[proofplan] We prove the result for a fixed row tuple $I$ and column tuple $J$. The key point is the null-Lagrangian divergence identity for gradient minors: after testing against a compactly supported smooth function, an $r$-minor can be written as a finite sum involving one undifferentiated component of $u_k$, one derivative of the [test function](/page/Test%20Function), and an $(r-1)$-minor. We then argue by induction on the size of the minor, using local compactness to get strong convergence of the undifferentiated factor and the induction hypothesis to get [weak convergence](/page/Weak%20Convergence) of the lower-order minor. The endpoint $p=r$ gives distributional convergence only; when $p>r$, boundedness in the [reflexive space](/page/Reflexive%20Space) $L^{p/r}$ upgrades the distributional convergence to weak $L^{p/r}$ convergence. [/proofplan]

[step:Fix the minor and record its integrability] Fix strictly increasing index tuples $I=(i_1,\dots,i_r)$ and $J=(j_1,\dots,j_r)$. For a map $v\in W^{1,p}(\Omega;\mathbb R^m)$ and an integer $1\le s\le r$, define \begin{align*} M_{I_s,J_s}(Jv)=\det\big(\partial_{x_{j_b}}v_{i_a}\big)_{a,b=1}^s, \end{align*} where $I_s=(i_1,\dots,i_s)$ and $J_s=(j_1,\dots,j_s)$. Since each [weak derivative](/page/Weak%20Derivative) $\partial_{x_{j_b}}v_{i_a}$ belongs to $L^p(\Omega)$, Holder's inequality applied to the $s$ factors in each determinant term gives \begin{align*} M_{I_s,J_s}(Jv)\in L^{\rho_s}(\Omega). \end{align*} Here \begin{align*} \rho_s=\frac{p}{s}. \end{align*} For every $1\le s\le r$, we have $\rho_s\ge 1$, so the minors are locally integrable. [/step]

[step:Derive the divergence identity for Sobolev minors] Let $\varphi\in C_c^\infty(\Omega)$, and choose an [open set](/page/Open%20Set) $V\subset\Omega$ with Lipschitz boundary such that $\operatorname{supp}\varphi\subset V$ and $\overline V\subset\Omega$. Let $2\le s\le r$, let $A=(a_1,\dots,a_s)$ be a strictly increasing row tuple with entries in $\{1,\dots,m\}$, and let $B=(b_1,\dots,b_s)$ be a strictly increasing column tuple with entries in $\{1,\dots,n\}$. For $v\in W^{1,p}(V;\mathbb R^m)$ and each $1\le b\le s$, let $B_b$ be the tuple obtained from $B$ by deleting $b_b$, and define the a.e.-defined cofactor minor map $C_b(v):V\to\mathbb R$ by \begin{align*} C_b(v)(x)=M_{(a_2,\dots,a_s),B_b}(Jv)(x). \end{align*} Then \begin{align*} \int_V \varphi(x)M_{A,B}(Jv)(x)\,d\mathcal L^n(x)=-\sum_{b=1}^s(-1)^{1+b}\int_V v_{a_1}(x)\partial_{x_{b_b}}\varphi(x)C_b(v)(x)\,d\mathcal L^n(x). \end{align*} Indeed, for $v\in C^\infty(V;\mathbb R^m)$, expansion of the determinant along its first row gives \begin{align*} M_{A,B}(Jv)=\sum_{b=1}^s(-1)^{1+b}\partial_{x_{b_b}}v_{a_1}C_b(v). \end{align*} The alternating structure of the determinant gives the Piola cancellation identity \begin{align*} \sum_{b=1}^s(-1)^{1+b}\partial_{x_{b_b}}C_b(v)=0. \end{align*} This follows by differentiating the [cofactor expansion](/theorems/398): every [second derivative](/page/Second%20Derivative) term appears twice with opposite signs, because mixed weak derivatives commute in the smooth case and the determinant changes sign when the two differentiated columns are interchanged. Therefore \begin{align*} M_{A,B}(Jv)=\sum_{b=1}^s(-1)^{1+b}\partial_{x_{b_b}}\big(v_{a_1}C_b(v)\big). \end{align*} Multiplying by $\varphi$ and integrating by parts over $V$ gives the displayed identity, with no boundary term because $\varphi$ has compact support in $V$. For general $v\in W^{1,p}(V;\mathbb R^m)$, choose smooth maps $v_\ell\in C^\infty(V;\mathbb R^m)$ with $v_\ell\to v$ in $W^{1,p}(V;\mathbb R^m)$, using the