Green s Functions and Distributions
|
|
- Norma Douglas
- 6 years ago
- Views:
Transcription
1 CHAPTER 9 Green s Functions and Distributions 9.1. Boundary Value Problems We would like to study, and solve if possible, boundary value problems such as the following: (1.1) u = f in U u = g on U, where U is open and bounded, with smooth boundary U. Now, we know how to solve the PDE on all space - we write the solution as a convolution with the fundamental solution: (1.2) u(x) = Φ(x y)f(y) dy. The idea of the Green s function is to find a formula like this that also takes into account the boundary conditions. Of course, the trouble is that since Φ(x) is set up to localize at x =, it cannot be expected to do the job in the interior of U at the same time as on the boundary. However, we can modify Φ so that it satisfies homogeneous boundary conditions. This is how the Green s function is set up, and in fact such a function allows us to solve general problems like (1.1), even for nonzero boundary data g. On the way to formulating Green s functions, we outline the theory of distributions (the Dirac delta function is an example of a distribution that is not a function in the usual sense), which is useful in the study of weak solutions of PDE. Let s start with a seemingly simple example, namely problem (1.1) in one dimension, with zero boundary conditions to start with: (1.3) u (x) = f(x) < x < 1 u() =, u(1) =. We solve this problem by integrating twice: u (x) = x f(y) dy + C; u(x) = where C, D are constants of integration. 115 x z f(y) dy dz + Cx + D,
2 GREEN S FUNCTIONS AND DISTRIBUTIONS Now we apply the boundary conditions. The boundary condition u() = lead immediately to D =. But the boundary condition at u(1) = gives an equation for C that is hard to understand. It becomes much more transparent when we change the order of integration. To do so, we note that the double integral occurs over a triangle (see Fig. 15.1). z x y z z y x y Figure 9.1. Area of Integration. Thus, x z f(y) dy dz = x x y f(y) dz dy = x (x y)f(y) dy Consequently, the solution we have so far is expressed as (1.4) u(x) = x The boundary condition at x = 1 becomes so that u(1) = 1 C = (x y)f(y) dy Cx. (1 y)f(y) dy C =, 1 (1 y)f(y) dy.
3 9.1. BOUNDARY VALUE PROBLEMS 117 Substituting back into (1.4) and rearranging terms, we arrive at u(x) = Thus, x (y x)f(y) dy+ 1 (1.5) u(x) = x(1 y)f(y) dy) = 1 G(x, y)f(y) dy, x y(1 x)f(y) dy+ where { y(1 x) if y x (1.6) G(x, y) = x(1 y) if x y. The graph of G(x, y) for fixed x (, 1) is shown in Figure The G(x,y) 1 x x(1 y)f(y) dy. x(1 - x) x 1 y Figure 9.2. The integral kernel G(x, y). function G(x, y) is called an integral kernel. It defines an integral operator G : C[, 1] X on the space of continuous functions on [, 1] that is the inverse of the differential operator L = d2 : X C[, 1], where X = {u dx 2 C[, 1] : u() = = u(1)} : (Gf)(x) = 1 G(x, y)f(y) dy. Note that the domain of L is the subspace of X consisting of differentiable functions. In fact, G is the Green s function for this boundary value problem - note that the formula (1.5) is similar to that of (1.2). Here are some properties of G : [, 1] [, 1] R. 1. G is non-negative: G(x, y) ;
4 GREEN S FUNCTIONS AND DISTRIBUTIONS 2. G is symmetric: G(x, y) = G(y, x). 3. G is continuous. 4. G is differentiable, except on the diagonal x = y. On the diagonal, G y has a jump discontinuity: [ ] G = 1, y x=y where [..] means the jump, or difference between the right limit and left limit. Now 2 G = except on the diagonal, where G y 2 y infinite negative slope. We will write 2 G (x, y) = δ(x y), y2 may be thought to have where δ(x) is the Dirac delta function. δ(x) is a measure that assigns mass one at x =, and zero mass elsewhere. We shall treat δ as a distribution, or generalized function, which leads us to the theory of distributions, a useful framework for considering PDEs, and Green s functions such as G(x, y) Distributions We start by considering smooth functions on R. Let D = Cc (R) denote the space of C functions with compact support; this is the space of test functions. To define the topology of D, we define what it means for a sequence of functions {φ n } to converge in the space. Let φ (j) denote the j-th derivative of φ. We say φ n φ D as n if: (a) There is a compact subset K of R such that supp φ n K for all n, and supp φ K. (b) φ (j) n φ (j) as n, uniformly on K, for each j : sup φ (j) n (x) φ (j) (x), as n. x K Similarly, we may define the space D( ) of test functions on.
