[proofplan] The proof realises the classical Direct Method of the Calculus of Variations. We pick a minimising sequence $(u_k) \subset \mathcal{A}$ — i.e. $\mathcal{F}[u_k] \to m := \inf_\mathcal{A} \mathcal{F}$ — and show it has a subsequence whose weak limit minimises $\mathcal{F}$. Three ingredients combine: (a) coercivity ($L \ge \theta|\xi|^p - C$) plus the Poincaré inequality forces the minimising sequence to be $W^{1,p}$-bounded; (b) reflexivity of $W^{1,p}$ for $1 < p < \infty$ ([Weak Compactness in $W^{1,p}$](/theorems/3104)) yields a weakly convergent subsequence with limit $u^* \in W^{1,p}(\Omega)$, and the [Rellich-Kondrachov Theorem](/theorems/64) upgrades the convergence to strong $L^r$ convergence for $r < p^*$; (c) convexity in $\xi$ together with the Carathéodory growth of $L$ implies weak lower semicontinuity of $\mathcal{F}$ on $W^{1,p}$. Combining (a), (b), (c) gives $\mathcal{F}[u^*] \le \liminf_j \mathcal{F}[u_{k_j}] = m$, and $u^* \in \mathcal{A}$ means $\mathcal{F}[u^*] \ge m$, hence equality and $u^*$ is a minimiser. The closure of $\mathcal{A}$ under weak convergence (it is a convex closed affine subspace) ensures $u^* \in \mathcal{A}$. The hypothesis $\inf \mathcal{F} > -\infty$ guarantees the minimising sequence has bounded $\mathcal{F}$-values; the hypothesis $\mathcal{A} \ne \varnothing$ guarantees the infimum is over a non-empty set so that minimising sequences exist. [/proofplan]

[step:Extract a minimising sequence and identify the target value] By hypothesis, $\mathcal{A} \ne \varnothing$ and $m := \inf_{u \in \mathcal{A}} \mathcal{F}[u] \in (-\infty, \infty]$ — in fact, since $\mathcal{A} \ne \varnothing$ and $\mathcal{F}[g] < \infty$ when $g \in \mathcal{A}$ (by the upper bound in (iii) below), we have $m \in \mathbb{R}$. By the definition of infimum, there exists a sequence $(u_k)_{k \ge 1} \subset \mathcal{A}$ with \begin{align*} \mathcal{F}[u_k] \to m \quad \text{as } k \to \infty. \end{align*} Such a sequence is called a *minimising sequence* for $\mathcal{F}$ on $\mathcal{A}$. Since $\mathcal{F}[u_k] \to m \in \mathbb{R}$, the sequence $(\mathcal{F}[u_k])$ is bounded; pick $K > 0$ such that $\mathcal{F}[u_k] \le K$ for all $k$. [/step]

[step:Bound the $W^{1,p}(\Omega)$ norm of the minimising sequence using coercivity and Poincaré]By the coercivity hypothesis (i), \begin{align*} \mathcal{F}[u_k] = \int_\Omega L(x, u_k(x), \nabla u_k(x))\, d\mathcal{L}^n(x) \ge \int_\Omega \bigl[\theta|\nabla u_k(x)|^p - C\bigr]\, d\mathcal{L}^n(x) = \theta\|\nabla u_k\|_{L^p(\Omega)}^p - C\,\mathcal{L}^n(\Omega). \end{align*} Rearranging and using $\mathcal{F}[u_k] \le K$, \begin{align*} \|\nabla u_k\|_{L^p(\Omega)}^p \le \frac{1}{\theta}\bigl[\mathcal{F}[u_k] + C\mathcal{L}^n(\Omega)\bigr] \le \frac{K + C\mathcal{L}^n(\Omega)}{\theta} =: M_1^p. \end{align*} Hence $\|\nabla u_k\|_{L^p(\Omega)} \le M_1$ for all $k$, with $M_1 = M_1(\theta, C, K, \Omega)$ depending only on the data. To bound $\|u_k\|_{L^p(\Omega)}$, we use the boundary condition $u_k - g \in W^{1,p}_0(\Omega)$ together with a Poincaré-type inequality. Since $\Omega$ is bounded with Lipschitz boundary, the Poincaré inequality on $W^{1,p}_0(\Omega)$ holds: there exists a constant $C_P = C_P(\Omega, p) > 0$ such that \begin{align*} \|w\|_{L^p(\Omega)} \le C_P \|\nabla w\|_{L^p(\Omega)} \quad \text{for all } w \in W^{1,p}_0(\Omega). \end{align*} This is a standard Poincaré inequality for zero-trace functions, derivable by extending $w$ by zero to $\mathbb{R}^n$ and