7 Calculus of Variations Ref: Evans, Sections 8.1, 8.2, 8.4 7.1 Motivation The calculus of variations is a technique in which a partial differential equation can be reformulated as a minimization problem. In the previous section, we saw an example of this technique. Letting vi denote the eigenfunctions of ½ ¡∆v = ¸v x 2 Ω (¤) v = 0 x 2 @Ω; and defining the class of functions 2 Yn = fw 2 C (Ω); w 6´ 0; w = 0 for x 2 @Ω; < w; vi >= 0; i = 1; : : : ; n ¡ 1g; we saw that if u 2 Yn is a minimizer of the functional 2 jjrwjjL2(Ω) I(w) = 2 jjwjjL2(Ω) over Yn, then u is an eigenfunction of (*) with corresponding eigenvalue 2 jjrujjL2(Ω) m ´ 2 : jjujjL2(Ω) Further, we showed that m is in fact the nth eigenvalue of (*). In other words to find a solution of an eigenvalue problem, we reformulated the problem in terms of minimizing a certain functional. Proving the existence of an eigenfunction is now equivalent to proving the existence of a minimizer of I over the class Yn. Proving the existence of a minimizer requires more sophisticated functional analysis. We will return to this idea later. The example above could be reformulated equivalently to say that we are trying to minimize the functional e 2 I(w) = jjrwjjL2(Ω) 2 over all functions w 2 Yn such that jjwjjL2(Ω) = 1. In particular, if v is in Yn, then the 2 normalized function ve ´ v=jjvjjL2(Ω) (which has the same Rayleigh quotient as v) is in Yn, 2 and, of course, jjvejjL2(Ω) = 1. Therefore, minimizing I over functions w 2 Yn is equivalent e 2 to minimizing I over functions w 2 Yn subject to the constraint jjwjjL(Ω) = 1. This type of minimization problem is called a constrained minimization problem. We begin by considering a simple example of how a partial differential equation can be rewritten as a minimizer of a certain functional over a certain class of admissible functions. 1 7.2 Dirichlet’s Principle In this section, we show that the solution of Laplace’s equation can be rewritten as a mini- mization problem. Let A ´ fw 2 C2(Ω); w = g for x 2 @Ωg: Let Z 1 I(w) ´ jrwj2 dx: 2 Ω Theorem 1. (Dirichlet’s Principle) Let Ω be an open, bounded subset of Rn. Consider Laplace’s equation on Ω with Dirichlet boundary conditions, ½ ∆u = 0 x 2 Ω (¤) u = g x 2 @Ω: The function u 2 A is a solution of (*) if and only if I(u) = min I(w): w2A Proof. First, we suppose u is a solution of (*). We need to show that I(u) · I(w) for all w 2 A. Let w 2 A. Then Z 0 = ∆u(u ¡ w) dx ΩZ Z = ¡ jruj2 dx + ru ¢ rw dx ZΩ ΩZ 1 ·¡ jruj2 dx + [jruj2 + jrwj2] dx 2 ΩZ ΩZ 1 1 = ¡ jruj2 dx + jrwj2 dx: 2 Ω 2 Ω Therefore, Z Z jruj2 dx · jrwj2 dx: Ω Ω But w is an arbitrary function in A. Therefore, I(u) = min I(w): w2A Next, suppose u minimizes I over all w 2 A. We need to show that u is a solution of (*). Let v be a C2 function such that v ´ 0 for x 2 @Ω. Therefore, for all ², u + ²v 2 A. Now let i(²) ´ I(u + ²v): By assumption, u is a minimizer of I. Therefore, i must have a minimum at ² = 0, and, therefore, i0(0) = 0. Now i(²) = I(u + ²v) Z = jr(u + ²v)j2 dx ZΩ = [jruj2 + 2²ru ¢ rv + ²2jrvj2] dx; Ω 2 implies Z i0(²) = [ru ¢ rv] + 2²jrvj2 dx: Ω Therefore, Z i0(0) = ru ¢ rv dx ΩZ Z @u = ¡ (∆u)v dx + v dS(x) @º ZΩ @Ω = ¡ (∆u)v dx: Ω 0 Now i (0) = 0 implies Z (∆u)v dx = 0: Ω Since this is true for all v 2 C2(Ω) such that v = 0 for x 2 @Ω, we can conclude that ∆u = 0, as claimed. 