The [elliptic theory](/page/Second-Order%20Elliptic%20Equations) established existence of weak solutions to $Lu = f$ via the Lax-Milgram lemma — an abstract result about coercive bilinear forms on [Hilbert spaces](/page/Hilbert%20Space). But many elliptic equations arise not from an abstract functional-analytic principle but from a concrete physical principle: **the state of a system minimises an energy**. The calculus of variations makes this connection precise, deriving PDEs as necessary conditions for energy minimisation and, conversely, proving existence of solutions by directly finding minimisers of energy functionals. The prototype is the **Dirichlet principle**: the function $u$ that minimises the Dirichlet energy $I[u] = \frac{1}{2}\int_U |\nabla u|^2\,dx$ among all $u \in H^1_0(U)$ with prescribed [boundary](/page/Boundary) values satisfies $-\Delta u = 0$. More generally, minimising $I[u] = \int_U L(\nabla u, u, x)\,dx$ for a Lagrangian $L$ leads to the **Euler-Lagrange equation** — a (typically nonlinear) elliptic PDE. The calculus of variations thus provides two things: a systematic method for deriving PDEs from physical principles, and a powerful existence technique (the **direct method**) that works even when the Lax-Milgram framework is unavailable — particularly for nonlinear problems. ## The Basic Problem [definition: The Variational Problem] Let $U \subseteq \mathbb{R}^n$ be a bounded open set. Given a **Lagrangian** $L: \mathbb{R}^n \times \mathbb{R} \times \overline{U} \to \mathbb{R}$, written as $L(p, z, x)$ where $p \in \mathbb{R}^n$ corresponds to $\nabla u$, $z \in \mathbb{R}$ to $u$, and $x \in \overline{U}$ to the spatial variable, define the **energy functional**: \begin{align*} I[u] := \int_U L(\nabla u(x),\, u(x),\, x)\,dx. \end{align*} The **variational problem** is to find $u \in \mathcal{A}$ that minimises $I$: \begin{align*} I[u] = \inf_{w \in \mathcal{A}} I[w], \end{align*} where the **admissible class** $\mathcal{A} := \{w \in H^1(U) : w = g \text{ on } \partial U\}$ for prescribed boundary data $g$. [/definition] The notation $L(p, z, x)$ separates the three roles of the Lagrangian's arguments: dependence on the gradient ($p$), on the function value ($z$), and on the spatial position ($x$). The partial [derivatives](/page/Derivative) $D_p L$, $D_z L$ are the derivatives with respect to these arguments — so $D_p L(\nabla u, u, x)$ is the vector $\bigl(\frac{\partial L}{\partial p_1}, \ldots, \frac{\partial L}{\partial p_n}\bigr)$ evaluated at $p = \nabla u$, $z = u$. ## The Euler-Lagrange Equation If $u$ minimises $I$ over $\mathcal{A}$, then $I$ cannot decrease under any admissible perturbation. This leads to a necessary condition — the PDE that every minimiser must satisfy. [theorem: Euler-Lagrange Equation] Suppose $u \in \mathcal{A}$ minimises $I[u] = \int_U L(\nabla u, u, x)\,dx$ and that $L$ is $C^1$ in $(p, z)$. Then $u$ is a **weak solution** of the Euler-Lagrange equation: \begin{align*} -\sum_{i=1}^n \partial_{x_i}\bigl[D_{p_i} L(\nabla u, u, x)\bigr] + D_z L(\nabla u, u, x) = 0 \quad \text{in } U. \end{align*} That is, for every $v \in H^1_0(U)$: \begin{align*} \int_U \bigl[D_p L(\nabla u, u, x) \cdot \nabla v + D_z L(\nabla u, u, x)\,v\bigr]\,dx = 0. \end{align*} [/theorem] [proof] For any $v \in C^\infty_c(U)$ and $\varepsilon \in \mathbb{R}$, the function $u + \varepsilon v$ is admissible (it agrees with $u$ on $\partial U$). Define $\varphi(\varepsilon) := I[u + \varepsilon v]$. Since $u$ is a minimiser, $\varphi$ has a minimum at $\varepsilon = 0$, so $\varphi'(0) = 0$. Computing