applying the Gagliardo-Nirenberg-Sobolev inequality ([61](/theorems/61)) on the extension, then restricting to $\Omega$ — or by a direct one-step argument on the bounded domain. Apply it with $w := u_k - g \in W^{1,p}_0(\Omega)$: \begin{align*} \|u_k - g\|_{L^p(\Omega)} \le C_P\|\nabla(u_k - g)\|_{L^p(\Omega)} \le C_P\bigl(\|\nabla u_k\|_{L^p(\Omega)} + \|\nabla g\|_{L^p(\Omega)}\bigr) \le C_P(M_1 + \|\nabla g\|_{L^p(\Omega)}). \end{align*} By the triangle inequality, \begin{align*} \|u_k\|_{L^p(\Omega)} \le \|u_k - g\|_{L^p(\Omega)} + \|g\|_{L^p(\Omega)} \le C_P(M_1 + \|\nabla g\|_{L^p(\Omega)}) + \|g\|_{L^p(\Omega)} =: M_0, \end{align*} with $M_0 = M_0(\theta, C, K, \Omega, p, g)$ depending only on the data and the boundary datum $g$. Combining, \begin{align*} \|u_k\|_{W^{1,p}(\Omega)} = \|u_k\|_{L^p(\Omega)} + \|\nabla u_k\|_{L^p(\Omega)} \le M_0 + M_1 =: M. \end{align*} The minimising sequence is uniformly bounded in $W^{1,p}(\Omega)$.[/step]

[guided]**Why coercivity gives a gradient bound.** The coercivity hypothesis (i) is the engine of the Direct Method. It says that $L$ grows at least as fast as $|\xi|^p$ in the gradient variable: $L(x, u, \xi) \ge \theta|\xi|^p - C$. Integrating over $\Omega$ gives \begin{align*} \mathcal{F}[u] \ge \theta\|\nabla u\|_{L^p(\Omega)}^p - C\mathcal{L}^n(\Omega), \end{align*} so the energy controls the gradient norm from below. For a minimising sequence $\mathcal{F}[u_k] \le K$, this yields the explicit bound \begin{align*} \|\nabla u_k\|_{L^p(\Omega)}^p \le \frac{K + C\mathcal{L}^n(\Omega)}{\theta}. \end{align*} The constant $\theta > 0$ is essential — without coercivity, the energy can decrease to $-\infty$ even with unbounded gradients. The constant $C \ge 0$ is harmless: it just shifts $\mathcal{F}$ by a constant and contributes the additive $C\mathcal{L}^n(\Omega)$ above. **Why we need the boundary condition for an $L^p$ bound.** A bound on $\|\nabla u_k\|_{L^p}$ alone does **not** imply a bound on $\|u_k\|_{L^p}$: adding any constant $c$ to $u_k$ leaves the gradient unchanged. To rule out this constant ambiguity, we use the affine boundary condition $u_k - g \in W^{1,p}_0(\Omega)$, which fixes the "constant part" via the trace. **Poincaré on $W^{1,p}_0$.** The Poincaré inequality for zero-trace functions on a bounded domain $\Omega$ is a standard result: $\|w\|_{L^p(\Omega)} \le C_P \|\nabla w\|_{L^p(\Omega)}$ for all $w \in W^{1,p}_0(\Omega)$. The constant $C_P$ depends on $\Omega$ and $p$; it is not the same constant as in [Poincaré Inequality on Balls](/theorems/3103), but is established by similar techniques (extension by zero to a containing ball, then GNS or direct integration). The hypothesis here — $\Omega$ bounded with Lipschitz boundary — is sufficient (in fact bounded $\Omega$ alone suffices for this form of Poincaré). Apply the inequality to $w := u_k - g \in W^{1,p}_0(\Omega)$: \begin{align*} \|u_k - g\|_{L^p(\Omega)} \le C_P\|\nabla(u_k - g)\|_{L^p(\Omega)} \le C_P(\|\nabla u_k\|_{L^p} + \|\nabla g\|_{L^p}) \le C_P(M_1 + \|\nabla g\|_{L^p}). \end{align*} Adding back $g$ via the triangle inequality bounds $\|u_k\|_{L^p}$, hence $\|u_k\|_{W^{1,p}}$. **Why $g \in W^{1,p}(\Omega)$ matters.** Implicit in the formulation of $\mathcal{A}$ is $g \in W^{1,p}(\Omega)$ (otherwise the affine condition $u - g \in W^{1,p}_0$ does not make sense within $W^{1,p}$). So $\|g\|_{L^p}$ and $\|\nabla g\|_{L^p}$ are finite, and the bound $M = M(\Omega, p, \theta, C, K, g)$ depends only on the problem data.[/guided]

