# [Calculus of Variations](/page/Calculus%20of%20Variations)
Many fundamental partial differential equations (PDEs) arise not from local force balances, but from global minimization principles. In physics, systems often settle into states that minimize potential energy or action. Geometrically, one seeks surfaces that minimize area given a [boundary](/page/Boundary). The **Calculus of Variations** provides the framework for finding [functions](/page/Function) that optimize a specific functional (usually an [integral](/page/Integral)). This approach converts the problem of minimizing an infinite-dimensional energy landscape into solving a corresponding differential equation, known as the Euler-Lagrange equation.
## Formal Definition
We define the general variational problem. We seek a function $u$ that minimizes an integral functional $I[u]$ over a class of admissible functions.
[definition: Variational Functional]
Let $n \in \mathbb{N}$ and let $U \subset \mathbb{R}^n$ be a bounded [open set](/page/Open%20Set).
Let $L: \mathbb{R}^n \times \mathbb{R} \times \bar{U} \to \mathbb{R}$ be a smooth function, called the **Lagrangian**. We denote the arguments of $L$ by $L(p, z, x)$, where $p \in \mathbb{R}^n$ corresponds to the gradient $\nabla u$, $z \in \mathbb{R}$ corresponds to the value $u(x)$, and $x \in \bar{U}$ is the position.
The **energy functional** $I$ is the map:
\begin{align*}
I: \mathcal{A} &\to \mathbb{R} \\
w &\mapsto \int_U L(\nabla w(x), w(x), x) \, dx
\end{align*}
where $\mathcal{A}$ is the **admissible class** of functions, typically defined by boundary conditions:
\begin{align*}
\mathcal{A} = \{ w \in C^2(\bar{U}) : w = g \text{ on } \partial U \}
\end{align*}
for a given boundary function $g$.
[/definition]
---
## Examples
The choice of the Lagrangian $L$ determines the physical or geometric nature of the problem.
[example: Dirichlet Principle]
Let $L(p, z, x) = \frac{1}{2} |p|^2$. The functional is the **Dirichlet energy**:
\begin{align*}
I[w] = \frac{1}{2} \int_U |\nabla w|^2 \, dx.
\end{align*}
Minimizers of this energy satisfy the Laplace equation $\Delta u = 0$.
[/example]
[example: Generalized Minimal Surfaces]
Let $L(p, z, x) = \sqrt{1 + |p|^2}$. The functional represents the **surface area** of the graph of $w$:
\begin{align*}
I[w] = \int_U \sqrt{1 + |\nabla w|^2} \, dx.
\end{align*}
Minimizers of this functional satisfy the Minimal Surface Equation.
[/example]
---
## Key Results
The central results in the calculus of variations connect the minimizers of $I[w]$ to solutions of specific PDEs and establish conditions under which such minimizers exist.
### 1. The Euler-Lagrange Equation
If a smooth minimizer exists, it must satisfy a specific PDE derived from the first variation of the functional.
[theorem: Euler Lagrange]
Let $u \in \mathcal{A}$ satisfy $I[u] = \min_{w \in \mathcal{A}} I[w]$.
If $u$ is smooth (specifically $u \in C^2(U)$), then $u$ satisfies the **Euler-Lagrange equation**:
\begin{align*}
-\sum_{i=1}^n \partial_{x_i} \left( L_{p_i}(\nabla u, u, x) \right) + L_z(\nabla u, u, x) = 0 \quad \text{for } x \in U.
\end{align*}
[/theorem]
### 2. The Second Variation
Just as the second derivative of a function must be non-negative at a minimum, the second variation of the functional imposes convexity conditions on the Lagrangian.
[theorem: Legendre Hadamard]
Let $u \in \mathcal{A}$ be a minimizer of $I[\cdot]$. Then the Lagrangian satisfies the **Legendre-Hadamard condition** (ellipticity condition) along the solution $u$. Specifically:
\begin{align*}
\sum_{i=1}^n \sum_{j=1}^n L_{p_i p_j}(\nabla u(x), u(x), x) \xi_i \xi_j \ge 0
\end{align*}
for all $\xi \in \mathbb{R}^n$ and all $x \in U$.
[/theorem]
### 3. Existence of Minimizers
To prove that a minimizer actually exists (Direct Method), we require two structural conditions on $L$: **Coercivity** (to prevent the minimizing [sequence](/page/Sequence) from escaping to infinity) and **Convexity** (to ensure lower semicontinuity).
[theorem: Direct Method Existence]
Assume the Lagrangian $L(p, z, x)$ satisfies:
1. **Convexity:** The map $p \mapsto L(p, z, x)$ is convex for all $z, x$.
2. **Coercivity:** There exist constants $\alpha > 0, \beta \ge 0$ such that $L(p, z, x) \ge \alpha |p|^q - \beta$ for some $q > 1$.
Then there exists a function $u$ in the [Sobolev space](/page/Sobolev%20Space) $W^{1,q}(U)$ such that:
\begin{align*}
I[u] = \inf_{w \in \mathcal{A}} I[w]
\end{align*}
where $\mathcal{A} = \{ w \in W^{1,q}(U) : w = g \text{ on } \partial U \}$.
[/theorem]
---
## Problems
[problem]
Let $n=1$ and $U = (0, 1)$. Consider the functional:
\begin{align*}
I[w] = \int_0^1 \left( \frac{1}{2} (w')^2 + \frac{1}{2} w^2 + w \right) \, dx.
\end{align*}
Find the Euler-Lagrange equation associated with this functional.
[/problem]
[solution]
The Lagrangian is $L(p, z, x) = \frac{1}{2}p^2 + \frac{1}{2}z^2 + z$.
We compute the partial [derivatives](/page/Derivative):
\begin{align*}
L_p = p, \quad L_z = z + 1.
\end{align*}
The Euler-Lagrange equation is:
\begin{align*}
-\frac{d}{dx} (L_p) + L_z &= 0 \\
-\frac{d}{dx} (u') + (u + 1) &= 0 \\
-u'' + u &= -1.
\end{align*}
Thus, the minimizer satisfies the ODE $u'' - u = 1$.
[/solution]
---
## References
1. Evans, L. C., *Partial Differential Equations* (2010).