standard density theorem for Sobolev maps by smooth maps on Lipschitz domains. We justify the passage to the limit by the multilinearity of determinants. Let $S_s$ denote the [symmetric group](/page/Symmetric%20Group) on $\{1,\dots,s\}$. For each $\sigma\in S_s$, the corresponding determinant term is \begin{align*} \operatorname{sgn}(\sigma)\prod_{a=1}^s \partial_{x_{b_{\sigma(a)}}}(v_\ell)_{a_a}. \end{align*} Subtracting the analogous product for $v$ and using the telescoping identity for products writes the difference as a sum of $s$ terms, each containing one factor $\partial_{x_{b_{\sigma(a)}}}(v_\ell-v)_{a_a}$ and $s-1$ factors chosen from derivatives of $v_\ell$ or $v$. Since $v_\ell\to v$ in $W^{1,p}(V;\mathbb R^m)$, the differentiated difference factor tends to $0$ in $L^p(V)$, and the remaining derivative factors are bounded in $L^p(V)$. Hölder's inequality applied with $s$ factors therefore gives \begin{align*} M_{A,B}(Jv_\ell)\to M_{A,B}(Jv) \quad \text{in } L^{p/s}(V). \end{align*} The same argument with $s-1$ factors gives \begin{align*} C_b(v_\ell)\to C_b(v) \quad \text{in } L^{p/(s-1)}(V). \end{align*} Combining this convergence with $v_{\ell,i_1}\to v_{i_1}$ in $L^p(V)$ and applying Hölder's inequality gives $v_{\ell,i_1}C_b(v_\ell)\to v_{i_1}C_b(v)$ in $L^{p/s}(V)$, hence in $L^1(V)$ because $V$ is bounded and $p/s\ge 1$. Passing to the limit gives the identity for $v$. [/step]

[step:Prove the distributional convergence by induction on the size of the minor]We prove the following assertion by induction on $s\in\{1,\dots,r\}$: for every increasing row tuple $A=(a_1,\dots,a_s)$ and column tuple $B=(b_1,\dots,b_s)$, \begin{align*} M_{A,B}(Ju_k)\to M_{A,B}(Ju) \end{align*} in $\mathcal D'(\Omega)$; if $p>s$, the convergence is weak convergence in $L^{p/s}(\Omega)$. For $s=1$, the minor is $\partial_{x_{b_1}}(u_k)_{a_1}$. Since $u_k\rightharpoonup u$ in $W^{1,p}(\Omega;\mathbb R^m)$, the weak derivatives satisfy \begin{align*} \partial_{x_{b_1}}(u_k)_{a_1}\rightharpoonup \partial_{x_{b_1}}u_{a_1} \end{align*} in $L^p(\Omega)$. This is weak convergence in $L^p(\Omega)$ and hence distributional convergence. Assume the assertion is known for minors of size $s-1$, where $2\le s\le r$. Fix increasing tuples $A=(a_1,\dots,a_s)$ and $B=(b_1,\dots,b_s)$, and fix $\varphi\in C_c^\infty(\Omega)$. Choose $V\subset\Omega$ as above with $\operatorname{supp}\varphi\subset V$ and $\overline V\subset\Omega$. We verify that the exponent needed below is admissible for compactness. Define \begin{align*} q_s=\frac{p}{p-s+1}. \end{align*} When $p=s$, this gives $q_s=s$, which is the conjugate exponent to $p/(s-1)$. Since $p\ge s$, we have $1\le q_s\le p$. If $p<n$, define \begin{align*} p^*=\frac{np}{n-p}. \end{align*} Then $p<p^*$, so $q_s<p^*$. If $p=n$, then $q_s<\infty$. If $p>n$, the Morrey case in the [Rellich Kondrachov compactness theorem](/theorems/8731) [citetheorem:8731] gives compact convergence into continuous representatives, hence into $L^{q_s}(V)$ because $V$ is bounded. Thus the Rellich Kondrachov [compactness theorem](/theorems/2748) [citetheorem:8731], applied componentwise on the bounded Lipschitz domain $V$, makes the restriction map from $W^{1,p}(V;\mathbb R^m)$ to $L^{q_s}(V;\mathbb R^m)$ compact. Since $u_k|_V\rightharpoonup u|_V$ in $W^{1,p}(V;\mathbb R^m)$, every subsequence has a further subsequence converging strongly in $L^{q_s}(V;\mathbb R^m)$ to $u|_V$. The standard subsequence contradiction criterion then gives convergence of the full sequence: \begin{align*} u_k|_V\to u|_V \quad \text{in } L^{q_s}(V;\mathbb R^m). \end{align*} For each $1\le b\le