[step:Extract a weakly convergent subsequence and improve to strong $L^r$ convergence]By the previous step, $\sup_k \|u_k\|_{W^{1,p}(\Omega)} \le M < \infty$. Apply the [Weak Compactness in $W^{1,p}$](/theorems/3104) theorem: since $1 < p < \infty$ and $\Omega$ is open and bounded, the bounded sequence $(u_k)$ has a subsequence $(u_{k_j})$ and a limit $u^* \in W^{1,p}(\Omega)$ with \begin{align*} u_{k_j} \rightharpoonup u^* \quad \text{weakly in } W^{1,p}(\Omega). \end{align*} In particular $u_{k_j} \rightharpoonup u^*$ weakly in $L^p(\Omega)$ and $\partial_{x_i} u_{k_j} \rightharpoonup \partial_{x_i} u^*$ weakly in $L^p(\Omega)$ for each $i = 1, \dots, n$. Now apply the [Rellich-Kondrachov Theorem](/theorems/64) to upgrade weak convergence to strong $L^r$ convergence: **Case $p < n$:** the embedding $W^{1,p}(\Omega) \hookrightarrow\hookrightarrow L^q(\Omega)$ is compact for every $q$ with $1 \le q < p^* = np/(n-p)$. Since $r < p^*$ by hypothesis (iii), the embedding $W^{1,p}(\Omega) \hookrightarrow\hookrightarrow L^r(\Omega)$ is compact. The bounded sequence $(u_{k_j})$ in $W^{1,p}(\Omega)$ therefore has a (further) subsequence converging strongly in $L^r(\Omega)$. Since the original $W^{1,p}$-weak limit is $u^*$, and strong $L^r$ convergence implies weak $L^r$ convergence — but we already know $u_{k_j} \rightharpoonup u^*$ weakly in $L^p \hookrightarrow L^r$ (for $r \le p$ and bounded $\Omega$, or generally) — uniqueness of weak limits identifies the strong $L^r$ limit as $u^*$. Hence (passing to a further subsequence, still denoted $(u_{k_j})$): \begin{align*} u_{k_j} \to u^* \quad \text{strongly in } L^r(\Omega). \end{align*} **Case $p \ge n$:** even simpler — the GNS embedding gives $W^{1,p}(\Omega) \hookrightarrow L^q(\Omega)$ for every $q \in [1, \infty)$ (when $p = n$) or $W^{1,p}(\Omega) \hookrightarrow C(\overline\Omega)$ (when $p > n$). The compactness of these embeddings via Rellich-Kondrachov gives a strongly $L^r$-convergent subsequence; the same argument applies. Pass to a further subsequence such that $u_{k_j} \to u^*$ $\mathcal{L}^n$-a.e. on $\Omega$ (every strongly $L^r$-convergent sequence has an a.e.-convergent subsequence — a standard consequence of $L^p$ convergence on $\sigma$-finite measure spaces). Henceforth $(u_{k_j})$ denotes this subsequence: \begin{align*} u_{k_j} &\rightharpoonup u^* \quad \text{weakly in } W^{1,p}(\Omega), \\ \nabla u_{k_j} &\rightharpoonup \nabla u^* \quad \text{weakly in } L^p(\Omega; \mathbb{R}^n), \\ u_{k_j} &\to u^* \quad \text{strongly in } L^r(\Omega), \\ u_{k_j} &\to u^* \quad \mathcal{L}^n\text{-a.e. on } \Omega. \end{align*} **Closure of $\mathcal{A}$.** The admissible class $\mathcal{A} = g + W^{1,p}_0(\Omega)$ is a closed affine subspace of $W^{1,p}(\Omega)$: $W^{1,p}_0(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in $W^{1,p}$, hence is itself closed; $\mathcal{A}$ is the translate of a closed linear subspace, hence is itself closed. As a closed convex (affine) subset of a Banach space, $\mathcal{A}$ is also weakly closed (Mazur's theorem: a convex norm-closed set is weakly closed). Since $u_{k_j} \in \mathcal{A}$ for all $j$ and $u_{k_j} \rightharpoonup u^*$ weakly, we conclude $u^* \in \mathcal{A}$.[/step]

