[proofplan]
We prove existence by the direct method. A minimizing sequence is bounded in $W^{1,p}(\Omega;\mathbb R^n)$ by the coercive energy bound, the positive-measure trace condition, and the subcritical loading estimate. Reflexivity and compact Sobolev embedding then give a weakly convergent subsequence with strong convergence of the deformations in $L^p$. Weak closedness keeps the limit admissible, [weak continuity of minors](/theorems/8750) identifies the weak limits of all minors, and polyconvex lower semicontinuity gives lower semicontinuity of the stored energy. Finally, weak continuity of the loading term gives the minimizing inequality.
[/proofplan]
[step:Reduce to a finite minimizing sequence]
Set
\begin{align*}
m:=\inf_{v\in\mathcal A}I[v]\in[-\infty,\infty].
\end{align*}
The coercive lower bound and the subcritical estimate on $\ell$ imply that $I$ is bounded below on $\mathcal A$ by [[Coercive Boundedness for Elastic Energies](/theorems/8756)][citetheorem:8756], applied with the present trace condition and loading estimate. Hence $m>-\infty$.
If $m=+\infty$, then every element of the nonempty set $\mathcal A$ minimizes $I$, and the theorem is proved. We therefore assume $m<+\infty$. Choose a minimizing sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A$ such that
\begin{align*}
I[u_k]\to m.
\end{align*}
After discarding finitely many terms, we may assume $I[u_k]\le m+1$ for every $k$.
[/step]
[step:Bound the minimizing sequence in $W^{1,p}(\Omega;\mathbb R^n)$]
For each $k\in\mathbb N$, define the stored-energy part
\begin{align*}
E[u_k]:=\int_\Omega W(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x).
\end{align*}
Since $I[u_k]=E[u_k]-\ell[u_k]$, the coercive lower bound on $W$ gives
\begin{align*}
I[u_k]\ge c\int_\Omega |\nabla u_k(x)|^p\,d\mathcal L^n(x)-C\mathcal L^n(\Omega)-|\ell[u_k]|.
\end{align*}
The loading estimate gives
\begin{align*}
|\ell[u_k]|\le a\|u_k\|_{W^{1,p}(\Omega)}^\theta+b
\end{align*}
with $\theta<p$. Because the class $\mathcal A$ has prescribed trace data on a boundary part of positive $\mathcal H^{n-1}$-measure, the Poincare-type trace coercivity used in [Coercive Boundedness for Elastic Energies][citetheorem:8756] converts control of $\|\nabla u_k\|_{L^p(\Omega)}$ together with the subcritical loading term into a uniform bound
\begin{align*}
\sup_{k\in\mathbb N}\|u_k\|_{W^{1,p}(\Omega)}<\infty.
\end{align*}
[guided]
The purpose of this step is to prevent the minimizing sequence from escaping to infinity in the Sobolev norm. For each $k\in\mathbb N$, define
\begin{align*}
E[u_k]:=\int_\Omega W(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x).
\end{align*}
Then $I[u_k]=E[u_k]-\ell[u_k]$. The growth hypothesis on $W$ says, for $\mathcal L^n$-a.e. $x\in\Omega$,
\begin{align*}
W(x,u_k(x),\nabla u_k(x))\ge c|\nabla u_k(x)|^p-C.
\end{align*}
Integrating this inequality with respect to $\mathcal L^n$ over $\Omega$ gives
\begin{align*}
E[u_k]\ge c\int_\Omega |\nabla u_k(x)|^p\,d\mathcal L^n(x)-C\mathcal L^n(\Omega).
\end{align*}
Therefore
\begin{align*}
I[u_k]\ge c\int_\Omega |\nabla u_k(x)|^p\,d\mathcal L^n(x)-C\mathcal L^n(\Omega)-|\ell[u_k]|.
\end{align*}
The loading term is allowed to be negative, so it could in principle destroy coercivity. The subcritical estimate prevents this: there are $a,b\ge 0$ and $0\le\theta<p$ such that
\begin{align*}
|\ell[u_k]|\le a\|u_k\|_{W^{1,p}(\Omega)}^\theta+b.
\end{align*}
Because $\theta<p$, the loading grows strictly slower than the $p$-power coercive term. The prescribed trace data on a boundary part of positive $\mathcal H^{n-1}$-measure supplies the Poincare-type estimate needed to control the full $W^{1,p}$ norm by the gradient term plus fixed trace data. Thus the hypotheses of [Coercive Boundedness for Elastic Energies][citetheorem:8756] are satisfied: $\Omega$ is bounded and Lipschitz, $p>1$, the admissible class carries trace data on a boundary part of positive surface measure, the stored energy has the coercive lower bound in $|\nabla u|^p$, and the loading has subcritical growth. Applying that theorem yields
\begin{align*}
\sup_{k\in\mathbb N}\|u_k\|_{W^{1,p}(\Omega)}<\infty.