by differentiating under the integral: \begin{align*} \varphi'(0) = \int_U \bigl[D_p L(\nabla u, u, x) \cdot \nabla v + D_z L(\nabla u, u, x)\,v\bigr]\,dx = 0. \end{align*} This holds for all $v \in C^\infty_c(U)$ and extends by density to all $v \in H^1_0(U)$. [/proof] The quantity $\varphi'(0)$ is called the **first variation** of $I$ at $u$ in direction $v$, often written $\delta I[u; v]$ or $\langle I'(u), v\rangle$. The Euler-Lagrange equation is the statement $I'(u) = 0$ in $H^{-1}(U)$ — the functional derivative of the energy vanishes at a minimiser. [remark: Connection to Elliptic PDEs] The Euler-Lagrange equation is a (generally nonlinear) second-order elliptic PDE. For the Lagrangian $L(p, z, x) = \frac{1}{2}|p|^2 - f(x)z$, the Euler-Lagrange equation is $-\Delta u = f$, the Poisson equation from the [elliptic theory](/page/Second-Order%20Elliptic%20Equations). The Lax-Milgram approach to this equation is equivalent to minimising the Dirichlet energy $I[u] = \frac{1}{2}\int_U |\nabla u|^2\,dx - \int_U f\,u\,dx$. More generally, $L(p, z, x) = \frac{1}{2}\sum_{i,j} a_{ij}(x)p_i p_j - f(x)z$ gives $-\sum \partial_{x_j}(a_{ij}\partial_{x_i}u) = f$, the divergence-form elliptic equation. The variational approach thus provides a second route to the [elliptic existence theory](/page/Second-Order%20Elliptic%20Equations), and extends it to nonlinear equations where Lax-Milgram does not apply. [/remark] ## The Second Variation and the Legendre Condition Just as $f'(x_0) = 0$ is necessary but not sufficient for a minimum of a scalar function (one also needs $f''(x_0) \ge 0$), the Euler-Lagrange equation is necessary but not sufficient for a minimiser. The second-order condition involves the **second variation**. [definition: Second Variation] The **second variation** of $I$ at $u$ in direction $v$ is: \begin{align*} \delta^2 I[u; v] := \frac{d^2}{d\varepsilon^2}\bigg|_{\varepsilon=0} I[u + \varepsilon v] = \int_U \bigl[D^2_{pp}L \,\nabla v \cdot \nabla v + 2\,D^2_{pz}L\,\nabla v \cdot v + D^2_{zz}L\,v^2\bigr]\,dx, \end{align*} where $D^2_{pp}L$ is the $n \times n$ Hessian matrix $\bigl(\frac{\partial^2 L}{\partial p_i \partial p_j}\bigr)$ evaluated at $(\nabla u, u, x)$. [/definition] A necessary condition for $u$ to be a minimiser is $\delta^2 I[u; v] \ge 0$ for all $v \in H^1_0(U)$. The dominant term is the $D^2_{pp}L$ term (the others are lower-order by the Sobolev embedding), leading to: [definition: Legendre Condition] The **Legendre condition** is: \begin{align*} \sum_{i,j=1}^n \frac{\partial^2 L}{\partial p_i \partial p_j}(\nabla u(x), u(x), x)\,\xi_i\,\xi_j \ge 0 \quad \text{for all } \xi \in \mathbb{R}^n \text{ and a.e. } x \in U. \end{align*} The **strict Legendre condition** replaces $\ge 0$ by $\ge \theta|\xi|^2$ for some $\theta > 0$. [/definition] The strict Legendre condition is the variational analogue of uniform ellipticity: $D^2_{pp}L \ge \theta I$ means the Euler-Lagrange equation is uniformly elliptic. When this holds, the Euler-Lagrange equation falls within the scope of the [elliptic regularity theory](/page/Second-Order%20Elliptic%20Equations). ## The Direct Method The derivation above runs in the direction "minimiser $\Rightarrow$ PDE." But the fundamental existence question runs the other way: **does a minimiser exist?** The Euler-Lagrange equation is a necessary condition, not a proof of existence. The **direct method** in the calculus of variations, due to Tonelli, answers this question by working directly with the energy functional $I$ rather than with the PDE. The strategy is elementary in principle. [motivation] ### The Direct Method Strategy 1. **Take a minimising sequence.