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Partial Differential Equations Partial Differential Equations William G. Faris May 17, 1999 Contents 1 The Laplace equation 5 1.1 Gradient and divergence . 5 1.2 Equilibrium conservation laws . 6 1.3 Polar coordinates . 7 1.4 The mean value property . 8 1.5 Approximate delta functions . 10 1.6 Regularity . 12 1.7 Liouville's theorem . 12 1.8 The maximum principle . 13 1.9 Differentiating an integral . 14 1.10 The fundamental solution . 15 1.11 Energy . 17 2 The heat equation 19 2.1 Conservation laws . 19 2.2 The fundamental solution . 20 2.3 Approach to equilibrium . 21 2.4 The mean value property . 23 2.5 The maximum principle . 25 3 The wave equation 27 3.1 Mechanical conservation laws . 27 3.2 The d'Alembert solution . 29 3.3 The Kirchoff solution . 30 3.4 The method of descent . 33 3.5 Solution in odd dimension . 33 3.6 Conservation of energy . 35 4 Hamilton-Jacobi equation 37 4.1 Characteristics . 37 4.2 Hamiltonian-Jacobi equations . 39 4.3 The Hopf-Lax solution . 41 4.4 Weak solutions . 43 1 4.5 The Hopf-Cole transformation for the quadratic Hamilton-Jacobi equation . 45 4.6 Laplace's method . 46 5 Conservation laws 48 5.1 Scalar conservation laws . 48 5.2 Conservation in one space dimension . 49 5.3 The Lax-Oleinik solution . 49 5.4 Distributions . 50 5.5 Weak solutions . 52 5.6 Entropy . 54 5.7 Entropy increase . 55 5.8 The Hopf-Cole transformation for the Burgers equation . 57 6 The Fourier transform 59 6.1 L1 theory . 59 6.2 The Plancherel theorem and L2 theory . 62 6.3 L2 derivatives . 64 6.4 The Poisson equation . 65 6.5 The heat equation . 67 6.6 The Schr¨odingerequation . 67 6.7 Hyperbolic equations . 69 6.8 L2 Sobolev inequalities . 71 7 The Dirichlet Laplacian 73 7.1 Sobolev spaces for open subsets . 73 7.2 The Laplacian with dissipation . 73 7.3 Dirichlet's principle and the Riesz representation theorem . 74 7.4 Finite elements . 76 7.5 Equivalent norms on the Sobolev space . 76 7.6 The Laplacian without dissipation . 78 7.7 Positive elliptic operators . 79 8 Sobolev inequalities 80 8.1 Lp spaces . 80 8.2 Convolution and Lp ......................... 84 8.3 Sobolev spaces . 85 8.4 Dirichlet boundary conditions . 86 8.5 The Gagliardo-Nirenberg-Sobolev inequality . 87 8.6 The Poincar´einequality . 89 9 Spectral theory and evolution equations: Discrete spectrum 90 9.1 Separation of variables . 90 9.2 Spectral theorem for compact self-adjoint operators . 92 9.3 Hilbert-Schmidt operators . 95 9.4 Compact embedding of Hilbert spaces . 96 2 9.5 Positive quadratic forms . 97 9.6 Evolution equations . 99 10 Spectral theory and evolution equations: Continuous spectrum101 10.1 Separation of variables . 101 10.2 Continuity lemma for bounded self-adjoint operators . 103 10.3 Spectral theorem for bounded self-adjoint operators . 105 10.4 Positive quadratic forms . 108 10.5 Evolution equations . 109 10.6 The role of the Fourier transform . 111 11 Energy and equilibrium 112 11.1 Least squares solutions . 112 11.2 Bilinear forms . 113 11.3 Equilibrium . 115 11.4 Boundedness and compactness . 116 12 Semigroup theory and evolution equations 118 12.1 Exponentials . 118 12.2 Dissipative operators . 119 12.3 The Hille-Yosida theorem . 120 13 Compactness 124 13.1 Total boundedness . 124 13.2 Compactness . 125 13.3 The Ascoli-Arzel`atotal boundedness theorem . 126 13.4 The Rellich-Kondrachov embedding theorem . 127 13.5 Almost uniform convergence . 129 14 Weak ∗ compactness 131 14.1 Weak ∗ compactness in the standard Banach spaces . 131 14.2 Compactness and minimization . 132 14.3 The lower topology . 133 14.4 Comparison of topologies . 134 14.5 Weak ∗ topology . 135 14.6 Metrizability . 137 14.7 Convexity . 138 15 Variational methods for nonlinear problems 140 15.1 The Euler-Lagrange equation . 140 15.2 Coercivity . 141 15.3 The weak ∗ topology on the Sobolev space . 142 15.4 Convex functionals with a derivative bound . 143 15.5 Convex functionals . 143 15.6 Functionals convex in the gradient . 144 15.7 Functionals without convexity . 145 3 16 Fixed points 148 16.1 Banach's fixed point theorem . 148 16.2 The Schauder-Leray fixed point theorem . 148 16.3 Semilinear elliptic PDE . 