[guided]**Why we need both weak and strong convergence.** The minimisation argument hinges on a *lower semicontinuity* property of $\mathcal{F}$ along the subsequence $(u_{k_j})$, which we will establish in the next step. To verify lower semicontinuity, we will need: - $\nabla u_{k_j} \rightharpoonup \nabla u^*$ **weakly** in $L^p$ — to handle the convex structure of $\xi \mapsto L(x, u, \xi)$, since convex functions on $L^p$ are weakly lower semicontinuous (a consequence of Mazur's theorem applied to epigraphs); - $u_{k_j} \to u^*$ **strongly** in $L^r$ — to handle the dependence of $L$ on $u$, which is only continuous (Carathéodory), so we cannot use weak convergence alone for the $u$-variable: weak convergence does not pass through continuous nonlinearities. **Reflexivity-driven extraction.** Theorem 3104 — [Weak Compactness in $W^{1,p}$](/theorems/3104) — gives a $W^{1,p}$-weakly convergent subsequence $u_{k_j} \rightharpoonup u^*$. We verified its hypotheses: $\Omega$ is open (bounded, in fact, but only openness is needed for 3104), and $1 < p < \infty$ by assumption. The mechanism behind 3104 is reflexivity of $W^{1,p}$ for $1 < p < \infty$: bounded sets in reflexive Banach spaces are weakly sequentially precompact (Eberlein-Šmulian). **Compactness of the Sobolev embedding.** The [Rellich-Kondrachov Theorem](/theorems/64) requires $\Omega$ to be a bounded open set with Lipschitz boundary — exactly our hypothesis. It asserts $W^{1,p}(\Omega) \hookrightarrow\hookrightarrow L^q(\Omega)$ for $1 \le q < p^*$ (case $p < n$); for $p = n$ the compact embedding extends to all $q \in [1, \infty)$, and for $p > n$ to $C(\overline{\Omega})$ via Morrey's embedding. By hypothesis (iii), $r < p^*$, so the embedding $W^{1,p}(\Omega) \hookrightarrow\hookrightarrow L^r(\Omega)$ is compact in every regime. Compactness means: bounded sequences in $W^{1,p}$ have $L^r$-strongly convergent subsequences. So from $\sup_j \|u_{k_j}\|_{W^{1,p}} \le M$, we extract a sub-subsequence with strong $L^r$ convergence to some limit $v \in L^r(\Omega)$. We must show $v = u^*$ a.e. **Identifying the strong limit with the weak $W^{1,p}$ limit.** Since $u_{k_j} \rightharpoonup u^*$ weakly in $W^{1,p}$, in particular $u_{k_j} \rightharpoonup u^*$ weakly in $L^p$. The strongly $L^r$-convergent sub-subsequence has, by the standard measure-theoretic fact, a further sub-subsequence converging $\mathcal{L}^n$-a.e. to its $L^r$ limit $v$. The same sub-subsequence — being bounded in $L^p$ and convergent a.e. — converges weakly in $L^p$ to $v$ (e.g. by the Brezis-Lieb-style argument or by combining a.e. convergence with $L^p$-boundedness via Fatou and the dominated convergence theorem applied to test pairings). Uniqueness of weak $L^p$ limits forces $v = u^*$ a.e. Hence the strong $L^r$ limit equals $u^*$. **A.e. convergence.** Strong $L^r$ convergence implies a.e. convergence along a further subsequence — a standard fact from measure theory: if $f_j \to f$ in $L^r(\Omega)$ then there is a subsequence with $f_{j_l} \to f$ a.e. (the diagonal extraction from $\sum_l \|f_{j_l} - f\|_{L^r} < \infty$ via Borel-Cantelli, or the standard $L^p$ argument). We pass to this further subsequence and continue to denote it $(u_{k_j})$. **Closure of $\mathcal{A}$ under weak convergence.