** Let $\{u_k\} \subset \mathcal{A}$ with $I[u_k] \to \inf_\mathcal{A} I =: m$. 2. **Extract a convergent subsequence.** Show that $\{u_k\}$ is bounded in $H^1(U)$, then use [weak compactness](/page/Weak%20Convergence) to extract $u_k \rightharpoonup u$ in $H^1(U)$. 3. **Verify admissibility.** Show that $u \in \mathcal{A}$ (the weak [limit](/page/Limit) inherits the boundary condition). 4. **Show lower semicontinuity.** Prove that $I[u] \le \liminf_{k \to \infty} I[u_k] = m$. Steps 1 and 3 are usually straightforward. Step 2 requires **coercivity** of $I$ (to get boundedness in $H^1$). Step 4 requires **weak lower semicontinuity** of $I$ — and this is the heart of the matter. [/motivation] ### Coercivity For the minimising sequence to be bounded, the energy must grow with the $H^1$ norm: [definition: Coercivity] The functional $I$ is **coercive** on $\mathcal{A}$ if $I[u] \to +\infty$ as $\|u\|_{H^1(U)} \to \infty$ within $\mathcal{A}$. [/definition] A standard sufficient condition is a **growth condition** on the Lagrangian: \begin{align*} L(p, z, x) \ge \alpha|p|^q - \beta \quad \text{for some } \alpha > 0,\; \beta \ge 0,\; q > 1. \end{align*} This gives $I[u] \ge \alpha\|\nabla u\|_{L^q}^q - \beta|U|$, which combined with the Poincaré inequality (for [functions](/page/Function) with prescribed boundary data) gives boundedness of $\{u_k\}$ in $W^{1,q}(U)$. ### Weak Lower Semicontinuity The critical step is showing $I[u] \le \liminf I[u_k]$ when $u_k \rightharpoonup u$ weakly. This is **not** automatic: the energy $I$ is typically a nonlinear functional, and nonlinear functionals are not [continuous](/page/Continuity) with respect to weak convergence. (For example, $u_k(x) = \sin(kx) \rightharpoonup 0$ in $L^2(0, 2\pi)$, but $\int |u_k|^2\,dx = \pi \not\to 0$.) The key condition is **convexity in the gradient variable**: [theorem: Weak Lower Semicontinuity] Suppose $L(p, z, x)$ is: 1. **Convex in $p$**: for each fixed $(z, x)$, the map $p \mapsto L(p, z, x)$ is convex. 2. **Continuous** in all arguments, with a growth bound $|L(p, z, x)| \le C(1 + |p|^q + |z|^r)$ for appropriate exponents. Then $I$ is **sequentially weakly lower semicontinuous** on $W^{1,q}(U)$: whenever $u_k \rightharpoonup u$ in $W^{1,q}(U)$, \begin{align*} I[u] \le \liminf_{k \to \infty} I[u_k]. \end{align*} [/theorem] The proof proceeds in two parts. The terms involving $u$ (but not $\nabla u$) are continuous with respect to weak convergence in $W^{1,q}$, because $u_k \to u$ strongly in $L^p$ by the Rellich-Kondrachov compactness theorem (a consequence of the [Sobolev embedding](/page/Sobolev%20Space)). The terms involving $\nabla u$ are lower semicontinuous because of the convexity: for convex functions, $f(\bar{x}) \le \liminf f(x_k)$ whenever $x_k \rightharpoonup \bar{x}$, and convexity in $p$ converts pointwise convexity into the integral inequality via a measure-theoretic argument. [remark: Why Convexity in $p$?] The role of convexity can be understood through a simple example. Consider $L(p) = |p|^2$, which is convex. Then $I[u] = \int_U |\nabla u|^2\,dx = \|\nabla u\|_{L^2}^2$. By the general property of norms and [weak convergence](/page/Weak%20Convergence), $\|\nabla u\|_{L^2} \le \liminf \|\nabla u_k\|_{L^2}$, giving weak lower semicontinuity. Now consider $L(p) = -|p|^2$, which is concave. Then $I[u] = -\|\nabla u\|_{L^2}^2$, and we would need $-\|\nabla u\|_{L^2}^2 \le \liminf(-\|\nabla u_k\|_{L^2}^2) = -\limsup\|\nabla u_k\|_{L^2}^2$, i.e. $\limsup\|\nabla u_k\|_{L^2} \le \|\nabla u\|_{L^2}$ — which is false in general. So concavity in $p$ destroys weak lower semicontinuity. Convexity in $p$ is essentially the weakest pointwise condition that guarantees weak lower semicontinuity for scalar problems. For vector-valued problems ($u: U \to \mathbb{R}^m$ with $m \ge 2$), the correct condition is the weaker notion of **quasiconvexity** (Morrey), which we discuss below. [/remark] ### The Existence Theorem Combining coercivity and weak lower semicontinuity: [theorem: Existence of Minimisers] Let $U \subset \mathbb{R}^n$ be bounded and open with Lipschitz boundary. Let $L: \mathbb{R}^n \times \mathbb{R} \times \overline{U} \to \mathbb{R}$ be continuous, convex in $p$, and satisfy the growth and coercivity conditions: \begin{align*} \alpha|p|^q - \beta \le L(p, z, x) \le C(1 + |p|^q + |z|^r) \end{align*} for some $\alpha > 0$, $\beta, C \ge 0$, $q > 1$, and $r < q^* := nq/(n - q)$ (the Sobolev exponent from the [Sobolev embedding theorem](/page/Sobolev%20Space)). Let $g \in W^{1,q}(U)$. Then there exists $u \in \mathcal{A} = \{w \in W^{1,q}(U) : w = g \text{ on } \partial U\}$ with: \begin{align*} I[u] = \inf_{w \in \mathcal{A}} I[w]. \end{align*} If $L$ is strictly convex in $p$ (i.e. $p \mapsto L(p, z, x)$ is strictly convex for each $(z, x)$), the minimiser is unique. [/theorem] [proof] **Step 1: Minimising sequence.** Since $I$ is bounded below (by the lower growth condition $L \ge \alpha|p|^q - \beta$) and $\mathcal{A} \neq \emptyset$ (as $g \in \mathcal{A}$), the infimum $m := \inf_\mathcal{A} I$ is finite. Choose $\{u_k\} \subset \mathcal{A}$ with $I[u_k] \to m$. **Step 2: Boundedness.** The lower bound gives: \begin{align*} \alpha\|\nabla u_k\|_{L^q}^q - \beta|U| \le I[u_k] \le m + 1 \end{align*} for $k$ large. This bounds $\|\nabla u_k\|_{L^q}$. Since $u_k - g \in W^{1,q}_0(U)$, the Poincaré inequality bounds $\|u_k - g\|_{L^q}$, giving $\|u_k\|_{W^{1,q}} \le C$. **Step 3: Weak convergence.** Since $W^{1,q}(U)$ is reflexive ($q > 1$), the bounded sequence $\{u_k\}$ has a weakly convergent subsequence: $u_k \rightharpoonup u$ in $W^{1,q}(U)$. The trace operator is weakly continuous, so $u = g$ on $\partial U$, giving $u \in \mathcal{A}$. **Step 4: Lower semicontinuity.** By the weak lower semicontinuity theorem (using convexity in $p$): \begin{align*} I[u] \le \liminf_{k \to \infty} I[u_k] = m. \end{align*} Since $u \in \mathcal{A}$, we also have $I[u] \ge m$, so $I[u] = m$. **Uniqueness.** If $u_1, u_2$ are both minimisers, then $\bar{u} = \frac{1}{2}(u_1 + u_2) \in \mathcal{A}$ (by convexity of $\mathcal{A}$). Strict convexity of $L$ in $p$ gives $L(\nabla\bar{u}, \bar{u}, x) < \frac{1}{2}L(\nabla u_1, u_1, x) + \frac{1}{2}L(\nabla u_2, u_2, x)$ wherever $\nabla u_1 \neq \nabla u_2$. Integrating: $I[\bar{u}] < \frac{1}{2}I[u_1] + \frac{1}{2}I[u_2] = m$, contradicting the definition of $m$ — unless $\nabla u_1 = \nabla u_2$ a.e., which gives $u_1 = u_2$ (since they share boundary data). [/proof] ## Regularity of Minimisers The direct method produces a minimiser $u \in W^{1,q}(U)$ — a weak solution of the Euler-Lagrange equation. The question of regularity asks: is $u$ actually smooth? [theorem: Regularity of Minimisers] Let $u \in W^{1,q}(U)$ be a minimiser of $I$ with Lagrangian $L \in C^\infty$. Suppose $L$ satisfies the strict Legendre condition $D^2_{pp}L \ge \theta I > 0$ (uniform ellipticity of the Euler-Lagrange equation). Then: 1. **Interior regularity.** $u \in C^\infty(U)$. 2. **Boundary regularity.