s$, define \begin{align*} C_{b,k}=M_{(a_2,\dots,a_s),B_b}(Ju_k),\qquad C_b=M_{(a_2,\dots,a_s),B_b}(Ju), \end{align*} where $B_b$ is the tuple obtained from $B$ by deleting $b_b$. Since $p>s-1$, the induction hypothesis gives \begin{align*} C_{b,k}\rightharpoonup C_b \quad \text{in } L^{\rho_{s-1}}(V). \end{align*} The exponents $q_s$ and $\rho_{s-1}$ are conjugate because \begin{align*} \rho_{s-1}=\frac{p}{s-1} \end{align*} and \begin{align*} \frac{1}{q_s}+\frac{1}{\rho_{s-1}}=1. \end{align*} Thus, since $(u_k)_{a_1}\partial_{x_{b_b}}\varphi\to u_{a_1}\partial_{x_{b_b}}\varphi$ strongly in $L^{q_s}(V)$ and $C_{b,k}\rightharpoonup C_b$ weakly in $L^{p/(s-1)}(V)$, we obtain \begin{align*} \lim_{k\to\infty}\int_V (u_k)_{a_1}(x)\partial_{x_{b_b}}\varphi(x)C_{b,k}(x)\,d\mathcal L^n(x)=\int_V u_{a_1}(x)\partial_{x_{b_b}}\varphi(x)C_b(x)\,d\mathcal L^n(x). \end{align*} Applying the Sobolev divergence identity to $u_k$ and to $u$, and then passing to the limit in each finite summand, gives \begin{align*} \lim_{k\to\infty}\int_\Omega \varphi(x)M_{A,B}(Ju_k)(x)\,d\mathcal L^n(x)=\int_\Omega \varphi(x)M_{A,B}(Ju)(x)\,d\mathcal L^n(x). \end{align*} This proves distributional convergence for size $s$. If $p>s$, then $\rho_s>1$ and the determinant estimates from the first step show that $(M_{A,B}(Ju_k))_{k=1}^{\infty}$ is bounded in $L^{\rho_s}(\Omega)$. By reflexivity of $L^{\rho_s}(\Omega)$, every subsequence has a further subsequence converging weakly in $L^{\rho_s}(\Omega)$ to some $h$. Weak $L^{\rho_s}$ convergence implies distributional convergence, so the distributional convergence just proved forces $h=M_{A,B}(Ju)$ as a distribution and hence as an element of $L^{\rho_s}(\Omega)$. Therefore the same subsequence criterion gives \begin{align*} M_{A,B}(Ju_k)\rightharpoonup M_{A,B}(Ju)\quad \text{in } L^{\rho_s}(\Omega). \end{align*}[/step]

[guided]The difficulty is that a determinant is a product of weakly convergent gradients, and products of weakly convergent sequences do not usually converge. The null-Lagrangian identity avoids this problem by moving one derivative onto the fixed test function and leaving one factor of $u_k$ undifferentiated, where compactness gives strong convergence. We prove the assertion by induction on the size $s$ of the minor. When $s=1$, the minor has the form $\partial_{x_{b_1}}(u_k)_{a_1}$. Since weak convergence in $W^{1,p}(\Omega;\mathbb R^m)$ includes weak convergence of every first weak derivative in $L^p(\Omega)$, we have \begin{align*} \partial_{x_{b_1}}(u_k)_{a_1}\rightharpoonup \partial_{x_{b_1}}u_{a_1} \end{align*} in $L^p(\Omega)$, hence also in distributions. Now assume the result has been proved for all minors of size $s-1$, with $2\le s\le r$. Fix a row tuple $A=(a_1,\dots,a_s)$, a column tuple $B=(b_1,\dots,b_s)$, and a test function $\varphi\in C_c^\infty(\Omega)$. Choose a bounded Lipschitz open set $V$ such that $\operatorname{supp}\varphi\subset V$ and $\overline V\subset\Omega$. This localization keeps all integrations away from the boundary of $\Omega$, so [integration by parts](/theorems/210) produces no boundary term. For each $1\le b\le s$, define the lower-order minor \begin{align*} C_{b,k}=M_{(a_2,\dots,a_s),B_b}(Ju_k), \end{align*} where $B_b$ is the column tuple obtained by deleting $b_b$. Define $C_b$ in the same way with $u$ in place of $u_k$. The induction hypothesis applies to $C_{b,k}$ because it is an $(s-1)$-minor. Since $p\ge s$, we have $p>s-1$, so the stronger part of the induction hypothesis gives \begin{align*} C_{b,k}\rightharpoonup C_b \quad \text{in } L^{\rho_{s-1}}(V), \qquad \rho_{s-1}=\frac{p}{s-1}. \end{align*} We