3 Laplace's Equation

Total Page:16

File Type:pdf, Size:1020Kb

3 Laplace's Equation 3 Laplace’s Equation We now turn to studying Laplace’s equation ∆u = 0 and its inhomogeneous version, Poisson’s equation, ¡∆u = f: We say a function u satisfying Laplace’s equation is a harmonic function. 3.1 The Fundamental Solution Consider Laplace’s equation in Rn, ∆u = 0 x 2 Rn: Clearly, there are a lot of functions u which satisfy this equation. In particular, any constant function is harmonic. In addition, any function of the form u(x) = a1x1 +:::+anxn for constants ai is also a solution. Of course, we can list a number of others. Here, however, we are interested in finding a particular solution of Laplace’s equation which will allow us to solve Poisson’s equation. Given the symmetric nature of Laplace’s equation, we look for a radial solution. That is, we look for a harmonic function u on Rn such that u(x) = v(jxj). In addition, to being a natural choice due to the symmetry of Laplace’s equation, radial solutions are natural to look for because they reduce a PDE to an ODE, which is generally easier to solve. Therefore, we look for a radial solution. If u(x) = v(jxj), then x u = i v0(jxj) jxj 6= 0; xi jxj which implies 1 x2 x2 u = v0(jxj) ¡ i v0(jxj) + i v00(jxj) jxj 6= 0: xixi jxj jxj3 jxj2 Therefore, n ¡ 1 ∆u = v0(jxj) + v00(jxj): jxj Letting r = jxj, we see that u(x) = v(jxj) is a radial solution of Laplace’s equation implies v satisfies n ¡ 1 v0(r) + v00(r) = 0: r Therefore, 1 ¡ n v00 = v0 r v00 1 ¡ n =) = v0 r =) ln v0 = (1 ¡ n) ln r + C C =) v0(r) = ; rn¡1 1 which implies ½ c1 ln r + c2 n = 2 v(r) = c1 (2¡n)rn¡2 + c2 n ¸ 3: From these calculations, we see that for any constants c1; c2, the function ½ c1 ln jxj + c2 n = 2 u(x) ´ c1 (3.1) (2¡n)jxjn¡2 + c2 n ¸ 3: for x 2 Rn, jxj 6= 0 is a solution of Laplace’s equation in Rn ¡ f0g. We notice that the function u defined in (3.1) satisfies ∆u(x) = 0 for x 6= 0, but at x = 0, ∆u(0) is undefined. We claim that we can choose constants c1 and c2 appropriately so that ¡∆xu = ±0 in the sense of distributions. Recall that ±0 is the distribution which is defined as follows. For all Á 2 D, (±0;Á) = Á(0): Below, we will prove this claim. For now, though, assume we can prove this. That is, assume we can find constants c1; c2 such that u defined in (3.1) satisfies ¡∆xu = ±0: (3.2) Let Φ denote the solution of (3.2). Then, define Z v(x) = Φ(x ¡ y)f(y) dy: Rn Formally, we compute the Laplacian of v as follows, Z ¡∆xv = ¡ ∆xΦ(x ¡ y)f(y) dy ZRn = ¡ ∆yΦ(x ¡ y)f(y) dy Z Rn = ±xf(y) dy = f(x): Rn That is, v is a solution of Poisson’s equation! Of course, this set of equalities above is entirely formal. We have not proven anything yet. However, we have motivated a solution formula for Poisson’s equation from a solution to (3.2). We now return to using the radial solution (3.1) to find a solution of (3.2). Define the function Φ as follows. For jxj 6= 0, let ½ 1 ¡ 2¼ ln jxj n = 2 Φ(x) = 1 1 (3.3) n(n¡2)®(n) jxjn¡2 n ¸ 3; where ®(n) is the volume of the unit ball in Rn. We see that Φ satisfies Laplace’s equation n on R ¡ f0g. As we will show in the following claim, Φ satisfies ¡∆xΦ = ±0. For this reason, we call Φ the fundamental solution of Laplace’s equation. 