5 Important example of a test function DISTRIBUTIONS 119 Let Ce 1 1 x 2 if x < 1 (2.7) η(x) = otherwise, where C = 1/ 1 x <1 e 1 x 2 dx is chosen so that η(x) dx = 1. To see that η is a test function, we observe that it has continuous derivatives of all orders, even where the definition is split, at x = 1. For example, with n = 1, the derivatives approach zero at x = 1, since every derivative is the product of a rational function and the exponential e 1 1 x 2 ; the exponential dominates the rational function. It is useful to rescale η using a parameter ɛ > : (2.8) η ɛ (x) = 1 ( x ) ɛ n η. ɛ Then: supp η ɛ = {x : x ɛ} and η ɛ (x) dx = 1. The space of distributions D is defined to be the space of continuous linear functionals on D. That is, f D means f : D R, and f has the properties (i) f is linear, i.e., f(aφ 1 +bφ 2 ) = af(φ 1 )+bf(φ 2 ) for each a, b R, φ 1, φ 2 D. (ii) f is continuous, i.e., φ n φ in D implies f(φ n ) f(φ) (as a sequence of numbers). Following the usual custom, we denote f(φ) using the more suggestive notation (f, φ). (Sometimes, f, φ is used, but not in these notes.) Examples of distributions 1. (f 1, φ) = R φ(x) dx (defined because φ is continuous and has compact support). 2. (f 2, φ) = φ(x) dx. 3. (f 3, φ) = φ(). 4. (f 4, φ) = φ ().
6 12 9. GREEN S FUNCTIONS AND DISTRIBUTIONS All of these are distributions, and the last three are related, as we shall see, by differentiation in the sense of distributions. The first two examples are associated with locally integrable functions: Let L loc 1 (Rn ) denote the space of locally integrable (in the sense of Lebesgue) functions on : g L loc 1 ( ) if g(x) dx < for any compact K. Examples. K 1. g(x) = 1 is not integrable on R, but it is in L loc 1 (R). 2. g(x) = x is unbounded on R, but is in L loc 1 (R). CLAIM If g L loc 1 (Rn ), then g defines a distribution f D by (f, φ) = g(x)φ(x) dx. Proof. Linearity of f is clear; continuity follows from the definition. However, not every distribution defines an L loc 1 function. We say a distribution f is regular if it has an L loc 1 representative f : (f, φ) = f(x)φ(x) dx for all φ D( ). Otherwise, the distribution f is called singular. Example f 3 is the Dirac δ-function: (δ, φ) = φ(); we show below that δ is a singular distribution. It will then follow that example f 4 is also singular. However, examples f 1 and f 2 define L loc 1 functions: For f 1 : take f(x) = 1. For f 2 : take f to be the Heaviside function H(x) = { if x < 1 if x Lemma 9.1. δ is a singular distribution. Proof. Suppose δ is regular, so that there is g L loc 1 such that (2.9) (δ, φ) = g(x)φ(x) dx for all φ D( ).
7 9.2. DISTRIBUTIONS 121 Now define a one-parameter family of test functions, modeled on η(x) : φ ɛ e 1 1 x ɛ 2 if x < ɛ (x) = if x ɛ. Now we calculate the effect of δ on φ ɛ in two ways. first, from the definition of δ : (δ, φ ɛ ) = φ ɛ () = 1 e Second, using the assumption (2.9): (δ, φ ɛ ) = g(x)φ ɛ (x) dx. But then 1 e = g(x)φ ɛ (x) dx g(x) 1 x ɛ e dx. But the latter integral approaches zero as ɛ zero, providing a contradiction for small enough ɛ. Note that the function φ ɛ used in the proof of the lemma is a mulitple (ɛ n /C to be precise) of η ɛ, but this multiple provides the very different behaviors of the two functions as ɛ. The function η ɛ is called a mollifier because it can be used to smooth rough functions. It does this through the convolution product: Recall that the convolution of two square integrable functions (i.e., in L 2 ( )) f and g is defined as g f(x) = g(x y)f(y) dy. Example: Compute g f for the functions { 1 if < x < 1 f(x) = g(x) = otherwise; and graph the convolution product g f. { x if x < 1 otherwise, Remark Convolution is symmetric: g f = f g, as can be seen by changing variables in the integral. Here is how the mollifier does its smoothing: Let f C( ), and define, for ɛ >, f ɛ (x) = η ɛ f(x). Then f ɛ C, and f ɛ (x) f(x) for all x, as ɛ.