** $\mathcal{A} = g + W^{1,p}_0(\Omega)$ is the affine translate of $W^{1,p}_0(\Omega)$. By definition, $W^{1,p}_0(\Omega) = \overline{C_c^\infty(\Omega)}^{W^{1,p}}$, the norm-closure of the test functions, which is a closed (hence convex closed) linear subspace. Translating by $g$ gives a closed convex (affine) subset of $W^{1,p}$. Mazur's theorem says: a convex set in a Banach space is norm-closed iff it is weakly closed. Hence $\mathcal{A}$ is weakly closed, and the weak limit $u^*$ of the sequence $(u_{k_j}) \subset \mathcal{A}$ lies in $\mathcal{A}$. Concretely: $u_{k_j} - g \in W^{1,p}_0(\Omega)$ for all $j$, and $u_{k_j} - g \rightharpoonup u^* - g$ weakly in $W^{1,p}$ (subtracting a fixed element preserves weak convergence). Since $W^{1,p}_0$ is weakly closed, $u^* - g \in W^{1,p}_0$, i.e. $u^* \in \mathcal{A}$. This is exactly the property we need: weak convergence inside the admissible class produces a weak limit that is itself admissible — without it, the limit candidate $u^*$ might fail the boundary condition and be disqualified.[/guided]

[step:Verify that $\mathcal{F}$ is sequentially weakly lower semicontinuous on $\mathcal{A}$]We show \begin{align*} \mathcal{F}[u^*] \le \liminf_{j \to \infty} \mathcal{F}[u_{k_j}]. \end{align*} The argument relies on the convexity hypothesis (ii) and is a standard application of the *Ioffe-Olech theorem* on lower semicontinuity of integral functionals with Carathéodory integrand convex in the gradient variable. We carry out the proof. **Linearisation via convexity.** For each fixed $(x, u) \in \Omega \times \mathbb{R}$ with $x$ in the full-measure set where $L(x, \cdot, \cdot)$ is convex in $\xi$ (which exists by hypothesis (ii)), and any $\xi, \eta \in \mathbb{R}^n$, the convexity of $\xi \mapsto L(x, u, \xi)$ implies the supporting-hyperplane inequality: there exists a subgradient $\zeta(x, u, \xi) \in \mathbb{R}^n$ such that \begin{align*} L(x, u, \eta) \ge L(x, u, \xi) + \zeta(x, u, \xi) \cdot (\eta - \xi). \end{align*} By the growth condition (iii), $\zeta(x, u, \xi)$ is bounded by $C(1 + |u|^{r-1} + |\xi|^{p-1})$ and is measurable in $(x, u, \xi)$ (the subdifferential of a Carathéodory function is a Carathéodory multifunction). The growth bound for $\zeta$ follows from a difference-quotient argument applied to the upper bound $L(x, u, \xi) \le C(1 + |u|^r + |\xi|^p)$: for any $\eta \in \mathbb{R}^n$ with $|\eta| = 1$, \begin{align*} \zeta \cdot \eta \le L(x, u, \xi + \eta) - L(x, u, \xi) \le C(1 + |u|^r + (|\xi| + 1)^p) + C(1 + |u|^r + |\xi|^p), \end{align*} and standard estimates on $(|\xi| + 1)^p - |\xi|^p \lesssim 1 + |\xi|^{p-1}$ produce the claimed bound. This is a textbook calculation; see Rockafellar, *Convex Analysis*, §23 (subgradient bounds for convex functions of $p$-growth), or the application of Mazur's theorem combined with $L^p$-boundedness in Dacorogna, *Direct Methods in the Calculus of Variations*, Lemma 3.2. **Apply the inequality at the candidate minimiser.** Set $\xi := \nabla u^*(x)$ and $\eta := \nabla u_{k_j}(x)$ in the above (with $u := u^*(x)$). For $\mathcal{L}^n$-a.e. $x \in \Omega$, \begin{align*} L(x, u^*(x), \nabla u_{k_j}(x)) \ge L(x, u^*(x), \nabla u^*(x)) + \zeta(x, u^*(x), \nabla u^*(x)) \cdot (\nabla u_{k_j}(x) - \nabla u^*(x)). \end{align*} Integrate over $\Omega$: \begin{align*} \int_\Omega L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n \ge \mathcal{F}[u^*] + \int_\Omega \zeta(x, u^*, \nabla u^*) \cdot (\nabla u_{k_j} - \nabla u^*)\, d\mathcal{L}^n. \end{align*} **Pass to the limit using weak convergence of $\nabla u_{k_j}$.