\end{align*}
[/guided]
[/step]
[step:Extract weak and strong convergent subsequences]
Since $1<p<\infty$, the space $W^{1,p}(\Omega;\mathbb R^n)$ is reflexive by [Reflexivity of Lebesgue and Sobolev Spaces][citetheorem:8729]. The bounded sequence $(u_k)$ therefore has a subsequence, not relabelled, and an element
\begin{align*}
u\in W^{1,p}(\Omega;\mathbb R^n)
\end{align*}
such that
\begin{align*}
u_k\rightharpoonup u \quad \text{in } W^{1,p}(\Omega;\mathbb R^n).
\end{align*}
Since $\Omega$ is bounded and Lipschitz and $p\ge n$, the [[Rellich Kondrachov Compactness Theorem](/theorems/8731)][citetheorem:8731] gives, after passing to a further subsequence,
\begin{align*}
u_k\to u \quad \text{in } L^p(\Omega;\mathbb R^n).
\end{align*}
Passing to another subsequence if necessary, the strong $L^p$ convergence also gives
\begin{align*}
u_k(x)\to u(x)
\end{align*}
for $\mathcal L^n$-a.e. $x\in\Omega$.
[/step]
[step:Use weak closedness to keep the limit admissible]
Each $u_k$ belongs to $\mathcal A$, and
\begin{align*}
u_k\rightharpoonup u \quad \text{in } W^{1,p}(\Omega;\mathbb R^n).
\end{align*}
Since $\mathcal A$ is sequentially weakly closed in $W^{1,p}(\Omega;\mathbb R^n)$, it follows that
\begin{align*}
u\in\mathcal A.
\end{align*}
By the definition of $\mathcal A$, this weak closedness includes the trace condition on $\Gamma$, the finite-energy determinant positivity condition encoded in $\mathcal C$, and the imposed non-interpenetration condition encoded in $\mathcal C$.
[/step]
[step:Identify the weak limits of the minors]
For $r\in\{2,\dots,n\}$, define the sequence
\begin{align*}
Z_{k,r}:\Omega\to\mathbb R^{N_r},\qquad Z_{k,r}(x):=M_r(\nabla u_k(x)).
\end{align*}
The uniform $W^{1,p}$ bound and the subcritical loading estimate imply
\begin{align*}
\sup_{k\in\mathbb N}|\ell[u_k]|<\infty.
\end{align*}
Since $I[u_k]\le m+1$ and $E[u_k]=I[u_k]+\ell[u_k]$, it follows that
\begin{align*}
\sup_{k\in\mathbb N}E[u_k]<\infty.
\end{align*}
Using the growth condition and discarding the nonnegative $|\nabla u_k|^p$ term gives, for each $r\in\{2,\dots,n\}$,
\begin{align*}
\sup_{k\in\mathbb N}\int_\Omega |Z_{k,r}(x)|^{q_r}\,d\mathcal L^n(x)<\infty.
\end{align*}
Since $q_r>1$, the space $L^{q_r}(\Omega;\mathbb R^{N_r})$ is reflexive by [Reflexivity of Lebesgue and Sobolev Spaces][citetheorem:8729]. Hence, after passing to a diagonal subsequence, there exist maps
\begin{align*}
Z_r\in L^{q_r}(\Omega;\mathbb R^{N_r})
\end{align*}
such that
\begin{align*}
Z_{k,r}\rightharpoonup Z_r \quad \text{in } L^{q_r}(\Omega;\mathbb R^{N_r})
\end{align*}
for every $r\in\{2,\dots,n\}$.
For $r=1$, the [weak convergence](/page/Weak%20Convergence) $u_k\rightharpoonup u$ in $W^{1,p}$ gives
\begin{align*}
M_1(\nabla u_k)=\nabla u_k\rightharpoonup \nabla u=M_1(\nabla u)
\quad \text{in } L^p(\Omega;\mathbb R^{n\times n}).
\end{align*}
For $r\ge 2$, since $p\ge n\ge r$, [Weak Continuity of Minors][citetheorem:8752] identifies the distributional weak limit of $M_r(\nabla u_k)$ with $M_r(\nabla u)$. On the bounded domain $\Omega$, every [test function](/page/Test%20Function) $\varphi\in C_c^\infty(\Omega;\mathbb R^{N_r})$ belongs to $L^{q_r'}(\Omega;\mathbb R^{N_r})$, where $q_r'$ is the Holder conjugate of $q_r$. Hence the weak convergence $Z_{k,r}\rightharpoonup Z_r$ in $L^{q_r}(\Omega;\mathbb R^{N_r})$ gives the same distributional limit as the sequence $M_r(\nabla u_k)$. Distributional limits are unique, so
\begin{align*}
Z_r=M_r(\nabla u)
\end{align*}
for each $r\in\{2,\dots,n\}$. Consequently,
\begin{align*}
M(\nabla u_k)\rightharpoonup M(\nabla u)
\end{align*}
componentwise in the product space
\begin{align*}
L^p(\Omega;\mathbb R^{N_1})\times\prod_{r=2}^n L^{q_r}(\Omega;\mathbb R^{N_r}).
