Partial Differential Equations

Partial Differential Equations Lecture Notes Erich Miersemann Department of Mathematics Leipzig University Version October, 2012 Contents Introduction 1.1 Examples 1.2 Equations from variational problems 1.2.1 Ordinary differential equations 1.2.2 Partial differential equations 1.3 Exercises Equations of first order 2.1 Linear equations 2.2 Quasilinear equations 2.2.1 A linearization method 2.2.2 Initial value problem of Cauchy 2.3 Nonlinear equations in two variables 2.3.1 Initial value problem of Cauchy 2.4 Nonlinear equations in Rn 2.5 Hamilton-Jacobi theory 2.6 Exercises 11 15 15 16 22 25 25 31 32 33 40 48 51 53 59 Classification 3.1 Linear equations of second order 3.1.1 Normal form in two variables 3.2 Quasilinear equations of second order 3.2.1 Quasilinear elliptic equations 3.3 Systems of first order 3.3.1 Examples 3.4 Systems of second order 3.4.1 Examples 3.5 Theorem of Cauchy-Kovalevskaya 3.5.1 Appendix: Real analytic functions 63 63 69 73 73 74 76 82 83 84 90 CONTENTS 3.6 Exercises 101 Hyperbolic equations 4.1 One-dimensional wave equation 4.2 Higher dimensions 4.2.1 Case n=3 4.2.2 Case n = 4.3 Inhomogeneous equation 4.4 A method of Riemann 4.5 Initial-boundary value problems 4.5.1 Oscillation of a string 4.5.2 Oscillation of a membrane 4.5.3 Inhomogeneous wave equations 4.6 Exercises 107 107 109 112 115 117 120 125 125 128 131 136 Fourier transform 141 5.1 Definition, properties 141 5.1.1 Pseudodifferential operators 146 5.2 Exercises 149 Parabolic equations 6.1 Poisson’s formula 6.2 Inhomogeneous heat equation 6.3 Maximum principle 6.4 Initial-boundary value problem 6.4.1 Fourier’s method 6.4.2 Uniqueness 6.5 Black-Scholes equation 6.6 Exercises Elliptic equations of second order 7.1 Fundamental solution 7.2 Representation formula 7.2.1 Conclusions from the representation formula 7.3 Boundary value problems 7.3.1 Dirichlet problem 7.3.2 Neumann problem 7.3.3 Mixed boundary value problem 7.4 Green’s function for 7.4.1 Green’s function for a ball 151 152 155 156 162 162 164 164 170 175 175 177 179 181 181 182 183 183 186 CONTENTS 7.5 7.6 7.4.2 Green’s function and conformal mapping 190 Inhomogeneous equation 190 Exercises 195 CONTENTS Preface These lecture notes are intented as a straightforward introduction to partial differential equations which can serve as a textbook for undergraduate and beginning graduate students For additional reading we recommend following books: W I Smirnov [21], I G Petrowski [17], P R Garabedian [8], W A Strauss [23], F John [10], L C Evans [5] and R Courant and D Hilbert[4] and D Gilbarg and N S Trudinger [9] Some material of these lecture notes was taken from some of these books CONTENTS Chapter Introduction Ordinary and partial differential equations occur in many applications An ordinary differential equation is a special case of a partial differential equation but the behaviour of solutions is quite different in general It is much more complicated in the case of partial differential equations caused by the fact that the functions for which we are looking at are functions of more than one independent variable Equation F (x, y(x), y (x), , y (n) ) = is an ordinary differential equation of n-th order for the unknown function y(x), where F is given An important problem for ordinary differential equations is the initial value problem y (x) = f (x, y(x)) y(x0 ) = y0 , where f is a given real function of two variables x, y and x0 , y0 are given real numbers Picard-Lindel¨ of Theorem Suppose (i) f (x, y) is