** Set $h := \zeta(\cdot, u^*, \nabla u^*) \in L^{p'}(\Omega; \mathbb{R}^n)$, where the $L^{p'}$ membership follows from the growth bound $|\zeta| \le C(1 + |u^*|^{r-1} + |\nabla u^*|^{p-1})$ together with $u^* \in L^r(\Omega)$ (so $|u^*|^{r-1} \in L^{r/(r-1)}$, with $r/(r-1) \ge p'$ when $r \le p$ — handled by Hölder in the bounded domain), $\nabla u^* \in L^p$ (so $|\nabla u^*|^{p-1} \in L^{p'}$). Since $\nabla u_{k_j} \rightharpoonup \nabla u^*$ weakly in $L^p(\Omega; \mathbb{R}^n)$ and $h \in L^{p'}(\Omega; \mathbb{R}^n)$ pairs with $L^p$ via $(L^p)^* = L^{p'}$, \begin{align*} \int_\Omega \zeta(x, u^*, \nabla u^*) \cdot (\nabla u_{k_j} - \nabla u^*)\, d\mathcal{L}^n \to 0 \quad \text{as } j \to \infty. \end{align*} Therefore \begin{align*} \liminf_{j \to \infty}\int_\Omega L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n \ge \mathcal{F}[u^*]. \end{align*} **Replace $u^*$ by $u_{k_j}$ in the first slot.** It remains to compare $\int L(x, u_{k_j}, \nabla u_{k_j})\, d\mathcal{L}^n = \mathcal{F}[u_{k_j}]$ with $\int L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n$. We claim \begin{align*} \int_\Omega \bigl[L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j})\bigr]\, d\mathcal{L}^n \to 0 \quad \text{as } j \to \infty. \end{align*} This is where the strong $L^r$ convergence $u_{k_j} \to u^*$ enters. By the Carathéodory hypothesis on $L$ and the growth (iii), $L(x, \cdot, \xi)$ is continuous for a.e. $x$ and uniformly fixed $\xi$, and $|L(x, u, \xi)| \le C(1 + |u|^r + |\xi|^p)$. From the [Rellich-Kondrachov Theorem](/theorems/64) and the bound $(u_{k_j})$ in $W^{1,p}$, we have strong $L^r$ convergence $u_{k_j} \to u^*$ — no equi-integrability of $|\nabla u_{k_j}|^p$ is needed. We exploit this directly via the modulus of continuity of $L$ in its second argument. For any $\eta > 0$, the bound $|L(x, u, \xi) - L(x, u', \xi)| \le \omega(x, |u - u'|; \xi)$ holds with a measurable modulus controlled by the growth: explicitly, the mean-value-type estimate together with the Carathéodory continuity of $L(x, \cdot, \xi)$ and the growth $|L| \le C(1 + |u|^r + |\xi|^p)$ gives \begin{align*} |L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j})| \le C\bigl(1 + |u_{k_j}|^{r-1} + |u^*|^{r-1} + |\nabla u_{k_j}|^p\bigr) \cdot \mathbf{1}_{\{|u_{k_j} - u^*| > 0\}} \cdot \chi_j(x) \end{align*} where $\chi_j(x) \to 0$ a.e. as $j \to \infty$ by Carathéodory continuity and the a.e. convergence $u_{k_j} \to u^*$. The pointwise bound $L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j}) \to 0$ a.e. is direct from this. To pass to $L^1$ convergence of the difference we use the strong $L^r$ convergence $u_{k_j} \to u^*$ combined with the Sobolev-Rellich strong convergence: the $u$-dependent terms $|u_{k_j}|^{r-1}$, $|u^*|^{r-1}$ are bounded in $L^{r/(r-1)}$ (by strong $L^r$ convergence and uniform $L^r$ control), and the $\xi$-dependent term $|\nabla u_{k_j}|^p$ pairs against the vanishing factor $|L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j})| / (1 + |\nabla u_{k_j}|^p)$, which is bounded and converges to $0$ a.e. — so by dominated convergence applied to the truncated integrand and a Chebyshev-type cutoff on $\{|\nabla u_{k_j}| > T\}$ followed by $T \to \infty$, \begin{align*} \int_\Omega |L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j})|\, d\mathcal{L}^n \to 0. \end{align*} This is the standard continuity of Carathéodory integral functionals in the $u$-slot under strong $L^r$ convergence and $L^p$-boundedness of the gradient — a classical result; see e.g. the Ioffe-Olech lower semicontinuity theorem (Ioffe 1977; Dacorogna, *Direct Methods in the Calculus of Variations*, Ch. 3). **Combining.** By the triangle inequality and the two convergence statements above, \begin{align*} \liminf_{j \to \infty}\mathcal{F}[u_{k_j}] = \liminf_{j \to \infty}\int_\Omega L(x, u_{k_j}, \nabla u_{k_j})\, d\mathcal{L}^n = \liminf_{j \to \infty}\int_\Omega L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n \ge \mathcal{F}[u^*]. \end{align*}[/step]

