rightarrow \infty } \int _\Omega \alpha _p |\mathcal {D}\phi ^k|^p {{\text {d}}}x\;. \end{aligned}$$
Because \(L^p(\Omega )\) has the Radon-Riesz property ([47, 2.5.26]), weak convergence and convergence of the norm guarantee strong convergence. Since \(\phi _k\) was assumed to converge uniformly, we have also \(\phi _k \rightarrow \phi \) in \(L^p\), and this shows that \(\phi _k \rightarrow \phi \) in \(W^{1,p}(\Omega ; \mathbb {R}^{n})\).
That \(\phi \) is a minimizer of the constrained problem follows directly ([8], Theorem 1.21) from the fact that the \(E_{\nu _k}\) are an equicoercive family of functionals, \(\Gamma \)-converging in the weak topology of \(W^{1,p}\). Indeed, equicoercivity follows easily from the above, while \(\Gamma \)-convergence is implied by the fact that \(E_{\nu _k}\) is an increasing sequence ([8], Remark 1.40), because \(\nu _k \rightarrow 0\) appears as a denominator in \(E_{\text {match}}\). \(\square \)
Remark 4.5
By the coercivity estimate (4.24) of Corollary 4.3, an entirely analogous result holds for minimizers with Neumann boundary conditions.
Remark 4.6
Contrary to what might be expected, the limit problem we have obtained is not a surface problem, since all the level sets are still coupled through the volume energy \(E_{\text {vol}}\). The line of reasoning above depends heavily on the fact that the coefficients of the volume term are held fixed, since the equicoercivity and uniform strict quasiconvexity (in the language of [25]) both require the presence of \(\Vert \mathcal {D}\phi \Vert ^p_{L^p(\Omega )}\) in the functional.
Oscillations and Lack of Rank-One Convexity for the Naive Approach
To model the tangential distortion energy we have considered a frame indifferent energy density with the argument \(\mathcal {D}_{\text {tg}}\phi +(\mathbf{{n}}_2 \circ \phi ) \otimes \mathbf{{n}}_1\). Let us now consider the case \(n=2\) and a simpler version of the membrane energy (3.6), where we use as an argument of the energy density directly the tangential Cauchy–Green strain tensor [cf (2.2)] \(({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x))^T ({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x)) + \mathbf{{n}}_1(x) \otimes \mathbf{{n}}_1(x)\), and define the membrane energy
$$\begin{aligned} {\tilde{E}}_\text {mem}[\phi ]:=\int _{\Omega } \eta _\sigma (\mathbf{{d}}_1(x)) W \left( \big ({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x)\big )^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x) + \mathbf{{n}}_1(x) \otimes \mathbf{{n}}_1(x) \right) {{\text {d}}}x, \end{aligned}$$
(4.28)
with \({{{\tilde{\mathcal {D}}}}}_{tg}\phi := \mathcal {D}\phi \text {P}_1\) defined as the tangential part of the derivative along \(T_x \mathcal {M}_1^{\mathbf{{d}}_1(x)}\), and \(W:\mathbb {R}^{2 \times 2}\rightarrow \mathbb {R}\) a frame indifferent energy density that has a strict minimum at \(\text {SO}(2)\). In fact, this energy is no longer lower semicontinuous and we will present counterexamples.