8 GREEN S FUNCTIONS AND DISTRIBUTIONS Note that f ɛ is defined and is C for f L loc 1. In that case, it takes a little measure theory (the Lebesgue dominated convergence theorem) to prove that f ɛ f almost everywhere (see [Evans, Appendix]). Convergence of distributions Let {f k } k=1 D ( ) be a sequence of distributions, and let f D ( ). We say if Examples. 1. Let n = 1. Then, for φ D(R), f k f in D in the sense of distributions (f k, φ) (f, φ) as k, for all φ D( ) (f k, φ) = f k (x) = = R 1 2k 1 2k { k if x 1 2k otherwise f k (x)φ(x) dx = φ(x) dx φ() as k = (δ, φ). 1 2k 1 2k Thus, f k δ in the sense of distribitions as k. kφ(x) dx 2. (Exercise): Prove that η ɛ δ in the sense of distributions, as ɛ Distributional derivatives. Let f C 1 ( ). Then f defines a distribution f D : (f, φ) = f(x)φ(x) dx. But f x i C( ) also defines a distribution: ( ) f, φ = x i integrating by parts over the support of φ. f x i (x)φ(x) dx = f(x) φ (x) dx x i ( = f, φ ), x i
9 9.2. DISTRIBUTIONS 123 This calculation suggests that for any distribution f we define its derivative, or distributional derivative f x i to be a distribution given by ( ) ( f, φ = f, φ ), for all φ D. x i x i Then every distribution is differentiable in the sense of distributions, hence has derivatives of all orders. It is not hard to show that f x i is indeed a distribution, by checking directly that it is continuous and linear. Examples with n = Consider the Heaviside function H(x). As we saw earlier, H is in L loc, and it acts on test functions by (H, φ) = φ(x) dx. Thus, for any test function φ, Therefore, 1 ( H (x), φ ) = (H, φ ) = φ (x) dx = φ() = (δ, φ), H = δ. 2. We can also differentiate the δ function directly from the definition of derivative: ( δ, φ ) = (δ, φ ) = φ () 3. Let f(x) = x. Then 1 if x < f (x) = 1 if x > is an L loc 1 function, hence defines a distribution. In fact, f = 2H 1, so f = 2δ. Here, we use the fact that differentiation is a linear operation on distributions, just as it is on differentiable functions. Here are two further properties, or more precisely, definitions: 1. Translation by y. If f D, we define f( + y) = f y D by (f y, φ) = (f, φ y ),
10 GREEN S FUNCTIONS AND DISTRIBUTIONS where φ y D is the test function defined by φ y (x) = φ(x y). To see that this makes sense, we simply check that it is consistent for f L loc 1 : (f y, φ) = f(x + y)φ(x) dx = f(z)φ(z y) dz = (f, φ y ). An example where this is useful is the δ function. We often write δ(x y) to mean the distribution δ y, and it is sometimes helpful to leave x in the arguments of both the distribution and the test function: (δ(x y), φ(x)) = φ(y). 2. Multiplication by a C function. Let c C, f D. Then we define cf D by (cf, φ) = (f, cφ), noting that cφ is a test function if φ is. Again, this definition is motivated by the case of a regular distribution f, in which case the formula makes sense as integrals. As an example of this definition, consider c(x) = x, f = δ. Then (xδ(x), φ(x)) = (δ(x), xφ(x)) =, where again we have left the argument x in the calculation for clarity. Since δ = H, we see that y(x) = H(x) is a solution of the differential equation x dy dx =. In fact, the general distributional solution of this equation (which is singular at x = ) is y(x) = ah(x) + b, for arbitrary constants a, b. Thus, we have a two parameter family of solutions of a first order equation Example. Here is a quick application to conservation laws: Consider the scalar conservation law (2.1) u t + f(u) x =, in which f : R R is a given C 1 function. We can interpret equation (2.1) in the sense of distributions, and in particular say what it means for a discontinuous function to be a solution. The equation simply states that if u D (R 2 ), and f(u) can be interpreted as a distribution, then the combination of distributional derivatives on the left hand side of the equation should be zero on every test function. But this implies for a test function φ(x, t) : (u, φ t ) + (f(u), φ x ) =.
