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The Direct Method Early Research Directions: Calculus of Variations The Direct Method Andrejs Treibergs University of Utah Friday, March 19, 2010 2. ERD Lectures on the Calculus of Variations Notes for the first of three lectures on Calculus of Variations. 1 Andrejs Treibergs, “The Direct Method,” March 19, 2010, 2 Predrag Krtolica, “Falling Dominoes,” April 2, 2010, 3 Andrej Cherkaev, “ ‘Solving’ Problems that Have No Solutions,” April 9, 2010. The URL for these Beamer Slides: “The Direct Method” http://www.math.utah.edu/~treiberg/DirectMethodSlides.pdf 3. References Richard Courant, Dirichlet’s Principle, Conformal Mapping and Minimal Surfaces, Dover (2005); origianlly pub. in series: Pure and Applied Mathematics 3, Interscience Publishers, Inc. (1950). Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Math. 19, American Mathematical Society (1998). Mario Giaquinta, Multiple Integrals in the Calculus of Variations, Annals of Math. Studies 105, Princeton University Press, 1983. Robert McOwen, Partial Differential Equations: Methods and Applications 2nd ed., Prentice Hall (2003). Richard Schoen & S. T. Yau, Lectures on Differential Geometry, Conf. Proc. & Lecture Notes in Geometry and Topology 1, International Press (1994). 4. Outline. Euler Lagrange Equations for Constrained Problems Solution of Minimization Problems Dirichlet’s principle and the Yamabe Problem. Outline of the Direct Method. Examples of failure of the Direct Method. Solve in a special case: Poisson’s Minimization Problem. Cast the minimization problem in Hilbert Space. Coercivity of the functional. Continuity of the functional. Differentiability of the functional. Convexity of the functional and uniqueness. Regularity 5. First Example of a Variational Problem: Shortest Length. To find the shortest curve γ(t) = (t, u(t)) in the Euclidean plane from (a, y1) to (b, y2) we seek the function u ∈ A, the admissible set 1 A = {w ∈ C ([a, b]) : w(a) = y1, w(b) = y2} that minimizes the length integral Z q L(u) = 1 +u ˙ 2(t) dt. Ω We saw that the minimizing curve would satisfy the Euler Equation d u˙ √ = 0 dt 1 +u ˙ 2 whose solution is u(t) = c1t + c2, a straight line. 6. Variational Problems with Constraints. w(a) = y1, 0 1 A = w ∈ C : w(b) = y2, R Ω u(t)dt = j0 This is the Isoperimetric Problem. We’ll see that the corresponding differential equations, the If we assume also that the curve Euler-Lagrange equations say that the curve has constant curvature, from (a, y1) to (b, y2) enclose a thus consists of an arc of a circle. fixed area j0 between a ≤ x ≤ b and between γ and the x-axis, then the But this may be unsolvable! For a admissible set becomes curves with given choice of endpoints and j0, the given area and now seek u ∈ A0 there may not be a circular arc that minimizes the length integral γ(t) = (t, u(t)) that connects the R p 2 endpoints and encloses an area j0. L(u) = Ω 1 +u ˙ (t) dt. 7. Formulation of the Euler Lagrange Equations. n More generally, let Ω ⊂ R be a bounded domain with smooth boundary, f (x, u, p), g(x, u, p) ∈ C2(Ω × R × Rn) and φ ∈ C1(Ω). Suppose that we seek to find u ∈ A = {x ∈ C1(Ω) : w = φ on ∂Ω} that achieves the extremum of the integral Z I (u) = f (x, u(x), Du(x)) dx Ω subject to the constraint that Z J(u) = g(x, u(x), Du(x)) dx = j0 Ω takes a prescribed value. 8. Formulation of the Euler Lagrange Equations.- 1 Choose two functions ηi , η2 ∈ C (Ω) such that η1(z) = η2(z) = 0 for all z ∈ ∂Ω. Then, assuming that u(x) is a solution of the constrained optimization problem, consider the two-parameter variation U(x, ε1, ε2) = u(x) + ε1η1(x) + ε2η2(x) which satisfies U = φ on ∂Ω so U ∈ A. Substitute U Z I (ε1, ε2) = f (x, U, DU) dx Ω Z J(ε1, ε2) = g(x, U, DU) dx Ω 2 I is extremized in R when ε1 = ε2 = 0 subject to the constraint J(ε1, ε2) = j0. 9. Formulation of the Euler Lagrange Equations.- - Using Lagrange Multipliers, there is a constant λ so that solution is the critical point of the Lagrange function Z L(ε1, ε2) = I (ε1, ε2) + λJ(ε1, ε2) = h(x, U, DU) dx Ω where h(x, U, DU) = f (x, U, DU) + λg(x, U, DU). The extremum satisfies ∂L ∂L = = 0 when ε1 = ε2 = 0. ∂ε1 ∂ε2 But for i = 1, 2, Z n 2 Z n ∂L ∂h ∂U X ∂h ∂ U ∂h X ∂h ∂ηi = + = ηi + ∂εi ∂U ∂εi ∂pj ∂εi ∂xj ∂U ∂pj ∂xj Ω j=1 Ω j=1 10. Formulation of the Euler Lagrange Equations.