also need a strongly convergent factor to pair with this weakly convergent lower-order minor. The factor supplied by the divergence identity is $(u_k)_{a_1}\partial_{x_{b_b}}\varphi$. By local compactness, [citetheorem:8731] gives strong convergence of $u_k$ to $u$ in the required local Lebesgue space after we check the exponent. The correct exponent is \begin{align*} q_s=\frac{p}{p-s+1}, \end{align*} because it is conjugate to the exponent $\rho_{s-1}$. Define \begin{align*} \rho_{s-1}=\frac{p}{s-1}. \end{align*} Then \begin{align*} \frac{1}{q_s}+\frac{s-1}{p}=1. \end{align*} Since $p\ge s$, one has $1\le q_s\le p$. If $p<n$, then $q_s\le p<p^*$; if $p=n$, then $q_s$ is finite; if $p>n$, the Morrey case of [citetheorem:8731] gives compact convergence into continuous representatives and hence into $L^{q_s}(V)$. Compactness first gives strong convergence for subsequences, but the weak limit in $W^{1,p}(V;\mathbb R^m)$ is fixed as $u|_V$, so every compactly convergent subsequence has strong limit $u|_V$. The subsequence contradiction criterion therefore yields \begin{align*} u_k|_V\to u|_V \quad \text{in } L^{q_s}(V;\mathbb R^m). \end{align*} Since $\partial_{x_{b_b}}\varphi$ is smooth and compactly supported in $V$, multiplication by $\partial_{x_{b_b}}\varphi$ preserves strong convergence in $L^{q_s}(V)$. Hence \begin{align*} (u_k)_{a_1}\partial_{x_{b_b}}\varphi\to u_{a_1}\partial_{x_{b_b}}\varphi \end{align*} strongly in $L^{q_s}(V)$. We now use the elementary weak-strong product principle: if $f_k\to f$ strongly in $L^{q_s}(V)$ and $g_k\rightharpoonup g$ weakly in $L^{p/(s-1)}(V)$, then \begin{align*} \int_V f_k(x)g_k(x)\,d\mathcal L^n(x)\to\int_V f(x)g(x)\,d\mathcal L^n(x). \end{align*} Indeed, write the difference as \begin{align*} \int_V (f_k-f)(x)g_k(x)\,d\mathcal L^n(x)+\int_V f(x)(g_k-g)(x)\,d\mathcal L^n(x). \end{align*} The first term tends to $0$ by Holder's inequality and boundedness of $(g_k)$ in $L^{p/(s-1)}(V)$; the second tends to $0$ by weak convergence of $g_k$ against the fixed function $f\in L^{q_s}(V)$. Applying this with \begin{align*} f_k=(u_k)_{a_1}\partial_{x_{b_b}}\varphi,\qquad g_k=C_{b,k}, \end{align*} gives convergence of every term in the divergence identity: \begin{align*} \lim_{k\to\infty}\int_V (u_k)_{a_1}(x)\partial_{x_{b_b}}\varphi(x)C_{b,k}(x)\,d\mathcal L^n(x)=\int_V u_{a_1}(x)\partial_{x_{b_b}}\varphi(x)C_b(x)\,d\mathcal L^n(x). \end{align*} We now record the divergence identity used for the tested $s$-minor. For each Sobolev map $v\in W^{1,p}(V;\mathbb R^m)$ and each $1\le b\le s$, define \begin{align*} D_b(v)=M_{(a_2,\dots,a_s),B_b}(Jv). \end{align*} For smooth $v$, expanding the determinant along the first row gives \begin{align*} M_{A,B}(Jv)=\sum_{b=1}^s(-1)^{1+b}\partial_{x_{b_b}}v_{a_1}D_b(v). \end{align*} The cofactor fields satisfy the Piola cancellation \begin{align*} \sum_{b=1}^s(-1)^{1+b}\partial_{x_{b_b}}D_b(v)=0, \end{align*} because each mixed second derivative term occurs twice with opposite signs after interchanging the two differentiated columns. Hence \begin{align*} M_{A,B}(Jv)=\sum_{b=1}^s(-1)^{1+b}\partial_{x_{b_b}}(v_{a_1}D_b(v)). \end{align*} Multiplying by $\varphi$ and integrating by parts over $V$ gives \begin{align*} \int_V \varphi(x)M_{A,B}(Jv)(x)\,d\mathcal L^n(x)=-\sum_{b=1}^s(-1)^{1+b}\int_V v_{a_1}(x)\partial_{x_{b_b}}\varphi(x)D_b(v)(x)\,d\mathcal L^n(x), \end{align*} with no boundary term because $\operatorname{supp}\varphi\subset V$ is compact. For $v\in W^{1,p}(V;\mathbb R^m)$, this follows by smooth approximation and the multilinear Hölder estimate for minors. Applying this identity