11 9.2. DISTRIBUTIONS 125 In this way, we can define distribution solutions of differential equations, even nonlinear equations. To see that this interpretation has some substance, consider a jump discontinuity u if x < st (2.11) u(x, t) = if x > st, u + where s, u ± are constants. Now f(u)(x, t) = f(u ) f(u + ) if x < st if x > st, But then u(x, t) = (u + u )H(x st)+u, f(u)(x, t) = (f(u + ) f(u ))H(x st)+f(u ). Thus, u t = (u + u )H (x st)( s); and H = δ. Equation (2.1) becomes x f(u) = (f(u +) f(u ))H (x st), ( s(u + u ) + (f(u + ) f(u )))δ(x st) =, in the sense of distributions. But then the constant s(u + u ) + f(u + ) f(u ) must be zero: s(u + u ) + f(u + ) f(u ) =. Thus, we have derived the Rankine-Hugoniot condition for shock wave solutions of (2.1). We are almost ready to describe Green s functions, but we first need to review integration by parts in. Consider a bounded open set U, with piecewise smooth boundary U. Consider functions f, g in C 1 (U) C(U), and let ν = (ν 1,..., ν n ) be the unit outward normal on U. Differentiating the product f(x)g(x) : U fg = f g + g f. x i x i x i Now integrate over U and apply the divergence theorem to F(x) = (,...,, fg,,..., ) : div F = fg. After rearranging, we get the formula for integration by x i parts: (2.12) f g dx = fg ν i ds g f dx. x i U U x i
12 GREEN S FUNCTIONS AND DISTRIBUTIONS 9.3. Green s Functions In this section, we first give a framework for Green s functions for a rather general differential operator, and then show how this applies when the operator is the Laplacian General Framework. Consider a linear partial differential operator L (L = for example), that acts on functions u : R, and consider solving a boundary value problem (3.13) Lu = f in U Bu = g on U. Here, f is a given function on U, g and given function on U, and U may be unbounded. Bu represents a linear combination of u and derivatives of u of lower order than the order of the partial differential operator L. Associated with L we have the fundamental solution Φ(x, y) : (3.14) LΦ(x, y) = δ(x y), x, y. In order to see how this works, consider Φ to be locally integrable in y for each x, and let f D( ). Then (3.15) v(x) = Φ(x, y)f(y) dy = (Φ(x, y), f(y)) satisfies Lv = f in : Lv(x) = L(Φ(x, y), f(y)) = (LΦ(x, y), f(y)) = (δ(x y), f(y)) = f(x). That is, the fundamental solution is the key to solving the inhomogeneous PDE. However, the formula (3.15) does not generally satisfy the boundary condition Bu = g. To satisfy the boundary condition, we add a solution w of the homogeneous equation Lw = so that u(x) = v(x) + w(x) satisfies the boundary condition Bu = g. But then w must satisfy the boundary condition Bw = g Bv. Thus, provided we have the fundamental solution Φ(x, y), the boundary value problem (3.13) is reduced to solving the problem Lw = in U; Bw = g Bv on U. The boundary condition for w is a bit clumsy, because it relies on applying the boundary operator to the integral (3.15) and restricting to the boundary. The way to express this more smoothly is to introduce the Green s function for the differential operator L with boundary operator B. For clarity, we consider L and B to have independent variable x U, and we let y U be independent.
13 9.3. GREEN S FUNCTIONS 127 The Green s function G = G(x, y) is defined as the solution of the problem (3.16) for each y U. LG(x, y) = δ y (x), x U BG(x, y) =, x U, We construct G using the fundamental solution, by defining a function φ y (x) : G(x, y) = Φ(x, y) φ y (x). Then φ y (x) satisfies Lφ y =, in U Bφ y = BΦ(, y), in U Green s Functions for the Laplacian. Let U be open and bounded, with piecewise smooth boundary U, and let f, g be continuous on U, U respectively. Consider the Dirichlet boundary value problem for Poisson s equation: (3.17) u = f, in U u = g, in U. The fundamental solution for is a function Φ(x y) of x y, since is translation invariant. (More generally, the fundamental solution for any constant coefficient L will be a function of x y.) The Green s function G(x, y) = Φ(x y) φ y (x) is then specified by φ y (x) = x U φ y (x) = Φ(x y), x U. In particular, φ y is a harmonic function in U. Theorem 9.2. If u C 2 (U) solves (3.17), then G (3.18) u(x) = G(x, y)f(y) dy (x, y)g(y) ds(y). ν y U Remark. Note that the first integral solves the inhomogeneous PDE, with zero boundary condition, while the second integral should be harmonic, and should satisfy the given boundary condition. That it is harmonic follows from the fact that G(x, y) is harmonic in x U for each y U, so that the normal derivative with respect to y is also harmonic. To verify the boundary condition, we could try to show that the normal derivative acts U