2 Claim 1. For Φ defined in (3.3), Φ satisfies ¡∆xΦ = ±0 in the sense of distributions. That is, for all g 2 D, Z ¡ Φ(x)∆xg(x) dx = g(0): Rn Proof. Let FΦ be the distribution associated with the fundamental solution Φ. That is, let FΦ : D! R be defined such that Z (FΦ; g) = Φ(x)g(x) dx Rn for all g 2 D. Recall that the derivative of a distribution F is defined as the distribution G such that (G; g) = ¡(F; g0) for all g 2 D. Therefore, the distributional Laplacian of Φ is defined as the distribution F∆Φ such that (F∆Φ; g) = (FΦ; ∆g) for all g 2 D. We will show that (FΦ; ∆g) = ¡(±0; g) = ¡g(0); and, therefore, (F∆Φ; g) = ¡g(0); which means ¡∆xΦ = ±0 in the sense of distributions. By definition, Z (FΦ; ∆g) = Φ(x)∆g(x) dx: Rn Now, we would like to apply the divergence theorem, but Φ has a singularity at x = 0. We get around this, by breaking up the integral into two pieces: one piece consisting of the ball of radius ± about the origin, B(0; ±) and the other piece consisting of the complement of this ball in Rn. Therefore, we have Z (FΦ; ∆g) = Φ(x)∆g(x) dx ZRn Z = Φ(x)∆g(x) dx + Φ(x)∆g(x) dx B(0;±) Rn¡B(0;±) = I + J: 3 We look first at term I. For n = 2, term I is bounded as follows, ¯ Z ¯ ¯Z ¯ ¯ ¯ ¯ ¯ ¯ 1 ¯ ¯ ¯ ¯¡ ln jxj∆g(x) dx¯ · Cj∆gjL1 ¯ ln jxj dx¯ B(0;±) 2¼ B(0;±) ¯Z Z ¯ ¯ 2¼ ± ¯ ¯ ¯ · C ¯ ln jrjr dr dθ¯ ¯Z0 0 ¯ ¯ ± ¯ ¯ ¯ · C ¯ ln jrjr dr¯ 0 · C ln j±j±2: For n ¸ 3, term I is bounded as follows, ¯Z ¯ Z ¯ 1 1 ¯ 1 ¯ ∆g(x) dx¯ · Cj∆gj 1 dx ¯ n¡2 ¯ L n¡2 B(0;±) n(n ¡ 2)®(n) jxj B(0;±) jxj Z µZ ¶ ± 1 · C n¡2 dS(y) dr 0 @B(0;r) jyj Z µZ ¶ ± 1 = n¡2 dS(y) dr 0 r @B(0;r) Z ± 1 = n®(n)rn¡1 dr rn¡2 0 Z ± n®(n) = n®(n) r dr = ±2: 0 2 Therefore, as ± ! 0+, jIj ! 0. Next, we look at term J. Applying the divergence theorem, we have Z Z Z @Φ Φ(x)∆xg(x) dx = ∆xΦ(x)g(x) dx ¡ g(x) dS(x) Rn Rn Rn @º ¡B(0;±) ¡ZB(0;±) @( ¡B(0;±)) @g + Φ(x) dS(x) Rn @º Z @( ¡B(0;±)) Z @Φ @g = ¡ g(x) dS(x) + Φ(x) dS(x) @(Rn¡B(0;±)) @º @(Rn¡B(0;±)) @º ´ J1 + J2: n using the fact that ∆xΦ(x) = 0 for x 2 R ¡ B(0; ±). We first look at term J1. Now, by assumption, g 2 D, and, therefore, g vanishes at 1. Consequently, we only need to calculate the integral over @B(0; ²) where the normal derivative º is the outer normal to Rn ¡ B(0; ±). By a straightforward calculation, we see that x r Φ(x) = ¡ : x n®(n)jxjn The outer unit normal to Rn ¡ B(0; ±) on B(0; ±) is given by x º = ¡ : jxj 4 Therefore, the normal derivative of Φ on B(0; ±) is given by µ ¶ µ ¶ @Φ x x 1 = ¡ ¢ ¡ = : @º n®(n)jxjn jxj n®(n)jxjn¡1 Therefore, J1 can be written as Z Z Z 1 1 ¡ n¡1 g(x) dS(x) = ¡ n¡1 g(x) dS(x) = ¡¡ g(x) dS(x): @B(0;±) n®(n)jxj n®(n)± @B(0;±) @B(0;±) Now if g is a continuous function, then Z ¡¡ g(x) dS(x) !¡g(0) as ± ! 0: Lastly, we look at term J2. Now using the fact that g vanishes as jxj ! +1, we only need to integrate over @B(0; ±). Using the fact that g 2 D, and, therefore, infinitely differentiable, we have ¯Z ¯ ¯ ¯ Z ¯ ¯ ¯ ¯ ¯ @g ¯ ¯@g ¯ ¯ Φ(x) dS(x)¯ · ¯ ¯ jΦ(x)j dS(x) @º @º 1 @B(0;±) Z L (@B(0;±)) @B(0;±) · C jΦ(x)j dS(x): @B(0;±) Now first, for n = 2, Z Z jΦ(x)j dS(x) = C j ln jxjj dS(x) @B(0;±) @B(0;±)Z · Cj ln j±jj dS(x) @B(0;±) = Cj ln j±jj(2¼±) · C±j ln j±jj: Next, for n ¸ 3, Z Z 1 jΦ(x)j dS(x) = C dS(x) jxjn¡2 @B(0;±) @BZ(0;±) C · n¡2 dS(x) ± @B(0;±) C = n®(n)±n¡1 · C±: ±n¡2 Therefore, we conclude that term J2 is bounded in absolute value by C±j ln ±j n = 2 C± n ¸ 3: Therefore, jJ2j ! 