Example 4.7
(Oscillation patterns) We construct an explicit sequence for which lower semicontinuity of the membrane energy \({\tilde{E}}_\text {mem}\) fails. Fix \(0<R<1\) and \(\mathcal {M}_1 = \mathbb {S}^1\) with the parametrization \(\xi \rightarrow e^{i \xi }\). Consider a sequence of deformations \(\varphi _k: \mathbb {S}^1 \rightarrow \mathbb {R}^2\) defined in polar coordinates of \((r, \theta )\) by the condition
$$\begin{aligned} \begin{aligned} \partial _\xi \varphi _k(\xi ) =&\left( R \sin k\xi \right) e_r\big ( r(\varphi _k( \xi ) ), \theta (\varphi _k( \xi ) ) \big )\\&+ \left( 1-R^2 \sin ^2 k \xi \right) ^{\frac{1}{2}} e_\theta \big (r(\varphi _k( \xi )), \theta (\varphi _k( \xi ) ) \big ), \end{aligned} \end{aligned}$$
(4.29)
where \(e_r=(\cos \theta , \sin \theta )^T, e_\theta =(-\sin \theta , \cos \theta )^T\) for given \(\phi _k(0)\). Note that for any k and \(\theta \) that \(|\partial _\theta \varphi _k(\theta )|=1\), so that the transformations are tangentially isometric. We define \(\varphi _k(0)\) via two integration constants \(r_0\) and \(\theta _0\) for the initialization of r and \(\theta \) at \(\xi =0\). We set \(\theta _0=0\) and choose \(r_0\) such that the curve \(\varphi _k\) is closed and simple, which imposes \(r_0=r(\varphi _k(0))=r(\varphi _k(2\pi ))\) since the first term in (4.29) has zero average. From the second term, taking into account that \(e_\theta (r, \theta )\) is independent of r, we get the condition
$$\begin{aligned} 2 \pi r_0 = \int _0^{2\pi }\left( 1-R^2 \sin ^2 k \xi \right) ^{\frac{1}{2}} {{\text {d}}}\xi =\frac{1}{k} \int _0^{2\pi k}\left( 1-R^2 \sin ^2 \zeta \right) ^{\frac{1}{2}} {{\text {d}}}\zeta , \end{aligned}$$
where we have applied the change of variables \(\zeta = k \xi \). By periodicity the right-hand side (an incomplete elliptic integral of the second kind with modulus R) is independent of k and thus determines \(r_0\). The resulting \(\varphi _k\) for several values of k are depicted in Fig. 3.
We observe that \(\partial _\theta \varphi _k(\theta ) \rightharpoonup r_0 e_\theta \text { in }L^p,\) for any \(1 \le p < \infty \) (and also weak-* in \(L^\infty \)). Therefore, the weak \(W^{1,p}\)-limit \(\varphi \) of the \(\varphi _k\) is the function defined by \(\varphi (\theta )=r_0 e_r\) and obviously not an isometry. Assuming \(0< \sigma < 1\) and extending \(\varphi _k, \varphi \) along the radial direction \(e_r\) to the annulus \(\{1-\sigma \le r \le 1+\sigma \}\), we obtain corresponding deformations given by
$$\begin{aligned} \phi ^k(r,\theta )&=\varphi _k(\theta )+(r-1)Q_{\frac{\pi }{2}}\partial _\theta \varphi _k(\theta ) \text { and }\\ \phi (r,\theta )&=\varphi (\theta )+(r-1)r_0 e_r=r \, r_0 e_r, \end{aligned}$$
where \(Q_{\frac{\pi }{2}}\) stands for clockwise rotation by \(\pi /2\), so that \(Q_{\frac{\pi }{2}}\partial _\theta \varphi _k(\theta )\) is the unit outward normal to \(\varphi _k(\mathbb {S}^1)\). Clearly also \(\phi ^k \rightharpoonup \phi \) in \(W^{1,p}\) on the annulus. We observe that \({\tilde{E}}_{\text {mem}}[\phi ^k]=0\), but \({\tilde{E}}_{\text {mem}}[\phi ]>0\). Hence, \({\tilde{E}}_\text {mem}\) is not weakly lower semicontinuous.
The celebrated Nash–Kuiper theorem [37, 51] states that it is possible to uniformly approximate any short \(C^\infty \) immersion by \(C^1\) isometric ones. Our explicit oscillations around \(r_0 \mathbb {S}^1\) is just one example of this phenomenon. Notice that a bending term of the type \(E_{\text {bend}}\) introduced in our model only compares the curvatures of \(\mathcal {M}_1^{\mathbf{{d}}_1(x)}\) and \(\mathcal {M}_2^{\mathbf{{d}}_2(\phi (x))}\). It therefore does not penalize oscillations, since it does not detect the curvature of \(\phi (\mathcal {M}_1)\) at all.