14 GREEN S FUNCTIONS AND DISTRIBUTIONS like a δ function as x approaches the boundary. In fact, the proof uses the divergence theorem to recover the boundary data. Proof of Theorem 9.2. Recall that G(x, y) has a singularity at y = x, because the fundamental solution has. As with our solution of Poisson s equation on all of space, using the fundamental solution, we exclude a small ball around x U, by integrating on the domain V ɛ = U B(x, ɛ). Let u C 2 (U) (not necessarily a solution). Then, using Green s identity, ( ) ( u(y) Φ(x y) Φ(x y) u(y) dy = u Φ Φ u ) ds(y). ν y ν y V ɛ Now Φ(x y) is harmonic away from y = x, so the first term on the left hand side is zero. On the right hand side, there are two terms, but there are also two parts of the boundary, namely U and B(x, ɛ). Let I ɛ = B(x,ɛ) V ɛ u(y) Φ ν y (x y) ds(y). As for the solution of Poisson s equation, I ɛ u(x) as ɛ. (See 8.2.) Similarly, let J ɛ = B(x,ɛ) Φ(x y) u ν y (y) ds(y). Now u is continuously differentiable, and Φ(x y) is constant on B(x, ɛ) (proportional to ln ɛ for n = 2, and to ɛ (n 2) for n > 2). Consequently, J ɛ as ɛ. Letting ɛ, we obtain Φ(x y) u(y) dy = u(x)+ U U ( u Φ (x y) Φ(x y) u ) ds(y). ν y ν y Similarly, since φ x is harmonic in U, has no singularity at x = y, and φ x (y) = Φ(x y) for y U, we have φ x u dy = (u φx Φ(x y) u ) ds(y). ν y ν y U U Subtracting, and using G(x, y) = Φ(x y) φ x (y), we obtain (3.19) G(x, y) u(y) dy = u(x) + u(y) G (x, y) ds(y). ν y (The other terms cancel.) U Finally, if u is a solution of (3.17), then we have proved the theorem. U
15 9.3. GREEN S FUNCTIONS The Method of Images. In some examples, we can use the socalled Method of Images to construct the Green s function. The idea is that Φ(x y) represents a field (more precisely, a potential; the study of solutions of Poisson s equation is sometimes referred to as Potential Theory) induced by a singularity at x = y, which induces a field on the boundary U. Suppose we can construct a point x outside U such that the function Φ(C( x y)) exactly cancels Φ(x y) on the boundary U, for some scale factor C, possibly depending on x. Then the new field is harmonic with respect to y U (since the Laplacian is invariant under scaling and translation by a constant), so we can set φ x (y) = Φ(C( x y)). The point x is the image of x that gives the technique its name. Here is how it works in two examples: (i) U is a half space, and (ii) U is a ball. (i) Let U = {x : x n > }. For x = (x 1,..., x n ) U, define the image x = (x 1,..., x n 1, x n ), and let G(x, y) = Φ(x y) Φ( x y), x n, y n. Since Φ( x y) is harmonic in U (with respect to y, for x U), we see that G(x, y) is the Green s function if we can show G(x, y) = for y n =, x n >. But for y n =, we have y x = n 1 (y j x j ) 2 + x 2 n = y x, j=1 so that Φ(x y) = Φ( x y). (See Figure ) (ii) Now consider U = B(, 1), the unit ball in. Here, the image x includes a scaling, in addition to reflection through the boundary. (See Figure ) We define x = x, and let x 2 G(x, y) = Φ(x y) Φ( x ( x y)). (Note that x = 1/ x.) To prove that G(x, y) = for x U, y U, we need to show that x y = x x y, x = x, y = 1. x 2 Proof. x 2 x y 2 = x x x y 2 = x 2 2y x + 1 = x y 2.
16 13 9. GREEN S FUNCTIONS AND DISTRIBUTIONS xn U x y x,. x 1 n-1 ~ x Figure 9.3. Green s function for a half plane. ~ x 1 x y Figure 9.4. Green s function for the unit ball.
MATH 425, FINAL EXAM SOLUTIONS
MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informatione (x y)2 /4kt φ(y) dy, for t > 0. (4)
Math 24A October 26, 2 Viktor Grigoryan Heat equation: interpretation of the solution Last time we considered the IVP for the heat equation on the whole line { ut ku xx = ( < x
More information2. Function spaces and approximation
2.1 2. Function spaces and approximation 2.1. The space of test functions. Notation and prerequisites are collected in Appendix A. Let Ω be an open subset of R n. The space C0 (Ω), consisting of the C
More informationSOLUTION OF POISSON S EQUATION. Contents
SOLUTION OF POISSON S EQUATION CRISTIAN E. GUTIÉRREZ OCTOBER 5, 2013 Contents 1. Differentiation under the integral sign 1 2. The Newtonian potential is C 1 2 3. The Newtonian potential from the 3rd Green
More informationSobolev spaces. May 18
Sobolev spaces May 18 2015 1 Weak derivatives The purpose of these notes is to give a very basic introduction to Sobolev spaces. More extensive treatments can e.g. be found in the classical references
More information+ 2x sin x. f(b i ) f(a i ) < ɛ. i=1. i=1
Appendix To understand weak derivatives and distributional derivatives in the simplest context of functions of a single variable, we describe without proof some results from real analysis (see [7] and
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationMATH 51H Section 4. October 16, Recall what it means for a function between metric spaces to be continuous:
MATH 51H Section 4 October 16, 2015 1 Continuity Recall what it means for a function between metric spaces to be continuous: Definition. Let (X, d X ), (Y, d Y ) be metric spaces. A function f : X Y is
More informationClass Meeting # 1: Introduction to PDEs
MATH 18.152 COURSE NOTES - CLASS MEETING # 1 18.152 Introduction to PDEs, Spring 2017 Professor: Jared Speck Class Meeting # 1: Introduction to PDEs 1. What is a PDE? We will be studying functions u =
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More informationMATH FALL 2014
MATH 126 - FALL 2014 JASON MURPHY Abstract. These notes are meant to supplement the lectures for Math 126 (Introduction to PDE) in the Fall of 2014 at the University of California, Berkeley. Contents 1.