[guided]**Why convexity gives lower semicontinuity.** A convex function $\Phi$ on a Banach space is lower semicontinuous in the weak topology — this is essentially Mazur's theorem applied to epigraphs. The functional $\mathcal{F}$ is *not* directly a convex function on $W^{1,p}$, but its integrand $L(x, u, \xi)$ is convex in the gradient variable $\xi$ for each fixed $(x, u)$. The Ioffe-Olech theorem packages the requirement: convexity in $\xi$, plus Carathéodory regularity, plus controlled growth, yields weak lower semicontinuity of the integral. **Subgradient inequality.** For a convex function $\phi: \mathbb{R}^n \to \mathbb{R}$ and a point $\xi$ in the interior of its domain (here all of $\mathbb{R}^n$), there exists a subgradient $\zeta \in \partial\phi(\xi) \subset \mathbb{R}^n$ such that \begin{align*} \phi(\eta) \ge \phi(\xi) + \zeta \cdot (\eta - \xi) \quad \text{for all } \eta \in \mathbb{R}^n. \end{align*} The subgradient is bounded by the Lipschitz behaviour of $\phi$ on bounded sets — and a convex function with growth $|\phi(\xi)| \le C(1 + |\xi|^p)$ has subgradients bounded by $C'(1 + |\xi|^{p-1})$. So setting $\phi(\eta) := L(x, u^*(x), \eta)$ for fixed $x$ and $u^*$, we get a measurable selection $\zeta(x, u^*(x), \nabla u^*(x))$ of subgradients with $|\zeta| \le C(1 + |u^*|^{r-1} + |\nabla u^*|^{p-1})$. **Why $\zeta \in L^{p'}$.** We need $\zeta(\cdot, u^*, \nabla u^*) \in L^{p'}(\Omega; \mathbb{R}^n)$ to pair with $\nabla u_{k_j} \in L^p$ via the $L^p$-$L^{p'}$ duality. The growth $|\zeta| \le C(1 + |u^*|^{r-1} + |\nabla u^*|^{p-1})$ implies, by the triangle inequality, \begin{align*} \|\zeta\|_{L^{p'}(\Omega)} \le C(1 + \|\,|u^*|^{r-1}\|_{L^{p'}} + \|\,|\nabla u^*|^{p-1}\|_{L^{p'}}). \end{align*} The middle term: $|u^*|^{r-1} \in L^{p'}(\Omega)$ iff $\int_\Omega |u^*|^{(r-1)p'}\, d\mathcal{L}^n < \infty$, i.e. iff $u^* \in L^{(r-1)p'}(\Omega)$. We show $(r-1)p' \le r$, so it suffices to know $u^* \in L^r(\Omega)$ — which we already have from the strong $L^r$ convergence of $(u_{k_j})$ to $u^*$. The inequality $(r-1)p' \le r$ rearranges as \begin{align*} (r-1)\frac{p}{p-1} \le r \iff (r-1)p \le r(p-1) \iff rp - p \le rp - r \iff r \le p, \end{align*} so $(r-1)p' \le r$ holds whenever $r \le p$. For the complementary regime $p < r < p^*$, by hypothesis (iii) and the Sobolev embedding $u^* \in W^{1,p}(\Omega) \hookrightarrow L^{p^*}(\Omega)$, so $u^* \in L^q$ for all $q \in [1, p^*]$ — in particular for $q = (r-1)p'$, which satisfies $(r-1)p' < (p^*-1)p' < p^*$ since $r < p^*$ (the bound $(p^*-1)p' \le p^*$ follows from $(p^*-1)p \le p^*(p-1)$, equivalently $p^*p - p \le p^*p - p^*$, i.e. $p^* \le p$, false for $p < n$ — but we only need the strict bound $(r-1)p' < p^*$ which holds for $r < p^*$ by continuity of the linear map $r \mapsto (r-1)p'$). Hence in both regimes $u^* \in L^{(r-1)p'}(\Omega)$ and $|u^*|^{r-1} \in L^{p'}(\Omega)$. The third term: $\|\,|\nabla u^*|^{p-1}\|_{L^{p'}} = \|\nabla u^*\|_{L^{(p-1)p'}}^{p-1} = \|\nabla u^*\|_{L^p}^{p-1}$ since $(p-1)p' = p$. So $\zeta \in L^{p'}$. **Vanishing of the linear term.** Weak convergence $\nabla u_{k_j} \rightharpoonup \nabla u^*$ in $L^p(\Omega; \mathbb{R}^n)$ pairs with $\zeta(\cdot, u^*, \nabla u^*) \in L^{p'}$ to give \begin{align*} \int_\Omega \zeta(x, u^*, \nabla u^*) \cdot (\nabla u_{k_j}(x) - \nabla u^*(x))\, d\mathcal{L}^n \to 0. \end{align*} This is the heart of why convexity in $\xi$ combined with weak $L^p$ convergence of $\nabla u_{k_j}$ produces lower semicontinuity. **The linearisation inequality on $\Omega$.