\end{align*}
[/step]
[step:Apply polyconvex lower semicontinuity to the stored energy]
Define
\begin{align*}
Y_k:\Omega\to\mathbb R^n,\qquad Y_k(x):=u_k(x),
\end{align*}
and
\begin{align*}
Z_k:\Omega\to\mathbb R^N,\qquad Z_k(x):=M(\nabla u_k(x)).
\end{align*}
Also define
\begin{align*}
Y:\Omega\to\mathbb R^n,\qquad Y(x):=u(x),
\end{align*}
and
\begin{align*}
Z:\Omega\to\mathbb R^N,\qquad Z(x):=M(\nabla u(x)).
\end{align*}
We have $Y_k\to Y$ strongly in $L^p(\Omega;\mathbb R^n)$ and a.e. on $\Omega$, hence also strongly in measure on $\Omega$. The previous step gives weak convergence of each minor component of $Z_k$ to the corresponding component of $Z$ in the required product of Lebesgue spaces. The coercive lower bound gives the integrable lower bound
\begin{align*}
G(x,u_k(x),M(\nabla u_k(x)))\ge -C
\end{align*}
for $\mathcal L^n$-a.e. $x\in\Omega$, and the constant function $x\mapsto -C$ belongs to $L^1(\Omega)$ because $\Omega$ is bounded.
The integrand $G$ is normal, lower semicontinuous in $(y,Z)$ for $\mathcal L^n$-a.e. $x$, and convex in $Z$ for each $y$. The minor orders are $S=\{1,\dots,n\}$. For each $r\in S$, the exponent condition in [[Ball Polyconvex Sequential Weak Lower Semicontinuity Theorem](/theorems/8754)][citetheorem:8754] requires $r\le p$ for the $r$-minor variable; this holds because $r\le n\le p$. The theorem also requires weak compactness of the lifted minor variables in the stated Lebesgue product space. For $r=1$ this is the weak convergence of $\nabla u_k$ in $L^p(\Omega;\mathbb R^{N_1})$, and for $2\le r\le n$ this is the weak convergence in $L^{q_r}(\Omega;\mathbb R^{N_r})$ obtained in the previous step, with $q_r>1$. The lifted deformation variables converge strongly in measure, the lifted minor variables converge weakly componentwise in the product Lebesgue space, and the preceding lower bound supplies the required integrable lower bound. Thus the hypotheses of [Ball Polyconvex Sequential Weak Lower Semicontinuity Theorem][citetheorem:8754] are satisfied for the lifted variables $(u_k,M(\nabla u_k))$. Hence
\begin{align*}
\int_\Omega G(x,u(x),M(\nabla u(x)))\,d\mathcal L^n(x)
\le \liminf_{k\to\infty}\int_\Omega G(x,u_k(x),M(\nabla u_k(x)))\,d\mathcal L^n(x).
\end{align*}
Using $W(x,y,F)=G(x,y,M(F))$, this becomes
\begin{align*}
\int_\Omega W(x,u(x),\nabla u(x))\,d\mathcal L^n(x)
\le \liminf_{k\to\infty}\int_\Omega W(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x).
\end{align*}
[/step]
[step:Pass the loading term to the limit and conclude minimality]
Since $\ell:\mathcal A\to\mathbb R$ is sequentially weakly continuous and $u_k\rightharpoonup u$ in $W^{1,p}(\Omega;\mathbb R^n)$, we have
\begin{align*}
\ell[u_k]\to \ell[u].
\end{align*}
Combining this convergence with the lower semicontinuity of the stored-energy part gives
\begin{align*}
I[u]\le \liminf_{k\to\infty} I[u_k].
\end{align*}
Because $(u_k)$ is a minimizing sequence,
\begin{align*}
\liminf_{k\to\infty}I[u_k]=m=\inf_{v\in\mathcal A}I[v].
\end{align*}
Since $u\in\mathcal A$, the opposite inequality
\begin{align*}
m\le I[u]
\end{align*}
holds by definition of the infimum. Therefore
\begin{align*}
I[u]=\inf_{v\in\mathcal A}I[v].
\end{align*}
Thus $u$ is a minimizer of the elastic energy on $\mathcal A$.
[/step]