0 as ± ! 0+. 5 Combining these estimates, we see that Z Φ(x)∆xg(x) dx = lim I + J1 + J2 = ¡g(0): Rn ±!0+ Therefore, our claim is proved. Solving Poisson’s Equation. We now return to solving Poisson’s equation ¡∆u = f x 2 Rn: From our discussion before the above claim, we expect the function Z v(x) ´ Φ(x ¡ y)f(y) dy Rn to give us a solution of Poisson’s equation. We now prove that this is in fact true. First, we make a remark. Remark. If we hope that the function v defined above solves Poisson’s equation, we must first verify that this integral actually converges. If we assume f has compact support on some bounded set K in Rn, then we see that Z Z Φ(x ¡ y)f(y) dy · jfjL1 jΦ(x ¡ y)j dy: Rn K If we additionally assume that f is bounded, then jfjL1 · C.
Recommended publications
  • Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions
    mathematics Article Best Subordinant for Differential Superordinations of Harmonic Complex-Valued Functions Georgia Irina Oros Department of Mathematics and Computer Sciences, Faculty of Informatics and Sciences, University of Oradea, 410087 Oradea, Romania; [email protected] or [email protected] Received: 17 September 2020; Accepted: 11 November 2020; Published: 16 November 2020 Abstract: The theory of differential subordinations has been extended from the analytic functions to the harmonic complex-valued functions in 2015. In a recent paper published in 2019, the authors have considered the dual problem of the differential subordination for the harmonic complex-valued functions and have defined the differential superordination for harmonic complex-valued functions. Finding the best subordinant of a differential superordination is among the main purposes in this research subject. In this article, conditions for a harmonic complex-valued function p to be the best subordinant of a differential superordination for harmonic complex-valued functions are given. Examples are also provided to show how the theoretical findings can be used and also to prove the connection with the results obtained in 2015. Keywords: differential subordination; differential superordination; harmonic function; analytic function; subordinant; best subordinant MSC: 30C80; 30C45 1. Introduction and Preliminaries Since Miller and Mocanu [1] (see also [2]) introduced the theory of differential subordination, this theory has inspired many researchers to produce a number of analogous notions, which are extended even to non-analytic functions, such as strong differential subordination and superordination, differential subordination for non-analytic functions, fuzzy differential subordination and fuzzy differential superordination. The notion of differential subordination was adapted to fit the harmonic complex-valued functions in the paper published by S.