More informationCHAPTER 2. Laplace s equation
18 CHAPTER 2 Laplace s equation There can be but one option as to the beauty and utility of this analysis by Laplace; but the manner in which it has hitherto been presented has seemed repulsive to the
More informationFriedrich symmetric systems
viii CHAPTER 8 Friedrich symmetric systems In this chapter, we describe a theory due to Friedrich [13] for positive symmetric systems, which gives the existence and uniqueness of weak solutions of boundary
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationUnbounded operators on Hilbert spaces
Chapter 1 Unbounded operators on Hilbert spaces Definition 1.1. Let H 1, H 2 be Hilbert spaces and T : dom(t ) H 2 be a densely defined linear operator, i.e. dom(t ) is a dense linear subspace of H 1.
More informationMeasure and Integration: Solutions of CW2
Measure and Integration: s of CW2 Fall 206 [G. Holzegel] December 9, 206 Problem of Sheet 5 a) Left (f n ) and (g n ) be sequences of integrable functions with f n (x) f (x) and g n (x) g (x) for almost
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationThe Mollifier Theorem
The Mollifier Theorem Definition of the Mollifier The function Tx Kexp 1 if x 1 2 1 x 0 if x 1, x R n where the constant K is chosen such that R n Txdx 1, is a test function on Rn. Note that Tx vanishes,
More informationMATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck
MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations
More informationPerron method for the Dirichlet problem.
Introduzione alle equazioni alle derivate parziali, Laurea Magistrale in Matematica Perron method for the Dirichlet problem. We approach the question of existence of solution u C (Ω) C(Ω) of the Dirichlet
More information1 Review of di erential calculus
Review of di erential calculus This chapter presents the main elements of di erential calculus needed in probability theory. Often, students taking a course on probability theory have problems with concepts
More informationChapter 1. Distributions
Chapter 1 Distributions The concept of distribution generalises and extends the concept of function. A distribution is basically defined by its action on a set of test functions. It is indeed a linear
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationPartial Differential Equations. William G. Faris
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....................
More informationChapter 7: Bounded Operators in Hilbert Spaces
Chapter 7: Bounded Operators in Hilbert Spaces I-Liang Chern Department of Applied Mathematics National Chiao Tung University and Department of Mathematics National Taiwan University Fall, 2013 1 / 84
More informationLECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More information1.5 Approximate Identities
38 1 The Fourier Transform on L 1 (R) which are dense subspaces of L p (R). On these domains, P : D P L p (R) and M : D M L p (R). Show, however, that P and M are unbounded even when restricted to these
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationThe Relativistic Heat Equation
Maximum Principles and Behavior near Absolute Zero Washington University in St. Louis ARTU meeting March 28, 2014 The Heat Equation The heat equation is the standard model for diffusion and heat flow,
More informationReal Analysis Problems
Real Analysis Problems Cristian E. Gutiérrez September 14, 29 1 1 CONTINUITY 1 Continuity Problem 1.1 Let r n be the sequence of rational numbers and Prove that f(x) = 1. f is continuous on the irrationals.
More informationTHE GREEN FUNCTION. Contents
THE GREEN FUNCTION CRISTIAN E. GUTIÉRREZ NOVEMBER 5, 203 Contents. Third Green s formula 2. The Green function 2.. Estimates of the Green function near the pole 2 2.2. Symmetry of the Green function 3
More informationare harmonic functions so by superposition
J. Rauch Applied Complex Analysis The Dirichlet Problem Abstract. We solve, by simple formula, the Dirichlet Problem in a half space with step function boundary data. Uniqueness is proved by complex variable
More information= 2 x x 2 n It was introduced by P.-S.Laplace in a 1784 paper in connection with gravitational
Lecture 1, 9/8/2010 The Laplace operator in R n is = 2 x 2 + + 2 1 x 2 n It was introduced by P.-S.Laplace in a 1784 paper in connection with gravitational potentials. Recall that if we have masses M 1,...,
More informationHARMONIC ANALYSIS. Date:
HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded
More informationt y n (s) ds. t y(s) ds, x(t) = x(0) +
1 Appendix Definition (Closed Linear Operator) (1) The graph G(T ) of a linear operator T on the domain D(T ) X into Y is the set (x, T x) : x D(T )} in the product space X Y. Then T is closed if its graph
More informationn 2 xi = x i. x i 2. r r ; i r 2 + V ( r) V ( r) = 0 r > 0. ( 1 1 ) a r n 1 ( 1 2) V( r) = b ln r + c n = 2 b r n 2 + c n 3 ( 1 3)
Sep. 7 The L aplace/ P oisson Equations: Explicit Formulas In this lecture we study the properties of the Laplace equation and the Poisson equation with Dirichlet boundary conditions through explicit representations
More informationMollifiers and its applications in L p (Ω) space
Mollifiers and its alications in L () sace MA Shiqi Deartment of Mathematics, Hong Kong Batist University November 19, 2016 Abstract This note gives definition of mollifier and mollification. We illustrate
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationfor all x,y [a,b]. The Lipschitz constant of f is the infimum of constants C with this property.
viii 3.A. FUNCTIONS 77 Appendix In this appendix, we describe without proof some results from real analysis which help to understand weak and distributional derivatives in the simplest context of functions
More informationVISCOSITY SOLUTIONS. We follow Han and Lin, Elliptic Partial Differential Equations, 5.