continuous in a rectangle Q = {(x, y) ∈ R2 : |x − x0 | < a, |y − y0 | < b} (ii) There is a constant K such that |f (x, y)| ≤ K for all (x, y) ∈ Q (ii) Lipschitz condition: There is a constant L such that |f (x, y2 ) − f (x, y1 )| ≤ L|y2 − y1 | 10 CHAPTER INTRODUCTION y y0 x0 x Figure 1.1: Initial value problem for all (x, y1 ), (x, y2 ) Then there exists a unique solution y ∈ C (x0 −α, x0 +α) of the above initial value problem, where α = min(b/K, a) The linear ordinary differential equation y (n) + an−1 (x)y (n−1) + a1 (x)y + a0 (x)y = 0, where aj are continuous functions, has exactly n linearly independent solutions In contrast to this property the partial differential uxx +uyy = in R2 has infinitely many linearly independent solutions in the linear space C (R2 ) The ordinary differential equation of second order y (x) = f (x, y(x), y (x)) has in general a family of solutions with two free parameters Thus, it is naturally to consider the associated initial value problem y (x) = f (x, y(x), y (x)) y(x0 ) = y0 , y (x0 ) = y1 , where y0 and y1 are given, or to consider the boundary value problem y (x) = f (x, y(x), y (x)) y(x0 ) = y0 , y(x1 ) = y1 Initial and boundary value problems play an important role also in the theory of partial differential equations A partial differential equation for 191 7.5 INHOMOGENEOUS EQUATION where f is given We need the following lemma concerning volume potentials We assume that Ω is bounded and sufficiently regular such that all the following integrals exist See [6] for generalizations concerning these assumptions Let for x ∈ Rn , n ≥ 3, f (y) V (x) = Ω dy |x − y|n−2 and set in the two-dimensional case f (y) ln V (x) = Ω |x − y| dy We recall that ωn = |∂B1 (0)| Lemma (i) Assume f ∈ C(Ω) Then V ∈ C (Rn ) and Vxi (x) = Vxi (x) = ∂ ∂xi ∂ f (y) ∂xi f (y) Ω Ω |x − y|n−2 ln |x − y| dy, if n ≥ 3, dy if n = (ii) If f ∈ C (Ω), then V ∈ C (Ω) and V V = −(n − 2)ωn f (x), x ∈ Ω, n ≥ = −2πf (x), x ∈ Ω, n = Proof To simplify the presentation, we consider the case n = (i) The first assertion follows since we can change differentiation with integration since the differentiate integrand is weakly singular, see an exercise (ii) We will differentiate at x ∈ Ω Let Bρ be a fixed ball such that x ∈ Bρ , ρ sufficiently small such that Bρ ⊂ Ω Then, according to (i) and since we have the identity ∂ ∂xi |x − y| |x − y| =− =− ∂ ∂yi |x − y| which implies that f (y) ∂ ∂xi ∂ ∂yi f (y) |x − y| + fyi (y) , |x − y| 192 CHAPTER ELLIPTIC EQUATIONS OF SECOND ORDER we obtain Vxi (x) = ∂ ∂xi dy |x − y| ∂ f (y) dy + ∂xi |x − y| f (y) Ω = Ω\Bρ = f (y) Ω\Bρ − + Bρ = Bρ f (y) Bρ ∂ ∂xi |x − y| dy dy |x − y| |x − y| ∂ ∂xi fyi (y) + |x − y| ∂ ∂yi f (y) Ω\Bρ ∂ ∂xi f (y) + fyi (y) |x − y| dy dy dy − |x − y| f (y) ∂Bρ ni dSy , |x − y| where n is the exterior unit normal at ∂Bρ It follows that the first and second integral is in C (Ω) The second integral is also in C (Ω) according to (i) and since f ∈ C (Ω) by assumption Because of x (|x − y|−1 ) = 0, x = y, it follows n V = Bρ i=1 fyi (y) ∂ ∂xi n − f (y) ∂Bρ i=1 |x − y| ∂ ∂xi |x − y| dy ni dSy Now we choose for Bρ a ball with the center at x, then V = I1 + I2 , where n I1 = Bρ (x) i=1 I2 = − fyi (y) f (y) ∂Bρ (x) yi − xi dy |x − y|3 dSy ρ2 We recall that n · (y − x) = ρ if y ∈ ∂Bρ (x) It is I1 = O(ρ) as ρ → and for I2 we obtain from the mean value theorem of the integral calculus that 193 7.5 INHOMOGENEOUS EQUATION for a y ∈ ∂Bρ (x) I2 = − f (y) ρ2 dSy ∂Bρ (x) = −ωn f (y), which implies that limρ→0 I2 = −ωn f (x) ✷ In the following we