    [Show full text]
  • Cloaking Via Change of Variables for Second Order Quasi-Linear Elliptic Differential Equations Maher Belgacem, Abderrahman Boukricha
    Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha To cite this version: Maher Belgacem, Abderrahman Boukricha. Cloaking via change of variables for second order quasi- linear elliptic differential equations. 2017. hal-01444772 HAL Id: hal-01444772 https://hal.archives-ouvertes.fr/hal-01444772 Preprint submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Cloaking via change of variables for second order quasi-linear elliptic differential equations Maher Belgacem, Abderrahman Boukricha University of Tunis El Manar, Faculty of Sciences of Tunis 2092 Tunis, Tunisia. Abstract The present paper introduces and treats the cloaking via change of variables in the framework of quasi-linear elliptic partial differential operators belonging to the class of Leray-Lions (cf. [14]). We show how a regular near-cloak can be obtained using an admissible (nonsingular) change of variables and we prove that the singular change- of variable-based scheme achieve perfect cloaking in any dimension d ≥ 2. We thus generalize previous results (cf. [7], [11]) obtained in the context of electric impedance tomography formulated by a linear differential operator in divergence form.
    [Show full text]
  • Harmonic Functions
    Lecture 1 Harmonic Functions 1.1 The Definition Definition 1.1. Let Ω denote an open set in R3. A real valued function u(x, y, z) on Ω with continuous second partials is said to be harmonic if and only if the Laplacian ∆u = 0 identically on Ω. Note that the Laplacian ∆u is defined by ∂2u ∂2u ∂2u ∆u = + + . ∂x2 ∂y2 ∂z2 We can make a similar definition for an open set Ω in R2.Inthatcase, u is harmonic if and only if ∂2u ∂2u ∆u = + =0 ∂x2 ∂y2 on Ω. Some basic examples of harmonic functions are 2 2 2 3 u = x + y 2z , Ω=R , − 1 3 u = , Ω=R (0, 0, 0), r − where r = x2 + y2 + z2. Moreover, by a theorem on complex variables, the real part of an analytic function on an open set Ω in 2 is always harmonic. p R Thus a function such as u = rn cos nθ is a harmonic function on R2 since u is the real part of zn. 1 2 1.2 The Maximum Principle The basic result about harmonic functions is called the maximum principle. What the maximum principle says is this: if u is a harmonic function on Ω, and B is a closed and bounded region contained in Ω, then the max (and min) of u on B is always assumed on the boundary of B. Recall that since u is necessarily continuous on Ω, an absolute max and min on B are assumed. The max and min can also be assumed inside B, but a harmonic function cannot have any local extrema inside B.
    [Show full text]
  • 23. Harmonic Functions Recall Laplace's Equation
    23. Harmonic functions Recall Laplace's equation ∆u = uxx = 0 ∆u = uxx + uyy = 0 ∆u = uxx + uyy + uzz = 0: Solutions to Laplace's equation are called harmonic functions. The inhomogeneous version of Laplace's equation ∆u = f is called the Poisson equation. Harmonic functions are Laplace's equation turn up in many different places in mathematics and physics. Harmonic functions in one variable are easy to describe. The general solution of uxx = 0 is u(x) = ax + b, for constants a and b. Maximum principle Let D be a connected and bounded open set in R2. Let u(x; y) be a harmonic function on D that has a continuous extension to the boundary @D of D. Then the maximum (and minimum) of u are attained on the bound- ary and if they are attained anywhere else than u is constant. Euivalently, there are two points (xm; ym) and (xM ; yM ) on the bound- ary such that u(xm; ym) ≤ u(x; y) ≤ u(xM ; yM ) for every point of D and if we have equality then u is constant. The idea of the proof is as follows. At a maximum point of u in D we must have uxx ≤ 0 and uyy ≤ 0. Most of the time one of these inequalities is strict and so 0 = uxx + uyy < 0; which is not possible. The only reason this is not a full proof is that sometimes both uxx = uyy = 0. As before, to fix this, simply perturb away from zero. Let > 0 and let v(x; y) = u(x; y) + (x2 + y2): Then ∆v = ∆u + ∆(x2 + y2) = 4 > 0: 1 Thus v has no maximum in D.
