[proofplan] We reduce the desired inequality to the general Ioffe lower semicontinuity principle for nonnegative normal integrands convex in the weakly convergent variable. First we pass to a subsequence whose energies converge to the liminf. Strong $L^p$ convergence gives [convergence in measure](/page/Convergence%20in%20Measure) of the state variables, while the [weak convergence](/page/Weak%20Convergence) hypothesis gives the required weak convergence of the gradient variables. Applying that prior principle to $(u_k,\nabla u_k)$ yields the inequality along the selected subsequence, which is exactly the original liminf. [/proofplan]

[step:Choose a subsequence realizing the lower energy limit] For each $k\in\mathbb N$, define the energy value $E_k\in[0,\infty]$ by \begin{align*}E_k:=\int_\Omega F(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x).\end{align*} The hypothesis gives $\sup_{k\in\mathbb N}E_k<\infty$, so \begin{align*}L:=\liminf_{k\to\infty}E_k\end{align*} is finite. By the definition of $\liminf$, there exists a strictly increasing map $j:\mathbb N\to\mathbb N$ such that \begin{align*} E_{j(k)}\to L. \end{align*} It is enough to prove \begin{align*}\int_\Omega F(x,u(x),\nabla u(x))\,d\mathcal L^n(x)\le L.\end{align*} because $L$ is the right-hand side in the theorem statement. [/step]

[step:Convert strong $L^p$ convergence into convergence in measure]The convergence \begin{align*}u_{j(k)}\to u \quad \text{in } L^p(\Omega;\mathbb R^m)\end{align*} implies convergence in $\mathcal L^n$-measure on $\Omega$. Indeed, for every $\varepsilon>0$, [Chebyshev's inequality](/theorems/1126) applied to the [measurable function](/page/Measurable%20Function) \begin{align*} x\mapsto |u_{j(k)}(x)-u(x)|^p \end{align*} gives \begin{align*}\mathcal L^n\big(\{x\in\Omega:|u_{j(k)}(x)-u(x)|>\varepsilon\}\big) \le \varepsilon^{-p}\int_\Omega |u_{j(k)}(x)-u(x)|^p\,d\mathcal L^n(x).\end{align*} The right-hand side tends to $0$ by strong $L^p$ convergence. Therefore \begin{align*} u_{j(k)}\to u \quad \text{in } \mathcal L^n\text{-measure on }\Omega. \end{align*}[/step]

[guided]The Ioffe lower semicontinuity principle is formulated with strong convergence in measure for the state variable. Our hypothesis is stronger: convergence in the norm of $L^p(\Omega;\mathbb R^m)$. We now make that implication explicit. Fix $\varepsilon>0$. Define the measurable exceptional set $A_k(\varepsilon)\subset\Omega$ by \begin{align*}A_k(\varepsilon):=\{x\in\Omega:|u_{j(k)}(x)-u(x)|>\varepsilon\}.\end{align*} On this set we have \begin{align*}\varepsilon^p\le |u_{j(k)}(x)-u(x)|^p.\end{align*} Multiplying by the indicator function $\mathbb 1_{A_k(\varepsilon)}$ and integrating with respect to $\mathcal L^n$, we obtain \begin{align*}\varepsilon^p\mathcal L^n(A_k(\varepsilon))\le \int_\Omega |u_{j(k)}(x)-u(x)|^p\,d\mathcal L^n(x).\end{align*} Dividing by $\varepsilon^p$ gives \begin{align*}\mathcal L^n(A_k(\varepsilon))\le \varepsilon^{-p}\int_\Omega |u_{j(k)}(x)-u(x)|^p\,d\mathcal L^n(x).\end{align*} The final integral tends to $0$ because $u_{j(k)}\to u$ in $L^p(\Omega;\mathbb R^m)$. Hence $\mathcal L^n(A_k(\varepsilon))\to0$ for every $\varepsilon>0$, which is exactly convergence in $\mathcal L^n$-measure.[/guided]

[step:Apply Ioffe lower semicontinuity to the state and gradient variables] Define the maps $z_k:\Omega\to\mathbb R^m$ and $P_k:\Omega\to\mathbb R^{m\times n}$ by $z_k(x):=u_{j(k)}(x)$ and $P_k(x):=\nabla u_{j(k)}(x)$. Define also $z:\Omega\to\mathbb R^m$ and $P:\Omega\to\mathbb R^{m\times n}$ by $z(x):=u(x)$ and $P(x):=\nabla u(x)$. From the previous step, $z_k\to z$ in $\mathcal L^n$-measure. From the weak convergence hypothesis and passage to a subsequence, \begin{align*}P_k\rightharpoonup P \quad \text{in } L^p(\Omega;\mathbb R^{m\times n}).\end{align*} We now use the general Ioffe lower semicontinuity theorem for normal integrands, viewed as the prior result from which the present Sobolev-gradient formulation is obtained. In the form needed here, it applies to a finite-measure Lebesgue domain $\Omega\subset\mathbb R^n$, an exponent $1<p<\infty$, measurable maps $z_k,z:\Omega\to\mathbb R^m$ with $z_k\to z$ in $\mathcal L^n$-measure, and maps $Q_k,Q:\Omega\to\mathbb R^{m\times n}$ with $Q_k\rightharpoonup Q$ in $L^p(\Omega;\mathbb R^{m\times n})$. For every nonnegative normal integrand $G:\Omega\times\mathbb R^m\times\mathbb R^{m\times n}\to[0,\infty]$ such that $Q\mapsto G(x,z,Q)$ is convex for $\mathcal L^n$-a.e. $x\in\Omega$ and every $z\in\mathbb R^m$, that theorem gives lower semicontinuity of the associated integral along such pairs $(z_k,Q_k)$. Applying this prior theorem with $G=F$, $Q_k=P_k$, and $Q=P$, all hypotheses are satisfied: $\Omega$ is bounded, so $\mathcal L^n(\Omega)<\infty$; $1<p<\infty$ by assumption; $F$ is a nonnegative normal integrand by assumption; $P\mapsto F(x,z,P)$ is convex for $\mathcal L^n$-a.e. $x\in\Omega$ and every $z\in\mathbb R^m$ by assumption; $z_k\to z$ in $\mathcal L^n$-measure by the previous step; and $P_k\rightharpoonup P$ in $L^p(\Omega;\mathbb R^{m\times n})$ by the weak convergence hypothesis after passing to the subsequence. Therefore \begin{align*}\int_\Omega F(x,u(x),\nabla u(x))\,d\mathcal L^n(x) \le \liminf_{k\to\infty}\int_\Omega F(x,u_{j(k)}(x),\nabla u_{j(k)}(x))\,d\mathcal L^n(x).\end{align*} [/step]

[step:Identify the subsequential lower limit with the original liminf] By construction of the subsequence, \begin{align*}\int_\Omega F(x,u_{j(k)}(x),\nabla u_{j(k)}(x))\,d\mathcal L^n(x)=E_{j(k)}\to L.\end{align*} Hence \begin{align*}\liminf_{k\to\infty}\int_\Omega F(x,u_{j(k)}(x),\nabla u_{j(k)}(x))\,d\mathcal L^n(x)=L.\end{align*} Combining this identity with the inequality from the previous step gives \begin{align*} \int_\Omega F(x,u(x),\nabla u(x))\,d\mathcal L^n(x) \le L = \liminf_{k\to\infty}\int_\Omega F(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x). \end{align*} This is the desired lower semicontinuity inequality. [/step]