with $v=u_k$ and with $v=u$, the tested $s$-minor is a finite sum of exactly the terms whose limits we have just computed. Therefore \begin{align*} \lim_{k\to\infty}\int_\Omega \varphi(x)M_{A,B}(Ju_k)(x)\,d\mathcal L^n(x)=\int_\Omega \varphi(x)M_{A,B}(Ju)(x)\,d\mathcal L^n(x). \end{align*} This proves distributional convergence for minors of size $s$. When $p>s$, set \begin{align*} \theta_s=\frac{p}{s}. \end{align*} Then $\theta_s>1$, so $L^{\theta_s}(\Omega)$ is reflexive by the reflexivity of Lebesgue spaces. The determinant estimate from the integrability step shows that $M_{A,B}(Ju_k)$ is bounded in $L^{\theta_s}(\Omega)$. Let $(M_{A,B}(Ju_{k_\ell}))_{\ell=1}^{\infty}$ be any subsequence. Reflexivity gives a further subsequence, not relabelled, and a function $h\in L^{\theta_s}(\Omega)$ such that \begin{align*} M_{A,B}(Ju_{k_\ell})\rightharpoonup h \quad \text{in } L^{\theta_s}(\Omega). \end{align*} Weak convergence in $L^{\theta_s}(\Omega)$ implies distributional convergence against every $\varphi\in C_c^\infty(\Omega)$, so the distributional convergence already proved forces $h=M_{A,B}(Ju)$ as a distribution and hence as an element of $L^{\theta_s}(\Omega)$. Thus every subsequence has a further weakly convergent subsequence with the same weak limit $M_{A,B}(Ju)$. If the full sequence did not converge weakly to $M_{A,B}(Ju)$ in $L^{\theta_s}(\Omega)$, some continuous linear functional on $L^{\theta_s}(\Omega)$ would separate a subsequence from that limit by a positive amount, contradicting the further-subsequence conclusion. Therefore \begin{align*} M_{A,B}(Ju_k)\rightharpoonup M_{A,B}(Ju) \quad \text{in } L^{p/s}(\Omega). \end{align*} This completes the induction step.[/guided]

[step:Upgrade distributional convergence to weak $L^{p/r}$ convergence when $p>r$] Assume now that $p>r$, and set \begin{align*} s_0=\frac{p}{r}. \end{align*} Then $s_0>1$, so $L^{s_0}(\Omega)$ is reflexive. By Holder's inequality applied to each determinant term, the sequence $M_{I,J}(Ju_k)$ is bounded in $L^{s_0}(\Omega)$. Let $(M_{I,J}(Ju_{k_\ell}))_{\ell=1}^\infty$ be any subsequence. Reflexivity gives a further subsequence, not relabelled, and a function $h\in L^{s_0}(\Omega)$ such that \begin{align*} M_{I,J}(Ju_{k_\ell})\rightharpoonup h \quad \text{in } L^{s_0}(\Omega). \end{align*} Weak convergence in $L^{s_0}$ implies distributional convergence, so the distributional convergence already proved forces \begin{align*} h=M_{I,J}(Ju) \end{align*} as distributions, and hence as elements of $L^{s_0}(\Omega)$. Therefore every subsequence has a further weakly convergent subsequence with weak limit $M_{I,J}(Ju)$. If the full sequence did not converge weakly to $M_{I,J}(Ju)$, there would be a functional in $(L^{s_0}(\Omega))^*$ and a subsequence on which the scalar evaluations stay a positive distance from their value at $M_{I,J}(Ju)$; applying the preceding further-subsequence conclusion to that subsequence gives a contradiction. Hence the bounded sequence itself converges weakly: \begin{align*} M_{I,J}(Ju_k)\rightharpoonup M_{I,J}(Ju)\quad \text{in } L^{p/r}(\Omega). \end{align*} [/step]

[step:Apply the argument to the prescribed $r\times r$ minor] Taking $s=r$ in the induction result gives \begin{align*} M_{I,J}(Ju_k)\to M_{I,J}(Ju) \end{align*} in $\mathcal D'(\Omega)$ for the fixed row tuple $I$ and column tuple $J$. Since $I$ and $J$ were arbitrary, every $r\times r$ minor of $Ju_k$ converges distributionally to the corresponding $r\times r$ minor of $Ju$. The preceding step gives the additional weak $L^{\rho_r}(\Omega)$ convergence whenever $p>r$. This proves the theorem. [/step]