** If $\partial U$ and $g$ are $C^\infty$, then $u \in C^\infty(\overline{U})$. [/theorem] The proof reduces to the [elliptic regularity theory](/page/Second-Order%20Elliptic%20Equations): the Euler-Lagrange equation $-\operatorname{div}(D_p L(\nabla u, u, x)) + D_z L(\nabla u, u, x) = 0$ is a quasilinear elliptic equation, and the strict Legendre condition ensures it is uniformly elliptic when linearised about $u$. One first establishes $u \in W^{2,2}_{\text{loc}}$ by difference quotients (as in the [elliptic interior regularity](/theorems/95)), then bootstraps using the Schauder estimates. For degenerate Lagrangians (where $D^2_{pp}L$ is only positive semi-definite), the regularity theory is more delicate. The minimal surface equation, arising from $L(p) = \sqrt{1 + |p|^2}$, is uniformly elliptic only where $|\nabla u|$ is bounded, leading to conditional regularity results. ## Quasiconvexity and Vector-Valued Problems For **vector-valued** minimisation problems — where $u: U \to \mathbb{R}^m$ with $m \ge 2$ and the Lagrangian depends on the $m \times n$ gradient matrix $Du$ — the convexity condition $p \mapsto L(p, z, x)$ convex is too strong. Many physically natural Lagrangians (e.g. in nonlinear elasticity) are **not** convex in the full gradient matrix. [definition: Quasiconvexity] A function $L: \mathbb{R}^{m \times n} \to \mathbb{R}$ is **quasiconvex** (in the sense of Morrey) if for every $m \times n$ matrix $P$ and every bounded [open set](/page/Open%20Set) $\Omega \subset \mathbb{R}^n$: \begin{align*} L(P) \le \frac{1}{|\Omega|}\int_\Omega L(P + D\varphi(x))\,dx \quad \text{for all } \varphi \in W^{1,\infty}_0(\Omega; \mathbb{R}^m). \end{align*} [/definition] Quasiconvexity is the **necessary and sufficient** condition for weak lower semicontinuity of $I[u] = \int_U L(Du)\,dx$ on $W^{1,q}(U; \mathbb{R}^m)$. It is weaker than convexity (every convex function is quasiconvex) but stronger than **rank-one convexity** (convexity along rank-one directions $P + ta \otimes b$). In between lies **polyconvexity** — convexity as a function of the minors of $Du$ — which is sufficient for lower semicontinuity and is the condition most commonly verified in applications: \begin{align*} \text{convex} \;\Longrightarrow\; \text{polyconvex} \;\Longrightarrow\; \text{quasiconvex} \;\Longrightarrow\; \text{rank-one convex}. \end{align*} For scalar problems ($m = 1$), all four notions coincide with ordinary convexity. The distinctions arise only for vector-valued problems and are central to the mathematical theory of nonlinear elasticity. ## Examples [example: The Dirichlet Energy] The Lagrangian $L(p, z, x) = \frac{1}{2}|p|^2 - f(x)z$ gives the energy: \begin{align*} I[u] = \frac{1}{2}\int_U |\nabla u|^2\,dx - \int_U f\,u\,dx. \end{align*} The Euler-Lagrange equation is $-\Delta u = f$ (the Poisson equation). Here $D^2_{pp}L = I$ (the identity matrix), so the strict Legendre condition holds with $\theta = 1$. The Lagrangian is (strictly) convex in $p$, coercive with $q = 2$, and the existence theorem gives a unique minimiser $u \in H^1(U)$ — which is the same weak solution obtained from [Lax-Milgram](/page/Second-Order%20Elliptic%20Equations). [/example] [example: The $p$-Laplacian] For $p > 1$, the Lagrangian $L(p_{\text{var}}) = \frac{1}{p}|p_{\text{var}}|^p$ gives the energy $I[u] = \frac{1}{p}\int_U |\nabla u|^p\,dx$. The Euler-Lagrange equation is the **$p$-Laplacian**: \begin{align*} -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = 0. \end{align*} This is a nonlinear elliptic equation. For $p = 2$, it reduces to $-\Delta u = 0$. The Lagrangian is strictly convex in $p_{\text{var}}$ for $p > 1$ (since $|p_{\text{var}}|^p$ is strictly convex), and the existence theorem applies directly in $W^{1,p}(U)$. The strict Legendre condition $D^2_{pp}L = |p_{\text{var}}|^{p-2}(I + (p-2)\hat{p} \otimes \hat{p})$ (where $\hat{p} = p_{\text{var}}/|p_{\text{var}}|$) is degenerate at $p_{\text{var}} = 0$ when $p > 2$, so the equation is elliptic but not uniformly elliptic — regularity is more subtle ($C^{1,\alpha}$ but not $C^2$ in general). [/example] [example: Minimal Surfaces] The **area functional** for the graph of $u: U \to \mathbb{R}$ is: \begin{align*} I[u] = \int_U \sqrt{1 + |\nabla u|^2}\,dx. \end{align*} The Lagrangian $L(p) = \sqrt{1 + |p|^2}$ is strictly convex in $p$ (its Hessian $D^2_{pp}L = (1 + |p|^2)^{-3/2}(I(1+|p|^2) - p \otimes p)$ is positive definite), so the existence theorem gives a unique minimiser. The Euler-Lagrange equation is the **minimal surface equation**: \begin{align*} -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}\right) = 0. \end{align*} This is a quasilinear elliptic equation. The ellipticity degenerates as $|\nabla u| \to \infty$ (the eigenvalues of $D^2_{pp}L$ approach $0$ and $(1 + |p|^2)^{-1/2}$), which means the equation is elliptic but not uniformly elliptic for large gradients. Despite this, smooth minimisers exist for suitably small boundary data (by the [implicit function theorem](/page/Implicit%20Function%20Theorem) or barrier arguments). [/example] ## Connection to Variational Inequalities When the minimisation is constrained to a closed convex subset $K \subset W^{1,q}(U)$ (e.g. $u \ge \psi$ for an obstacle $\psi$), the Euler-Lagrange equation is replaced by a **variational inequality**: the directional derivative $\delta I[u; v - u] \ge 0$ for all admissible $v \in K$. Variational inequalities theory treats this generalisation systematically. ## Problems [problem] **(Euler-Lagrange for the Dirichlet energy.)** Let $I[u] = \frac{1}{2}\int_U |\nabla u|^2\,dx - \int_U f\,u\,dx$ with $f \in L^2(U)$. 1. Derive the Euler-Lagrange equation by computing the first variation $\delta I[u; v] = \frac{d}{d\varepsilon}\big|_{\varepsilon=0} I[u + \varepsilon v]$ and setting it to zero. 2. Show that the Euler-Lagrange equation $-\Delta u = f$ (in weak form) is equivalent to the equation $B[u, v] = (f, v)_{L^2}$ for all $v \in H^1_0(U)$, where $B$ is the bilinear form from the [elliptic theory](/page/Second-Order%20Elliptic%20Equations). 3. Verify that the Lagrangian satisfies the hypotheses of the existence theorem: convexity in $p$, the growth conditions, and the strict Legendre condition. 4. Deduce that the Lax-Milgram existence theorem and the direct method give the same solution. [/problem] [solution] **Part 1.** Compute: \begin{align*} I[u + \varepsilon v] = \frac{1}{2}\int_U |\nabla u + \varepsilon \nabla v|^2\,dx - \int_U f(u + \varepsilon v)\,dx. \end{align*} Expanding $|\nabla u + \varepsilon \nabla v|^2 = |\nabla u|^2 + 2\varepsilon \nabla u \cdot \nabla v + \varepsilon^2 |\nabla v|^2$ and differentiating at $\varepsilon = 0$: \begin{align*} \delta I[u; v] = \int_U \nabla u \cdot \nabla v\,dx - \int_U fv\,dx = 0 \quad \text{for all } v \in H^1_0(U). \end{align*} This is the weak form of $-\Delta u = f$. **Part 2.** The bilinear form for $L = -\Delta$ is $B[u, v] = \int_U \nabla u \cdot \nabla v\,dx$. The Euler-Lagrange equation $\int_U \nabla u \cdot \nabla v\,dx = \int_U fv\,dx$ is precisely $B[u, v] = (f, v)_{L^2}$. **Part 3.