Example 4.8
(Lack of rank-one convexity) We present an additional example of a configuration for which the integrand of an energy of the type \({\tilde{E}}_{\text {mem}}\) is not rank-one convex. Rank-one convexity of the complete energy density, i.e., , convexity in \(t \in \mathbb {R}\) when composed with the function \(A+tB\) for any matrix A and any rank-one matrix B, is known to be a necessary condition for quasiconvexity ([20], Theorem 5.3). Quasiconvexity, in turn, is necessary for weak lower semicontinuity of integral functionals in Sobolev spaces ([20], Theorem 8.1 and Remark 8.2).
Let \(\Omega =(-2,2)^2\), and \(\mathcal {M}_1\) be a closed \(C^2\) curve such that \(\mathcal {M}_1 \cap (-1,1)\times (0,2)=(-1,1)\times \left\{ 1\right\} \). At any point \(x_0 \in (-1,1)\times \left\{ 1\right\} \), the tangential derivatives are just partial derivatives along the first coordinate, yielding
$$\begin{aligned} {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)= & {} \mathcal {D}\phi (x_0) \text {P}(e_2)=\left( \begin{array}{cc} \partial _1 \phi _1(x_0) &{} 0 \\ \partial _1 \phi _2(x_0) &{} 0 \end{array}\right) , \text { and }\\ {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)= & {} \left( \begin{array}{cc} \left( \partial _1 \phi _1(x_0)\right) ^2 + \left( \partial _1 \phi _2(x_0)\right) ^2 &{} 0 \\ 0 &{} 0 \end{array}\right) . \end{aligned}$$
Hence, the tangential area distortion measure reduces to
$$\begin{aligned} \begin{aligned} {\mathrm {tr}}({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0))&=\det ({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)+e_2 \otimes e_2)\\&=\left( \partial _1 \phi _1(x_0)\right) ^2 + \left( \partial _1 \phi _2(x_0)\right) ^2, \end{aligned} \end{aligned}$$
(4.30)
where \(e_2=(0,1)^T\). Defining now the convex function
$$\begin{aligned} F(a,d)=\frac{1}{2} a + \frac{1}{2} d + d^{-1}-2, \end{aligned}$$
which has a unique minimum with value 0 for \(a=d=1\), we have that the energy density
$$\begin{aligned} W_F(B)=F\left( {\mathrm {tr}}(B^TB), \det (B^TB+e_2 \otimes e_2) \right) \end{aligned}$$
has a pointwise minimum, with value zero, whenever \(\mathcal {D}\phi \) is such that \(\left( \partial _1 \phi _1\right) ^2 + \left( \partial _1 \phi _2\right) ^2=1\).
Consider now, for \(0\le \lambda \le 1\), the family of matrices
$$\begin{aligned} B(\lambda )=\left( \begin{array}{cc} \lambda &{} 0 \\ (1-\lambda ) &{} 0 \end{array}\right) =\lambda \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 0 \end{array}\right) +(1-\lambda ) \left( \begin{array}{cc} 0 &{} 0 \\ 1 &{} 0 \end{array}\right) . \end{aligned}$$
(4.31)
Clearly \(B(\lambda )\) is rank one. But we have \(W_F(B(\lambda ))=\lambda ^2+(1-\lambda )^2+\frac{1}{\lambda ^2+(1-\lambda )^2}-2\) and therefore
$$\begin{aligned} W_F(B(0))=F(B(1))=0, \text { but }W_F( B(1/2))=\frac{1}{2}, \end{aligned}$$
which demonstrates that \(W_F\) is not rank-one convex.