7.3 Euler-Lagrange Equations Laplace’s equation is an example of a class of partial differential equations known as Euler- Lagrange equations. These equations are defined as follows. Let Ω be an open, bounded subset of Rn. Let L be a smooth function such that L : Rn £ R £ Ω ! R: We will write L = L(p; z; x) where p 2 Rn, z 2 R and x 2 Ω. Associated with a function L, we define the Euler-Lagrange equation Xn ¡ (Lpi (ru; u; x))xi + Lz(ru; u; x) = 0: i=1 The function L is known as the Lagrangian. Example 2. Let 1 L(p; z; x) = jpj2: 2 The associated Euler-Lagrange equation is just Laplace’s equation ∆u = 0: ¦ Example 3. Let 1 Xn L(p; z; x) = aij(x)p p ¡ zf(x) 2 i j i;j=1 3 where aij = aji. The associated Euler-Lagrange equation is Xn ij ¡ (a uxj )xi = f; i;j=1 a generalization of Poisson’s equation. ¦ Recall that Dirichlet’s principle stated that a solution of ½ ∆u = 0 x 2 Ω ½ Rn u = g x 2 @Ω is a minimizer of Z 1 I(w) = jrwj2 dx 2 Ω over C2 functions which satisfy the boundary condition. In other words, a harmonic function on Ω is a minimizer of Z I(w) = L(rw; w; x) dx Ω 1 2 where L is the associated Lagrangian given by L(p; z; x) = 2 jpj . For a given Lagrangian L define Z IL(w) = L(rw; w; x) dx: Ω We will now show that if u is a minimizer of IL(w) over an admissible class of functions A, then u is a solution of the associated Euler-Lagrange equation Xn ¡ (Lpi (ru; u; x))xi + Lz(ru; u; x) = 0: i=1 As in the proof of Dirichlet’s principle, suppose u is a minimizer of Z IL(w) = L(rw; w; x) dx Ω over an admissible class of functions A. Let v 2 C1(Ω) such that v has compact support 1 within Ω. We denote this space of functions by Cc (Ω). Define i(²) = I(u + ²v): If u is a minimizer of I, then i0(0) = 0. i(²) = I(u + ²v) Z = L(ru + ²rv; u + ²v; x) dx: Ω 4 Therefore, Z Xn 0 i (²) = Lpi (ru + ²rv; u + ²v; x)vxi + Lz(ru + ²rv; u + ²v; x)v dx: Ω i=1 Now i0(0) = 0 implies Z Xn 0 0 = i (0) = Lpi (ru; u; x)vxi + Lz(ru; u; x)v dx: Ω i=1 Integrating by parts and using the fact that v = 0 for x 2 @Ω, we conclude that " # Z Xn ¡ (Lpi (ru; u; x))xi + Lz(ru; u; x) v dx = 0: Ω i=1 1 Since this is true for all v 2 Cc (Ω), we conclude that u is a solution of the Euler-Lagrange equation associated with the Lagrangian L. Consequently, to solve Euler-Lagrange equa- tions, we can reformulate these partial differential equations as minimization problems of the functionals Z IL(w) = L(rw; w; x) dx: Ω Above we showed how solving certain partial differential equations could be rewritten as minimization problems. Sometimes, however, the minimization problem is the physical problem which we are interested in solving. Example 4. (Minimal Surfaces) Let w :Ω ! R. The surface area of the graph of w is given by Z I(w) = (1 + jrwj2)1=2 dx: Ω The problem is to look for the minimal surface, the surface with the least surface area, which satisfies the boundary condition w = g for x 2 @Ω. Alternatively, this minimization problem can be written as a partial differential equation. In particular, the Lagrangian associated with I is L(p; z; x) = (1 + jpj2)1=2: The associated Euler-Lagrange equation is µ ¶ Xn u xi = 0: (1 + jruj2)1=2 i=1 xi This equation is known as the minimal surface equation. ¦ 7.4 Existence of Minimizers We now discuss the existence of a minimizer of Z I(w) = L(rw; w; x) dx Ω over some admissible class of functions A. We will discuss existence under two assumptions on the Lagrangian L: convexity and coercivity. We discuss these issues now. 5 7.4.1 Convexity Assume u is a minimizer of Z I(w) = L(rw; w; x) dx: Ω 1 We discussed earlier that if u is a minimizer of I, then for any v 2 Cc (Ω), the function i(²) = I(u + ²v) has a local minimum at ² = 0, and, therefore, i0(0) = 0. In addition, if i has a minimum at ² = 0, then i00(0) ¸ 0. We now calculate i00(0) explicitly to see what this implies about I and L. By a straightforward calculation, we see that " Z Xn 00 i (²) = Lpipj (ru + ²rv; u + ²v; x)vxi vxj Ω i;j=1 # Xn 2 +2 Lpiz(ru + ²rv; u + ²v; x)vxi v + Lzz(ru + ²rv; u + ²v; x)v dx: i=1 Therefore, we conclude that " Z Xn Xn 00 0 · i (0) = Lpipj (ru; u; x)vxi vxj + 2 Lpiz(ru; u; x)vxi v Ω i;j=1 i=1 (7.1) 2¤ +Lzz(ru; u; x)v dx 1 for all v 2 Cc (Ω). By an approximation argument, we can show that (7.1) is also valid for the function µ ¶ x ¢ » v(x) = ±½ ³(x) ± n 1 where ± > 0, » 2 R , ³ 2 Cc (Ω) and ½ : R ! R is the periodic ”zig-zag” function 8 1 <> x 0 · x · ½(x) = 2 > 1 : 1 ¡ x · x · 1 2 and ½(x + 1) = ½(x) elsewhere.