    [Show full text]
  • Chapter 8 Change of Variables, Parametrizations, Surface Integrals
    Chapter 8 Change of Variables, Parametrizations, Surface Integrals x0. The transformation formula In evaluating any integral, if the integral depends on an auxiliary function of the variables involved, it is often a good idea to change variables and try to simplify the integral. The formula which allows one to pass from the original integral to the new one is called the transformation formula (or change of variables formula). It should be noted that certain conditions need to be met before one can achieve this, and we begin by reviewing the one variable situation. Let D be an open interval, say (a; b); in R , and let ' : D! R be a 1-1 , C1 mapping (function) such that '0 6= 0 on D: Put D¤ = '(D): By the hypothesis on '; it's either increasing or decreasing everywhere on D: In the former case D¤ = ('(a);'(b)); and in the latter case, D¤ = ('(b);'(a)): Now suppose we have to evaluate the integral Zb I = f('(u))'0(u) du; a for a nice function f: Now put x = '(u); so that dx = '0(u) du: This change of variable allows us to express the integral as Z'(b) Z I = f(x) dx = sgn('0) f(x) dx; '(a) D¤ where sgn('0) denotes the sign of '0 on D: We then get the transformation formula Z Z f('(u))j'0(u)j du = f(x) dx D D¤ This generalizes to higher dimensions as follows: Theorem Let D be a bounded open set in Rn;' : D! Rn a C1, 1-1 mapping whose Jacobian determinant det(D') is everywhere non-vanishing on D; D¤ = '(D); and f an integrable function on D¤: Then we have the transformation formula Z Z Z Z ¢ ¢ ¢ f('(u))j det D'(u)j du1::: dun = ¢ ¢ ¢ f(x) dx1::: dxn: D D¤ 1 Of course, when n = 1; det D'(u) is simply '0(u); and we recover the old formula.
    [Show full text]
  • Introduction to the Modern Calculus of Variations
    MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result.
    [Show full text]
  • Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds
    Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds Xiaolong Hans Han Abstract We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hy- perbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock and Dunfield. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by defining two functionals on orientable closed and cusped hyperbolic 3-manifolds, and formulate several questions and conjectures. Contents 1 Introduction2 1.1 Motivation and previous results . .2 1.2 A glimpse at the non-compact case . .3 1.3 Main theorem . .4 1.4 Topology and the Thurston norm of harmonic forms . .5 1.5 Minimal surfaces and the least area norm . .5 1.6 Sharpness and the interaction between harmonic forms and minimal surfaces6 1.7 Acknowledgments . .6 2 L2 Harmonic Forms and Compactly Supported Cohomology7 2.1 Basic definitions of L2 harmonic forms . .7 2.2 Hodge theory on cusped hyperbolic 3-manifolds . .9 2.3 Topology of L2-harmonic 1-forms and the Thurston Norm . 11 2.4 How much does the hyperbolic metric come into play? . 14 arXiv:2011.14457v2 [math.GT] 23 Jun 2021 3 Minimal Surface and Least Area Norm 14 3.1 Truncation of M and the definition of Mτ ................. 16 3.2 The least area norm and the L1-norm . 17 4 L1-norm, Main theorem and The Proof 18 4.1 Bounding L1-norm by L2-norm .