VISCOSITY SOLUTIONS PETER HINTZ We follow Han and Lin, Elliptic Partial Differential Equations, 5. 1. Motivation Throughout, we will assume that Ω R n is a bounded and connected domain and that a ij C(Ω)
More informationA review: The Laplacian and the d Alembertian. j=1
Chapter One A review: The Laplacian and the d Alembertian 1.1 THE LAPLACIAN One of the main goals of this course is to understand well the solution of wave equation both in Euclidean space and on manifolds
More informationMathematical Methods
Mathematical Methods Lecture 11. 1/12-28 1 Adjoint and self-adjoint operators We again define the adjoint operator L or L through (Lf, g) = (f, L g). We should think in the distribution-sense, so that
More informationSolutions: Problem Set 4 Math 201B, Winter 2007
Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial ifferential Equations «Viktor Grigoryan 3 Green s first identity Having studied Laplace s equation in regions with simple geometry, we now start developing some tools, which will lead
More informationElliptic PDEs of 2nd Order, Gilbarg and Trudinger
Elliptic PDEs of 2nd Order, Gilbarg and Trudinger Chapter 2 Laplace Equation Yung-Hsiang Huang 207.07.07. Mimic the proof for Theorem 3.. 2. Proof. I think we should assume u C 2 (Ω Γ). Let W be an open
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationTOPICS IN FOURIER ANALYSIS-IV. Contents
TOPICS IN FOURIER ANALYSIS-IV M.T. NAIR Contents 1. Test functions and distributions 2 2. Convolution revisited 8 3. Proof of uniqueness theorem 13 4. A characterization of distributions 14 5. Distributions
More informationImmerse Metric Space Homework
Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps
More informationDistributions. Chapter INTRODUCTION
Chapter 6 Distributions 6.1 INTRODUCTION The notion of a weak derivative is an indispensable tool in dealing with partial differential equations. Its advantage is that it allows one to dispense with subtle
More informationProblem Set 5. 2 n k. Then a nk (x) = 1+( 1)k
Problem Set 5 1. (Folland 2.43) For x [, 1), let 1 a n (x)2 n (a n (x) = or 1) be the base-2 expansion of x. (If x is a dyadic rational, choose the expansion such that a n (x) = for large n.) Then the
More informationRiemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,
Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of
More informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationMATH 425, HOMEWORK 3 SOLUTIONS
MATH 425, HOMEWORK 3 SOLUTIONS Exercise. (The differentiation property of the heat equation In this exercise, we will use the fact that the derivative of a solution to the heat equation again solves the
More informationg(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0)
Mollifiers and Smooth Functions We say a function f from C is C (or simply smooth) if all its derivatives to every order exist at every point of. For f : C, we say f is C if all partial derivatives to
More informationDivision of the Humanities and Social Sciences. Supergradients. KC Border Fall 2001 v ::15.45
Division of the Humanities and Social Sciences Supergradients KC Border Fall 2001 1 The supergradient of a concave function There is a useful way to characterize the concavity of differentiable functions.
More informationFundamental Solutions and Green s functions. Simulation Methods in Acoustics
Fundamental Solutions and Green s functions Simulation Methods in Acoustics Definitions Fundamental solution The solution F (x, x 0 ) of the linear PDE L {F (x, x 0 )} = δ(x x 0 ) x R d Is called the fundamental
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationMATH 825 FALL 2014 ANALYSIS AND GEOMETRY IN CARNOT-CARATHÉODORY SPACES
MATH 825 FALL 2014 ANALYSIS AND GEOMETRY IN CARNOT-CARATHÉODORY SPACES Contents 1. Introduction 1 1.1. Vector fields and flows 2 1.2. Metrics 2 1.3. Commutators 3 1.4. Consequences of Hörmander s condition
More informationANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.
ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n
More informationMATH 426, TOPOLOGY. p 1.
MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 15 Heat with a source So far we considered homogeneous wave and heat equations and the associated initial value problems on the whole line, as
More informationENGI 4430 PDEs - d Alembert Solutions Page 11.01
ENGI 4430 PDEs - d Alembert Solutions Page 11.01 11. Partial Differential Equations Partial differential equations (PDEs) are equations involving functions of more than one variable and their partial derivatives
More informationMath Tune-Up Louisiana State University August, Lectures on Partial Differential Equations and Hilbert Space
Math Tune-Up Louisiana State University August, 2008 Lectures on Partial Differential Equations and Hilbert Space 1. A linear partial differential equation of physics We begin by considering the simplest
More informationWhat can you really measure? (Realistically) How to differentiate nondifferentiable functions?
I. Introduction & Motivation I.1. Physical motivation. What can you really measure? (Realistically) T (x) vs. T (x)ϕ(x) dx. I.2. Mathematical motivation. How to differentiate nondifferentiable functions?
More informationIntroductory Analysis I Fall 2014 Homework #9 Due: Wednesday, November 19
Introductory Analysis I Fall 204 Homework #9 Due: Wednesday, November 9 Here is an easy one, to serve as warmup Assume M is a compact metric space and N is a metric space Assume that f n : M N for each
More informationMATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
More informationAsymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates
Asymptotic Behavior of Fragmentation-drift Equations with Variable Drift Rates Daniel Balagué joint work with José A. Cañizo and Pierre GABRIEL (in preparation) Universitat Autònoma de Barcelona SIAM Conference
More informationP(E t, Ω)dt, (2) 4t has an advantage with respect. to the compactly supported mollifiers, i.e., the function W (t)f satisfies a semigroup law:
Introduction Functions of bounded variation, usually denoted by BV, have had and have an important role in several problems of calculus of variations. The main features that make BV functions suitable
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationNotes on Partial Differential Equations. John K. Hunter. Department of Mathematics, University of California at Davis
Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous
More informationRegularity for Poisson Equation
Regularity for Poisson Equation OcMountain Daylight Time. 4, 20 Intuitively, the solution u to the Poisson equation u= f () should have better regularity than the right hand side f. In particular one expects
More informationSobolev Spaces. Chapter 10
Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p
More informationEXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018
EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationElementary Theory and Methods for Elliptic Partial Differential Equations. John Villavert
Elementary Theory and Methods for Elliptic Partial Differential Equations John Villavert Contents 1 Introduction and Basic Theory 4 1.1 Harmonic Functions............................... 5 1.1.1 Mean Value
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 6. Solve the completely inhomogeneous diffusion problem on the half-line
Strauss PDEs 2e: Section 3.3 - Exercise 2 Page of 6 Exercise 2 Solve the completely inhomogeneous diffusion problem on the half-line v t kv xx = f(x, t) for < x
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationClass Meeting # 9: Poisson s Formula, Harnack s Inequality, and Liouville s Theorem
MATH 8.52 COUSE NOTES - CLASS MEETING # 9 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 9: Poisson s Formula, Harnack s Inequality, and Liouville s Theorem. epresentation Formula
More informationMath 140A - Fall Final Exam
Math 140A - Fall 2014 - Final Exam Problem 1. Let {a n } n 1 be an increasing sequence of real numbers. (i) If {a n } has a bounded subsequence, show that {a n } is itself bounded. (ii) If {a n } has a
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationLebesgue s Differentiation Theorem via Maximal Functions
Lebesgue s Differentiation Theorem via Maximal Functions Parth Soneji LMU München Hütteseminar, December 2013 Parth Soneji Lebesgue s Differentiation Theorem via Maximal Functions 1/12 Philosophy behind
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationMultivariable Calculus
2 Multivariable Calculus 2.1 Limits and Continuity Problem 2.1.1 (Fa94) Let the function f : R n R n satisfy the following two conditions: (i) f (K ) is compact whenever K is a compact subset of R n. (ii)
More informationfor all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true
3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO
More informationMath 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.3.5, 4.3.7, 4.3.8, 4.3.9,
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationHere we used the multiindex notation:
Mathematics Department Stanford University Math 51H Distributions Distributions first arose in solving partial differential equations by duality arguments; a later related benefit was that one could always
More informationApplications of the Maximum Principle
Jim Lambers MAT 606 Spring Semester 2015-16 Lecture 26 Notes These notes correspond to Sections 7.4-7.6 in the text. Applications of the Maximum Principle The maximum principle for Laplace s equation is
More informationProblem Set 5: Solutions Math 201A: Fall 2016
Problem Set 5: s Math 21A: Fall 216 Problem 1. Define f : [1, ) [1, ) by f(x) = x + 1/x. Show that f(x) f(y) < x y for all x, y [1, ) with x y, but f has no fixed point. Why doesn t this example contradict
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationBOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET
BOUNDARY VALUE PROBLEMS ON A HALF SIERPINSKI GASKET WEILIN LI AND ROBERT S. STRICHARTZ Abstract. We study boundary value problems for the Laplacian on a domain Ω consisting of the left half of the Sierpinski
More information