assume that Green’s function exists for the domain Ω, which is the case if Ω is a ball Theorem 7.3 Assume f ∈ C (Ω) ∩ C(Ω) Then u(x) = G(x, y)f (y) dy Ω is the solution of the inhomogeneous problem (7.14), (7.15) Proof For simplicity of the presentation let n = We will show that u(x) := G(x, y)f (y) dy Ω is a solution of (7.4), (7.5) Since G(x, y) = + φ(x, y), 4π|x − y| where φ is a potential function with respect to x or y, we obtain from the above lemma that 1 f (y) dy + 4π |x − y| Ω = −f (x), u = x φ(x, y)f (y) dy Ω where x ∈ Ω It remains to show that u achieves its boundary values That is, for fixed x0 ∈ ∂Ω we will prove that lim x→x0 , x∈Ω u(x) = Set u(x) = I1 + I2 , 194 CHAPTER ELLIPTIC EQUATIONS OF SECOND ORDER where G(x, y)f (y) dy, I1 (x) = Ω\Bρ (x0 ) I2 (x) = G(x, y)f (y) dy Ω∩Bρ (x0 ) Let M = maxΩ |f (x)| Since G(x, y) = 1 + φ(x, y), 4π |x − y| we obtain, if x ∈ Bρ (x0 ) ∩ Ω, |I2 | ≤ ≤ M 4π M 4π dy + O(ρ2 ) |x − y| Ω∩Bρ (x0 ) dy + O(ρ2 ) B2ρ(x) |x − y| = O(ρ2 ) as ρ → Consequently for given |I2 | < there is a ρ0 = ρ0 ( ) > such that for all < ρ ≤ ρ0 For each fixed ρ, < ρ ≤ ρ0 , we have lim x→x0 , x∈Ω I1 (x) = since G(x0 , y) = if y ∈ Ω \ Bρ (x0 ) and G(x, y) is uniformly continuous in x ∈ Bρ/2 (x0 ) ∩ Ω and y ∈ Ω \ Bρ (x0 ), see Figure 7.8 ✷ Remark For the proof of (ii) in the above lemma it is sufficient to assume that f is H¨older continuous More precisely, let f ∈ C λ (Ω), < λ < 1, then V ∈ C 2,λ (Ω), see for example [9] 195 7.6 EXERCISES x0 ρ y x Figure 7.8: Proof of Theorem 7.3 7.6 Exercises , y ∈ Ω Show that Let γ(x, y) be a fundamental solution to − γ(x, y) Ω Φ(x) dx = Φ(y) for all Φ ∈ C02 (Ω) Hint: See the proof of the representation formula Show that |x|−1 sin(k|x|) is a solution of the Helmholtz equation u + k u = in Rn \ {0} Assume u ∈ C (Ω), Ω bounded and sufficiently regular, is a solution of u = u3 in Ω u = on ∂Ω Show that u = in Ω Let Ωα = {x ∈ R2 : x1 > 0, < x2 < x1 tan α}, < α ≤ π Show that π π u(x) = r α k sin kθ α is a harmonic function in Ωα satisfying u = on ∂Ωα , provided k is an integer Here (r, θ) are polar coordinates with the center at (0, 0) 196 CHAPTER ELLIPTIC EQUATIONS OF SECOND ORDER Let u ∈ C (Ω) be a solution of u = on the quadrangle Ω = (0, 1) × (0, 1) satisfying the boundary conditions u(0, y) = u(1, y) = for all y ∈ [0, 1] and uy (x, 0) = uy (x, 1) = for all x ∈ [0, 1] Prove that u ≡ in Ω Let u ∈ C (Rn ) be a solution of i e., Rn u2 (x) dx < ∞ Show that u ≡ in Rn u = in Rn satisfying u ∈ L2 (Rn ), Hint: Prove BR (0) |∇u|2 dx ≤ const R2 B2R (0) |u|2 dx, where c is a constant independent of R To show this inequality, multiply the differential equation by ζ := η u, where η ∈ C is a cut-off function with properties: η ≡ in BR (0), η ≡ in the exterior of B2R (0), ≤ η ≤ 1, |∇η| ≤ C/R Integrate the product, apply integration by parts and use the formula 2ab ≤ a2 + b2 , > Show that a bounded harmonic function defined on Rn must be a constant (a theorem of Liouville) Assume u ∈ C (B1 (0)) ∩ C(B1 (0) \ {(1, 0)}) is a solution of u = in B1 (0) u = on ∂B1 (0) \ {(1, 0)} Show that there are at least two solutions Hint: Consider u(x, y) = − (x2 + y ) (1 − x)2 + y Assume Ω ⊂ Rn is bounded and u, v ∈ C (Ω)∩C(Ω) satisfy u = v and max∂Ω |u − v| ≤ for given > Show that maxΩ |u − v| ≤ 10 Set Ω = Rn \ B1 (0) and let u ∈ C (Ω) be a harmonic function in Ω satisfying lim|x|→∞ u(x) = Prove that max |u| = max |u| Ω ∂Ω Hint: Apply the maximum principle to Ω ∩ BR (0), R large 197 7.6 EXERCISES 11 Let Ωα = {x ∈ R2 : x1 > 0, < x2 < x1 tan α}, < α ≤ π, Ωα,R = Ωα ∩ BR (0), and assume f is given and bounded on Ωα,R Show that for each solution u ∈ C (Ωα,R ) ∩ C (Ωα,R ) of Ωα,R satisfying u = on ∂Ωα,R ∩ BR (0), holds: For given u = f in > there is a constant C( ) such that π |u(x)| ≤ C( ) |x| α − in Ωα,R Hint: (a) Comparison principle (a consequence from the maximum principle): Assume Ω is bounded, u, v ∈ C (Ω) ∩ C(Ω) satisfying − u ≤ − v in Ω and u ≤ v on ∂Ω Then u ≤ v in Ω (b) An appropriate comparison function is π v = Ar α − sin(B(θ + η)) , A, B, η appropriate constants, B, η positive 12 Let Ω be the quadrangle (−1, 1) × (−1, 1) and u ∈ C (Ω) ∩ C(Ω) a solution of the boundary value problem − u = in Ω, u = on ∂Ω Find a lower and an upper bound for u(0, 0) Hint: Consider the comparison function v = A(x2 + y ), A = const 13 Let u ∈ C (Ba (0)) ∩ C(Ba (0)) satisfying u ≥ 0, Prove (Harnack’s inequality): u = in Ba (0) an−2 (a − |ζ|) an−2 (a + |ζ|) u(0) ≤ u(ζ) ≤ u(0) (a + |ζ|)n−1 (a − |ζ|)n−1 Hint: Use the formula (see Theorem 7.2) u(y) = a2 − |y|2 aωn |x|=a u(x) dSx |x − y|n for y = ζ and y = 14 Let φ(θ) be a 2π-periodic C -function with the Fourier series ∞ φ(θ) = (an cos(nθ) + bn sin(nθ)) n=0 Show that u= ∞ (an cos(nθ) + bn sin(nθ)) rn n=0 solves the Dirichlet problem in B1 (0) 198 CHAPTER ELLIPTIC EQUATIONS OF SECOND ORDER 15 Assume u ∈ C (Ω) satisfies u = in Ω Let Ba (ζ) be a ball such that its closure is in Ω Show that |Dα u(ζ)| ≤ M |α|γn a |α| , where M = supx∈Ba (ζ) |u(x)| and γn = 2nωn−1 /((n − 1)ωn ) Hint: Use the formula of Theorem 7.2, successively to the k th derivatives in balls with radius a(|α| − k)/m, k = o, 1, , m − 16 Use the result of the previous exercise to show that u ∈ C (Ω) satisfying u = in Ω is real analytic in Ω Hint: Use Stirling’s formula n! = nn e−n √ 2πn + O √ n as n → ∞, to show that u is in the class CK,r (ζ), where K = cM and r = a/(eγn ) The constant c is the constant in the estimate nn ≤ cen n! which follows from Stirling’s formula See Section 3.5 for the definition of a real analytic function 17 Assume Ω is connected and u ∈ C (Ω) is a solution of u = in Ω Prove that u ≡ in Ω if Dα u(ζ) = for all α, for a point ζ ∈ Ω In particular, u ≡ in Ω if u ≡ in an open subset of Ω 18 Let Ω = {(x1 , x2 , x3 ) ∈ R3 : x3 > 0}, which is a half-space of R3 Show that 1 G(x, y) = − , 4π|x − y| 4π|x − y| where y = (y1 , y2 , −y3 ), is the Green function to Ω 19 Let Ω = {(x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 < R2 , x3 > 0}, which is half of a ball in R3 Show that G(x, y) = R − 4π|x − y| 4π|y||x − y | R − + , 4π|x − y| 4π|y||x − y | where y = (y1 , y2 , −y3 ), y = R2 y/(|y|2 ) and y = R2 y/(|y|2 ), is the Green function to Ω 199 7.6 EXERCISES 20 Let Ω = {(x1 , x2 , x3 ) ∈ R3 : x2 > 0, x3 > 0}, which is a wedge in R3 Show that 1 G(x, y) = − 4π|x − y| 4π|x − y| 1 − + , 4π|x − y | 4π|x − y | where y = (y1 , y2 , −y3 ), y = (y1 , −y2 , y3 ) and y = (y1 , −y2 , −y3 ), is the Green function to Ω 21 Find Green’s function for the exterior of a disk, i e., of the domain Ω = {x ∈ R2 : |x| > R} 22 Find Green’s function for the angle domain Ω = {z ∈ C : < arg z < απ}, < α < π 23 Find Green’s function for the slit domain Ω = {z ∈ C : < arg z < 2π} 24 Let for a sufficiently regular domain Ω ∈ Rn , a ball or a quadrangle for example, K(x, y) dy, F (x) = Ω where K(x, y) is continuous in Ω × Ω where x = y, and which satisfies c |K(x, y)| ≤ |x − y|α with a constants c and α, α < n Show that F (x) is continuous on Ω 25 Prove (i) of the lemma of Section 7.5 Hint: Consider the case n ≥ Fix a function η ∈ C (R) satisfying ≤ η ≤ 1, ≤ η ≤ 2, η(t) = for t ≤ 1, η(t) = for t ≥ and consider for > the regularized integral f (y)η V (x) := Ω dy , |x − y|n−2 where η = η(|x − y|/ ) Show that V converges uniformly to V on compact subsets of Rn as → 0, and that ∂V (x)/∂xi converges uniformly on compact subsets of Rn to f (y) Ω ∂ ∂xi |x − y|n−2 dy 200 CHAPTER ELLIPTIC EQUATIONS OF SECOND ORDER as → 26 Consider the inhomogeneous Dirichlet problem − u = f in Ω, u = φ on ∂Ω Transform this problem into a Dirichlet problem for the Laplace equation Hint: Set u = w + v, where w(x) := Ω s(|x − y|)f (y) dy Bibliography [1] M Abramowitz and I A Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables Vol 55, National Bureau of Standards Applied Mathematics Series, U.S Government Printing Office, Washington, DC, 1964 Reprinted by Dover, New York, 1972 [2] S Bernstein, Sur un th´eor`eme de g´eom´etrie et son application aux d´eriv´ees partielles du type elliptique Comm Soc Math de Kharkov (2)15,(1915–1917), 38–45 German translation: Math Z 26 (1927), 551–558 [3] E Bombieri, E De Giorgi and E Giusti, Minimal cones and the Bernstein problem Inv Math (1969), 243–268 [4] R Courant und D Hilbert, Methoden der Mathematischen Physik Band und Band Springer-Verlag, Berlin, 1968 English translation: Methods of Mathematical Physics Vol and Vol 2, Wiley-Interscience, 1962 [5] L C Evans, Partial Differential Equations Graduate Studies in Mathematics , Vol 19, AMS, Providence, 1991 [6] L C Evans and R F Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992 [7] R Finn, Equilibrium Capillary Surfaces Grundlehren, Vol 284, Springer-Verlag, New York, 1986 [8] P R Garabedian, Partial Differential Equations Chelsia Publishing Company, New York, 1986 [9] D Gilbarg and N S Trudinger, Elliptic Partial Differential Equations of Second Order Grundlehren, Vol 224, Springer-Verlag, Berlin, 1983 201 202 BIBLIOGRAPHY [10] F John, Partial Differential Equations Springer-Verlag, New York, 1982 [11] K K¨onigsberger, Analysis Springer-Verlag, Berlin, 1993 [12] L D Landau and E M Lifschitz, Lehrbuch der Theoretischen Physik Vol 1., Akademie-Verlag, Berlin, 1964 German translation from Russian English translation: Course of Theoretical Physics Vol 1, Pergamon Press, Oxford, 1976 [13] R Leis, Vorlesungen u ¨ber partielle Differentialgleichungen zweiter Ordnung B I.-Hochschultaschenb¨ ucher 165/165a, Mannheim, 1967 [14] J.-L Lions and E Magenes, Probl´emes aux limites non homog´enes et applications Dunod, Paris, 1968 [15] E Miersemann, Kapillarfl¨ achen Ber Verh S¨achs Akad Wiss Leipzig, Math.-Natur Kl 130 (2008), Heft 4, S Hirzel, Leipzig, 2008 [16] Z Nehari, Conformal Mapping Reprinted by Dover, New York, 1975 [17] I G Petrowski, Vorlesungen u ¨ber Partielle Differentialgleichungen Teubner, Leipzig, 1955 Translation from Russian Englisch translation: Lectures on Partial Differential Equations Wiley-Interscience, 1954 [18] H Sagan, Introduction to the Calculus of Variations Dover, New York, 1992 [19] J Simons, Minimal varieties in riemannian manifolds Ann of Math(2) 88 (1968), 62–105 [20] W I Smirnow, Lehrgang der H¨ oheren Mathematik., Teil II VEB Verlag der Wiss., Berlin, 1975 Translation from Russian English translation: Course of Higher Mathematics, Vol 2., Elsevier, 1964 [21] W I Smirnow, Lehrgang der H¨ oheren Mathematik., Teil IV VEB Verlag der Wiss., Berlin, 1975 Translation from Russian English translation: Course of Higher Mathematics, Vol 4., Elsevier, 1964 [22] A Sommerfeld, Partielle Differentialgleichungen Geest & Portig, Leipzig, 1954 [23] W A Strauss, Partial Differential equations An Introduction Second edition, Wiley-Interscience, 2008 German translation: Partielle Differentialgleichungen Vieweg, 1995 BIBLIOGRAPHY 203 [24] M E Taylor, Pseudodifferential operators Princeton, New Jersey, 1981 [25] G N Watson, A treatise on the Theory of Bessel Functions Cambridge, 1952 [26] P Wilmott, S Howison and J Dewynne, The Mathematics of Financial Derivatives, A Student Introduction Cambridge University Press, 1996 [27] K Yosida, Functional Analysis Grundlehren, Vol 123, Springer-Verlag, Berlin, 1965 Index d’Alembert formula 108 asymptotic expansion 84 basic lemma 16 Beltrami equations Black-Scholes equation 164 Black-Scholes formulae 165, 169 boundary condition 14, 15 capillary equation 21 Cauchy-Kowalevskaya theorem 63, 84 Cauchy-Riemann equations 13 characteristic equation 28, 33, 41, 47, 74 characteristic curve 28 characteristic strip 47 classification linear equations second order 63 quasilinear equations second order 73 systems first order 74 cylinder surface 29 diffusion 163 Dirac distribution 177 Dirichlet integral 17 Dirichlet problem 181 domain of dependence 108, 115 domain of influence 109 elliptic 73, 75 nonuniformly elliptic 73 second order 175 system 75, 82 uniformly elliptic 73 Euler-Poisson-Darboux equation 111 Euler equation 15, 17 first order equations 25 two variables 40 Rn 51 Fourier transform 141 inverse Fourier transform 142 Fourier’s method 126, 162 functionally dependent 13 fundamental solution 175, 176 Gamma function 176 gas dynamics 79 Green’s function 183 ball 186 conformal mapping 190 Hamilton function 54 Hamilton-Jacobi theory 53 harmonic function 179 heat equation 14 inhomogeneous 155 heat kernel 152, 153 helicoid 30 hyperbolic equation 107 inhomogeneous equation 117 one dimensional 107 higher dimension 109 system 74 initial conditions 15 initial-boundary value problem uniqueness 134 204 205 INDEX string 125 membrane 128, 129 initial value problem Cauchy 33, 48 integral of a system 28 Jacobi theorem 55 Kepler 56 Laplace equation 13, 20 linear elasticity 83 linear equation 11, 25 second order 63 maximum principle heat equation 156 parabolic 161 harmonic function 180 Maxwell equations 76 mean value formula 180 minimal surface equation 18 Monge cone 42, 43 multi-index 90 multiplier 12 Navier Stokes 83 Neumann problem 20, 182 Newton potential 13 normal form 69 noncharacteristic curve 33 option call 165 put 169 parabolic equation 151 system 75 Picard-Lindel¨of theorem Poisson’s formula 152 Poisson’s kernel 187 pseudodifferential operators 146, 147 quasiconform mapping 76 quasilinear equation 11, 31 real analytic function 90 resonance 134 Riemann’s method 120 Riemann problem 61 Schr¨odinger equation 137 separation of variables 126 singularity function 176 speed plane 78 relative 81 surface 81, 83 sound 82 spherical mean 110 strip condition 46 telegraph equation 78 wave equation 14, 107, 131 wave front 51 volume potential 191 ... 1.2 Equations from variational problems 1.2.1 Ordinary differential equations 1.2.2 Partial differential equations 1.3 Exercises Equations of first order 2.1 Linear equations. .. Chapter Introduction Ordinary and partial differential equations occur in many applications An ordinary differential equation is a special case of a partial differential equation but the behaviour... = for all t ≥ 1.2 Equations from variational problems A large class of ordinary and partial differential equations arise from variational problems 1.2.1 Ordinary differential equations Set b f