- - - At the critical point when εi = 0 for i = 1, 2 so U = u, Z ∂L ∂h = ηi + Dph • Dx ηi dx = 0, ∂ε ∂u i η1=ε2=0 Ω which is the weak form of the Euler Lagrange equations. Integrating the second term by parts (assuming it is OK to do so), using ηi = 0 on the boundary Z ∂h ηi − div (Dph) dt = 0. Ω ∂u As both ηi are arbitrary, we obtain the Euler-Lagrange equations ∂h − div (D h) = 0. ∂u p 11. Application to the Classical Isoperimetric Problem. In this case, p h = f + λg = 1 +u ˙ 2 + λu. Thus ∂h d ∂h 0 = − ∂u dt ∂u˙ d u˙ = λ − √ dt 1 +u ˙ 2 u¨ = λ − (1 +u ˙ 2)3/2 = λ − κ where κ is the curvature of the curve γ. Thus solutions of the E-L equations are curves of constant curvature which are arcs of circles. 12. Riemann assumes the Dirichlet Principle. The great strides of Riemann’s Thesis (1851) and memoir on Abelian Functions (1857) relied on the statement that there exists a minimizer of an electrostatics problem which he did not prove! Riemann had learned it from Dirichlet’s lectures. On a surface, which he thought of as a thin conducting sheet, he imagined the stationary electric current generated by connecting arbitrary points of the surface with the poles of an electric battery. The potential of such a current will be the solution of a boundary value problem. The corresponding variational problem is to find among all possible flows the one that produces the least quantity of heat. (Dirichlet’s Principle) Let G ⊂ R2 (or in a smooth surface) be a compact domain and φ ∈ C(∂G). Then there is u ∈ C1(G) ∩ C(G) that satisfies u = φ on ∂G and minimizes the Dirichlet Integral Z D[u] = |Du|2 dA. G Moreover, ∆u = 0 on G. 13. Weierstrass objects. Hilbert proves it. Because D[u] ≥ 0 it was assumed that minimizers exist. Weierstrass found the flaw in the argument and published his objections in 1869. Dirichlet’s Principle became the stimulus of much work. Finally, Hilbert proved the Dirichlet Principle in 1900 assuming appropriate smoothness. 14. Hilbert’s 19th and 20th problems. Hilbert challenged mathematicians to solve what he considered to be the most important problems of the 20th century in his address to the International Congress of Mathematicians in Paris, 1900. 20th “Has not every variational problem a solution, provided certain assumptions regarding the given boundary conditions are satisfied, and provided also that if need be that the notion of solution be suitably extended?” 19th “Are the solutions of regular problems in the Calculus of Variations always necessarily analytic?” These questions have stimulated enormous effort and now things are much better understood. The existence and regularity theory for elliptic PDE’s and variational problems is one of the greatest achievements of 20th century mathematics. 15. Variational Problem from Geometry: Yamabe Problem. In 1960, Yamabe tried to prove the Poincar´eConjecture that every smooth closed orientable simply connected three manifold M is the three sphere using a variational method. He tried to find an Einstein metric (Ric(g) = cg) where Ric is the Ricci curvature and c is a constant, which would have to have constant sectional curvature in three dimensions. Consider the functional on the space M of all smooth Riemannian metrics on M. For g ∈ M, R M Rg dµg Q(g) = n−2 R n M dµg where Rg = scalar curvature and µg = volume form for g. The Einstein Metrics are maxi-min critical points. g0 is Einstein if it is critical so Q(g0) = sup inf Q(g) g1∈M g∈Cg1 ∞ Cg1 = {ρg1 : ρ ∈ C (M): ρ > 0} are all metrics conformal to g1. 16. Yamabe Problem-. The “mini” part for n ≥ 3 is called the Yamabe Problem. For n = 2 this is equivalent to the Uniformization Theorem: every closed surface carries a constant curvature metric. Theorem (Yamabe Problem.) n Let (M , g0) be an n ≥ 3 dimensional smooth compact Riemannian manifold without boundary. Then there is ρ ∈ C∞(M) with ρ > 0 so that Q(ρg0) = inf Q(g). g∈Cg0 Moreover, g˜ = ρg0 has constant scalar curvature Rg˜. The Yamabe Problem was resolved in steps taken by Yamabe (1960), Trudinger (1974), Aubin (1976) and Schoen (1984). The Poincar´eConjecture has since been practically resolved using Ricci Flow by Hamilton and Perelman (2004). 17. Minimization Problem. We illustrate with a simple minimization problem. Let Ω ⊂ Rn be a connected bounded domain with smooth boundary. Let us assume that f (x, u, p) ∈ C1(Ω × R × Rn). For u ∈ C1(Ω) we define the functional Z F(u) = f x, u(x), Du(x) dx Ω If φ ∈ C1(Ω), then we define the set of admissible functions to be A = u ∈ C1(Ω) : u = φ on ∂Ω .
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