150 4 Chapter 1 The Laplace equation 1.1 Gradient and divergence In these lectures we follow the notation suggested by Evans. If u is a scalar function, then Du is the gradient, the vector of partial derivatives. If J is a vector function, then DJ is the matrix of partial derivatives of the components of J. The trace of this matrix is div J = tr(DJ), the divergence of the vector field. (Similarly, the determinant of this matrix is the Jacobian.) The matrix of second derivatives of u is D2u, the Hessian matrix. This is a symmetric matrix. The trace of the Hessian matrix is the Laplacian: ∆u = tr(D2u) = div Du: (1.1) We recall some basic calculus ideas. Let J be a vector field with divergence div J. The divergence theorem says that for bounded open sets V with smooth boundary @V we have Z Z div J dy = J · ν dS; (1.2) V @V where ν is the unit normal vector pointing outward, and dS is the surface measure. Let s be a scalar function with gradient Ds. Then we have the product rule for differentiating in the form div(sJ) = Ds · J + s div J: (1.3) If we apply the divergence theorem to the left hand side of this equation, we get the fundamental integration by parts result: Z Z Z Ds · J dy + s div J dy = sJ · ν dS: (1.4) V V @V This will be used over and over again. 5 1.2 Equilibrium conservation laws We begin with the most classical of partial differential equations, the Laplace equation. This equation is linear of second order, and is both translation and rotation invariant. It describes equilibrium in space. We will see in this first chapter that even though the equation only involves second derivatives, every solution automatically has all derivatives. Now we proceed to the derivation of the Poisson and Laplace equations. Let J be a vector field on an open subset U of Rn. This is the current. Let f be a function on U. This is the source (the rate of production of some quantity). An equilibrium conservation law is an equation of the form Z Z J · ν dS = f dx: (1.5) @V V Here V is supposed to range over suitable bounded open subsets of U. The boundary @V of each V is supposed to be a smooth subset of U. This equation says that the amount of substance flowing out of the region V is equal to the rate of production. If we apply the divergence theorem, we obtain Z Z div J dx = f dx: (1.6) V V Since this is assumed to hold for all subregions V , we have the differential form of the equilibrium conservation law: div J = f: (1.7) Now assume that the current is proportional to the negative of the gradient of some scalar function u defined on U. Thus J = −Du: (1.8) We get the Poisson equation −4u = − div Du = f: (1.9) When there is equilibrium with no source, then this is the Laplace equation 4u = 0: (1.10) Even if one is interested in the Poisson equation, the Laplace equation is impor- tant, since the difference of two solutions of the Poisson equation is a solution of the Laplace equation. In the following we will usually think of the Poisson or Laplace equation being satisfied for a function u that is C2 on some open set U. In applications u could be temperature, density, or electric potential. The corresponding current J would be heat flux, diffusive current, or electric field. 6 The source f could be a source of heat, a source of diffusing particles, or an electric charge density. If the Laplace equation is satisfied in U, then we have Z J · ν dS = 0 (1.11) @V for every suitable subregion V of U. This says that there is no net flow into or out of the region V . 1.3 Polar coordinates It will sometime be convenient to calculate integrals in polar coordinates. Thus Z Z 1 Z f(x) dx = f(x) dSdr: (1.12) n R 0 @B(x0;r) Here dS represents surface measure on the n−1 dimensional sphere @B(x0; r) of radius r centered at x0. The total surface measure of the sphere is proportional to rn−1 and the proportionality constant will be taken so that it is by definition nα(n)rn−1. Thus, for example, nα(n) in dimensions n = 1; 2; 3 has the values 2, 2π, and 4π. In dimensions n = 1; 2; 3 these numbers represent the count of two points, the length of a unit circle, and the area of a unit sphere. 2 As an example, we can take f(x) = exp(−x ) and x0 = 0. Then Z Z 1 exp(−x2) dx = nα(n) exp(−r2)rn−1 dr: (1.13) 0 Here the total surface measure of the ball is defined to be nα(n)rn−1. We can also.
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