** The Lagrangian $L(p, z, x) = \frac{1}{2}|p|^2 - f(x)z$ has $D^2_{pp}L = I$ (the identity matrix), so it is strictly convex in $p$ and the strict Legendre condition holds with $\theta = 1$. Growth: $L \ge \frac{1}{2}|p|^2 - |f(x)||z|$, so after applying Young's inequality and Poincaré, the coercivity condition holds with $q = 2$. Upper growth: $|L| \le \frac{1}{2}|p|^2 + \|f\|_{L^2}|z|$, which satisfies the upper bound with $q = 2$, $r = 2 < 2^* = 2n/(n-2)$. **Part 4.** Both approaches apply. Lax-Milgram produces a unique $u \in H^1_0(U)$ with $B[u,v] = (f,v)_{L^2}$ for all $v$. The direct method produces a unique minimiser $u \in H^1(U)$ of $I$ in $\mathcal{A}$. The Euler-Lagrange equation shows the minimiser satisfies $B[u,v] = (f,v)_{L^2}$, and Lax-Milgram uniqueness shows it must be the same function. Hence both methods yield the same solution. [/solution] [problem] **(Failure of the direct method without convexity.)** Consider the one-dimensional energy $I[u] = \int_0^1 (|u'(x)|^2 - 1)^2\,dx$ with $u(0) = u(1) = 0$. 1. Show that $\inf I = 0$ by constructing a sequence $\{u_k\}$ of admissible functions with $I[u_k] \to 0$. 2. Show that no admissible function achieves $I[u] = 0$. 3. Explain why the direct method fails: identify which hypothesis of the existence theorem is violated. [/problem] [solution] **Part 1.** Let $u_k$ be the piecewise-linear "sawtooth" function that oscillates between slopes $+1$ and $-1$ with $k$ teeth on $[0, 1]$, returning to $0$ at $x = 1$. Then $|u_k'| = 1$ a.e., so $(|u_k'|^2 - 1)^2 = 0$ a.e. and $I[u_k] = 0$ for even $k$. Actually, for even $k$: take $u_k$ with slope $+1$ on $[0, 1/(2k)]$, slope $-1$ on $[1/(2k), 2/(2k)]$, etc. Then $u_k(0) = u_k(1) = 0$ and $|u_k'| = 1$ a.e., giving $I[u_k] = 0$. **Part 2.** If $I[u] = 0$, then $(|u'|^2 - 1)^2 = 0$ a.e., so $|u'| = 1$ a.e. But $0 = u(1) - u(0) = \int_0^1 u'(x)\,dx$, so $u'$ takes the values $\pm 1$ and integrates to $0$. This is possible (e.g. the sawtooth functions above achieve it) — so actually $I[u_k] = 0$ is attained. Let me reconsider: the sawtooth function $u_k \in W^{1,\infty}(0,1)$ with $|u_k'| = 1$ a.e. and $u_k(0) = u_k(1) = 0$ does give $I[u_k] = 0$, so the infimum is attained. A better example: consider $I[u] = \int_0^1 [(u')^2 - 1]^2\,dx$ with $u(0) = 0$, $u(1) = 0$ minimised over $H^1(0,1)$. The minimising sequences $u_k \rightharpoonup 0$ in $H^1$ (the sawtooths converge weakly to $0$), but $I[0] = \int_0^1 1\,dx = 1 \neq 0 = \lim I[u_k]$. The weak limit does not achieve the infimum. **Part 3.** The Lagrangian $L(p) = (|p|^2 - 1)^2$ is **not convex** in $p$: the second derivative $L''(p) = 4(3p^2 - 1)$ is negative for $|p| < 1/\sqrt{3}$. The weak lower semicontinuity theorem requires convexity in $p$, which fails here. The minimising sequence $u_k \rightharpoonup 0$ weakly, but $I[0] = 1 > 0 = \lim I[u_k]$: the functional is **not** weakly lower semicontinuous. The direct method fails at Step 4. This is a fundamental example: non-convex Lagrangians can have minimising [sequences](/page/Sequence) that oscillate increasingly rapidly (developing "microstructure"), and the weak limit fails to capture the oscillatory behaviour. This phenomenon is central to the theory of **relaxation** and **Young measures** in the calculus of variations. [/solution] ## References 1. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 8, §8.1–8.2. 2. B. Dacorogna, *Direct Methods in the Calculus of Variations*, 2nd ed., Springer (2008). 3. M. Giaquinta and S. Hildebrandt, *Calculus of Variations I*, Springer (1996). 4. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 8, §8.2.4 (quasiconvexity and polyconvexity). 5. C. B. Morrey, *Multiple [Integrals](/page/Integral) in the Calculus of Variations*, Springer (1966).