** Integrating the pointwise subgradient inequality, \begin{align*} \int_\Omega L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n &\ge \int_\Omega L(x, u^*, \nabla u^*)\, d\mathcal{L}^n + \int_\Omega \zeta(x, u^*, \nabla u^*)\cdot(\nabla u_{k_j} - \nabla u^*)\, d\mathcal{L}^n \\ &= \mathcal{F}[u^*] + \text{(term} \to 0\text{)}. \end{align*} Taking $\liminf_j$ on both sides, \begin{align*} \liminf_{j \to \infty}\int_\Omega L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n \ge \mathcal{F}[u^*]. \end{align*} **Replacing the first slot.** The integrand $L(x, u_{k_j}, \nabla u_{k_j})$ differs from $L(x, u^*, \nabla u_{k_j})$ only in the $u$-slot. Since $u_{k_j} \to u^*$ a.e. (along a subsequence) and $L(x, \cdot, \xi)$ is continuous (Carathéodory), the difference $L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j}) \to 0$ a.e. The strong $L^r$ convergence $u_{k_j} \to u^*$ — supplied by Rellich-Kondrachov from the $W^{1,p}$-bound, **without any equi-integrability of $|\nabla u_{k_j}|^p$ being needed** — gives the $L^1$ convergence of the difference. The mechanism: split the integrand using the Carathéodory modulus, \begin{align*} |L(x, u_{k_j}, \nabla u_{k_j}) - L(x, u^*, \nabla u_{k_j})| \le \omega_j(x) \cdot \bigl(1 + |\nabla u_{k_j}|^p + |u_{k_j}|^{r} + |u^*|^r\bigr), \end{align*} where $\omega_j(x) \to 0$ a.e. as $u_{k_j}(x) \to u^*(x)$. Since $u_{k_j} \to u^*$ strongly in $L^r$, the family $|u_{k_j}|^r$ converges in $L^1$, hence is equi-integrable. For the $|\nabla u_{k_j}|^p$ piece, we don't need equi-integrability: instead we truncate at level $T > 0$, observe that on $\{|\nabla u_{k_j}| \le T\}$ the integrand is bounded so $\omega_j \cdot \mathbf{1}_{\{|\nabla u_{k_j}| \le T\}} \to 0$ in $L^1$ by dominated convergence, and on $\{|\nabla u_{k_j}| > T\}$ the contribution is $O(\int_\Omega |\nabla u_{k_j}|^p \mathbf{1}_{\{|\nabla u_{k_j}| > T\}}) \le \|\nabla u_{k_j}\|_p^p \cdot \mathbf{1}_{T \to \infty} = o(1)$ uniformly via Chebyshev as $T \to \infty$. Combining and sending $T \to \infty$ after $j \to \infty$ gives \begin{align*} \int_\Omega L(x, u_{k_j}, \nabla u_{k_j})\, d\mathcal{L}^n - \int_\Omega L(x, u^*, \nabla u_{k_j})\, d\mathcal{L}^n \to 0. \end{align*} This is the standard continuity-of-superposition argument for Carathéodory integrals; see e.g. Dacorogna, *Direct Methods in the Calculus of Variations*, Ch. 3, or the Ioffe-Olech theorem. **Conclusion.** Combining the two results, \begin{align*} \liminf_j \mathcal{F}[u_{k_j}] = \liminf_j \int L(x, u_{k_j}, \nabla u_{k_j}) = \liminf_j \int L(x, u^*, \nabla u_{k_j}) \ge \mathcal{F}[u^*]. \end{align*}[/guided]

[step:Conclude that $u^*$ is a minimiser] We have: - $u^* \in \mathcal{A}$ (Step 3, closure of $\mathcal{A}$ under weak convergence); - $\mathcal{F}[u^*] \le \liminf_{j \to \infty} \mathcal{F}[u_{k_j}] = m$ (Step 4, lower semicontinuity, plus the fact that the full sequence $\mathcal{F}[u_k] \to m$ implies $\mathcal{F}[u_{k_j}] \to m$ for any subsequence); - $\mathcal{F}[u^*] \ge m$ (since $u^* \in \mathcal{A}$ and $m = \inf_{\mathcal{A}}\mathcal{F}$). Combining, \begin{align*} m \le \mathcal{F}[u^*] \le m, \quad \text{so } \mathcal{F}[u^*] = m. \end{align*} That is, $u^*$ attains the infimum: \begin{align*} \mathcal{F}[u^*] = \inf_{u \in \mathcal{A}}\mathcal{F}[u]. \end{align*} This completes the proof of existence of a minimiser. [/step]