    [Show full text]
  • 7. Transformations of Variables
    Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 7. Transformations of Variables Basic Theory The Problem As usual, we start with a random experiment with probability measure ℙ on an underlying sample space. Suppose that we have a random variable X for the experiment, taking values in S, and a function r:S→T . Then Y=r(X) is a new random variabl e taking values in T . If the distribution of X is known, how do we fin d the distribution of Y ? This is a very basic and important question, and in a superficial sense, the solution is easy. 1. Show that ℙ(Y∈B) = ℙ ( X∈r −1( B)) for B⊆T. However, frequently the distribution of X is known either through its distribution function F or its density function f , and we would similarly like to find the distribution function or density function of Y . This is a difficult problem in general, because as we will see, even simple transformations of variables with simple distributions can lead to variables with complex distributions. We will solve the problem in various special cases. Transformed Variables with Discrete Distributions 2. Suppose that X has a discrete distribution with probability density function f (and hence S is countable). Show that Y has a discrete distribution with probability density function g given by g(y)=∑ f(x), y∈T x∈r −1( y) 3. Suppose that X has a continuous distribution on a subset S⊆ℝ n with probability density function f , and that T is countable.
    [Show full text]
  • Complex Analysis
    Complex Analysis Andrew Kobin Fall 2010 Contents Contents Contents 0 Introduction 1 1 The Complex Plane 2 1.1 A Formal View of Complex Numbers . .2 1.2 Properties of Complex Numbers . .4 1.3 Subsets of the Complex Plane . .5 2 Complex-Valued Functions 7 2.1 Functions and Limits . .7 2.2 Infinite Series . 10 2.3 Exponential and Logarithmic Functions . 11 2.4 Trigonometric Functions . 14 3 Calculus in the Complex Plane 16 3.1 Line Integrals . 16 3.2 Differentiability . 19 3.3 Power Series . 23 3.4 Cauchy's Theorem . 25 3.5 Cauchy's Integral Formula . 27 3.6 Analytic Functions . 30 3.7 Harmonic Functions . 33 3.8 The Maximum Principle . 36 4 Meromorphic Functions and Singularities 37 4.1 Laurent Series . 37 4.2 Isolated Singularities . 40 4.3 The Residue Theorem . 42 4.4 Some Fourier Analysis . 45 4.5 The Argument Principle . 46 5 Complex Mappings 47 5.1 M¨obiusTransformations . 47 5.2 Conformal Mappings . 47 5.3 The Riemann Mapping Theorem . 47 6 Riemann Surfaces 48 6.1 Holomorphic and Meromorphic Maps . 48 6.2 Covering Spaces . 52 7 Elliptic Functions 55 7.1 Elliptic Functions . 55 7.2 Elliptic Curves . 61 7.3 The Classical Jacobian . 67 7.4 Jacobians of Higher Genus Curves . 72 i 0 Introduction 0 Introduction These notes come from a semester course on complex analysis taught by Dr. Richard Carmichael at Wake Forest University during the fall of 2010. The main topics covered include Complex numbers and their properties Complex-valued functions Line integrals Derivatives and power series Cauchy's Integral Formula Singularities and the Residue Theorem The primary reference for the course and throughout these notes is Fisher's Complex Vari- ables, 2nd edition.
    [Show full text]
  • The Laplace Operator in Polar Coordinates in Several Dimensions∗
    The Laplace operator in polar coordinates in several dimensions∗ Attila M´at´e Brooklyn College of the City University of New York January 15, 2015 Contents 1 Polar coordinates and the Laplacian 1 1.1 Polar coordinates in n dimensions............................. 1 1.2 Cartesian and polar differential operators . ............. 2 1.3 Calculating the Laplacian in polar coordinates . .............. 2 2 Changes of variables in harmonic functions 4 2.1 Inversionwithrespecttotheunitsphere . ............ 4 1 Polar coordinates and the Laplacian 1.1 Polar coordinates in n dimensions Let n ≥ 2 be an integer, and consider the n-dimensional Euclidean space Rn. The Laplace operator Rn n 2 2 in is Ln =∆= i=1 ∂ /∂xi . We are interested in solutions of the Laplace equation Lnf = 0 n 2 that are spherically symmetric,P i.e., is such that f depends only on r=1 xi . In order to do this, we need to use polar coordinates in n dimensions. These can be definedP as follows: for k with 1 ≤ k ≤ n 2 k 2 define rk = i=1 xi and put for 2 ≤ k ≤ n write P (1) rk−1 = rk sin φk and xk = rk cos φk. rk−1 = rk sin φk and xk = rk cos φk. The polar coordinates of the point (x1,x2,...,xn) will be (rn,φ2,φ3,...,φn). 2 2 2 In case n = 2, we can write y = x1, x = x2. The polar coordinates (r, θ) are defined by r = x + y , (2) x = r cos θ and y = r sin θ, so we can take r2 = r and φ2 = θ.
    [Show full text]
  • THE IMPACT of RIEMANN's MAPPING THEOREM in the World
    THE IMPACT OF RIEMANN'S MAPPING THEOREM GRANT OWEN In the world of mathematics, scholars and academics have long sought to understand the work of Bernhard Riemann. Born in a humble Ger- man home, Riemann became one of the great mathematical minds of the 19th century. Evidence of his genius is reflected in the greater mathematical community by their naming 72 different mathematical terms after him. His contributions range from mathematical topics such as trigonometric series, birational geometry of algebraic curves, and differential equations to fields in physics and philosophy [3]. One of his contributions to mathematics, the Riemann Mapping Theorem, is among his most famous and widely studied theorems. This theorem played a role in the advancement of several other topics, including Rie- mann surfaces, topology, and geometry. As a result of its widespread application, it is worth studying not only the theorem itself, but how Riemann derived it and its impact on the work of mathematicians since its publication in 1851 [3]. Before we begin to discover how he derived his famous mapping the- orem, it is important to understand how Riemann's upbringing and education prepared him to make such a contribution in the world of mathematics. Prior to enrolling in university, Riemann was educated at home by his father and a tutor before enrolling in high school. While in school, Riemann did well in all subjects, which strengthened his knowl- edge of philosophy later in life, but was exceptional in mathematics. He enrolled at the University of G¨ottingen,where he learned from some of the best mathematicians in the world at that time.
    [Show full text]
  • 21 Laplace's Equation and Harmonic Func- Tions
    21 Laplace's Equation and Harmonic Func- tions 21.1 Introductory Remarks on the Laplacian operator Given a domain Ω ⊂ Rd, then r2u = div(grad u) = 0 in Ω (1) is Laplace's equation defined in Ω. If d = 2, in cartesian coordinates @2u @2u r2u = + @x2 @y2 or, in polar coordinates 1 @ @u 1 @2u r2u = r + : r @r @r r2 @θ2 If d = 3, in cartesian coordinates @2u @2u @2u r2u = + + @x2 @y2 @z2 while in cylindrical coordinates 1 @ @u 1 @2u @2u r2u = r + + r @r @r r2 @θ2 @z2 and in spherical coordinates 1 @ @u 1 @2u @u 1 @2u r2u = ρ2 + + cot φ + : ρ2 @ρ @ρ ρ2 @φ2 @φ sin2 φ @θ2 We deal with problems involving most of these cases within these Notes. Remark: There is not a universally accepted angle notion for the Laplacian in spherical coordinates. See Figure 1 for what θ; φ mean here. (In the rest of these Notes we will use the notation r for the radial distance from the origin, no matter what the dimension.) 1 Figure 1: Coordinate angle definitions that will be used for spherical coordi- nates in these Notes. Remark: In the math literature the Laplacian is more commonly written with the symbol ∆; that is, Laplace's equation becomes ∆u = 0. For these Notes we write the equation as is done in equation (1) above. The non-homogeneous version of Laplace's equation, namely r2u = f(x) (2) is called Poisson's equation. Another important equation that comes up in studying electromagnetic waves is Helmholtz's equation: r2u + k2u = 0 k2 is a real, positive parameter (3) Again, Poisson's equation is a non-homogeneous Laplace's equation; Helm- holtz's equation is not.
    [Show full text]