Lecture notes for Partial Differential equations pdf

what is elliptic partial differential equations and what are homogeneous partial differential equations, partial differential equations questions and solutions pdf free download
Dr.JakeFinlay Profile Pic
Published Date:22-07-2017
Your Website URL(Optional)
Notes on Partial Differential Equations John K. Hunter 1 Department of Mathematics, University of California at Davis 1 Revised 6/18/2014. Thanks to Kris Jenssen and Jan Koch for corrections. Supported in part by NSF Grant DMS-1312342.CHAPTER 1 Preliminaries In this chapter, we collect various definitions and theorems for future use. Proofs may be found in the references e.g. 4, 11, 24, 37, 42, 44. 1.1. Euclidean space n LetR be n-dimensional Euclidean space. We denote the Euclidean norm of a n vectorx= (x ,x ,...,x )∈R by 1 2 n  1/2 2 2 2 x = x +x +···+x 1 2 n and the inner product of vectorsx= (x ,x ,...,x ), y = (y ,y ,...,y ) by 1 2 n 1 2 n x·y =x y +x y +···+x y . 1 1 2 2 n n n We denote Lebesgue measure onR by dx, and the Lebesgue measure of a set n E⊂R by E. n c n IfE is a subset ofR , we denote the complement by E =R \E, the closure ◦ ◦ by E, the interior by E and the boundary by ∂E = E\E . The characteristic n function χ :R →R ofE is defined by E  1 if x∈E, χ (x) = E 0 if x∈/ E. A set E is bounded ifx:x∈E is bounded inR. A set is connected if it is not the disjoint union of two nonempty relatively open subsets. We sometimes refer to a connected open set as a domain. ′ We say that a (nonempty) open set Ω is compactly contained in an open set ′ ′ ′ ′ Ω, written Ω ⋐ Ω, if Ω ⊂ Ω and Ω is compact. If Ω ⋐ Ω, then ′ ′ dist(Ω,∂Ω)= infx−y:x∈Ω,y∈∂Ω0. 1.2. Spaces of continuous functions n Let Ω be an open set inR . We denote the space of continuous functions u : Ω →R by C(Ω); the space of functions with continuous partial derivatives in k Ω of order less than or equal to k ∈N by C (Ω); and the space of functions with ∞ continuous derivatives of all orders by C (Ω). Functions in these spaces need not ∞ be bounded even if Ω is bounded; for example, (1/x)∈C (0,1). n If Ω is a bounded open set inR , we denote by C(Ω) the space of continuous functions u : Ω →R. This is a Banach space with respect to the maximum, or supremum, norm kuk = supu(x). ∞ x∈Ω n We denote the support of a continuous function u :Ω→R by suppu =x∈Ω:u(x) = 6 0. 12 1. PRELIMINARIES We denote byC (Ω) the space of continuous functions whose support is compactly c ∞ contained in Ω, and by C (Ω) the space of functions with continuous derivatives c of all orders and compact support in Ω. We will sometimes refer to such functions as test functions. n n ThecompletionofC (R )withrespecttotheuniformnormisthespaceC (R ) c 0 ofcontinuousfunctionsthatapproachzeroatinfinity. (Notethatinmanyplacesthe ∞ notationC andC is used to denote the spaces of compactly supported functions 0 0 ∞ that we denote by C and C .) c c k Ω →R belongs to C (Ω) If Ω is bounded, then we say that a function u : if it is continuous and its partial derivatives of order less than or equal to k are uniformly continuous in Ω, in which case they extend to continuous functions on k Ω. The space C (Ω) is a Banach space with respect to the norm X α kuk = sup∂ u k C (Ω) Ω α≤k where we use the multi-index notation for partial derivatives explained in Sec- α tion 1.8. This norm is finite because the derivatives ∂ u are continuous functions on the compact set Ω. m k A vector fieldX :Ω→R belongs toC (Ω) if each of its components belongs k to C (Ω). 1.3. H¨older spaces The definition of continuity is not a quantitative one, because it does not say how rapidly the values u(y) of a function approach its value u(x) as y → x. The modulus of continuity ω : 0,∞ → 0,∞ of a general continuous function u, satisfying u(x)−u(y)≤ω(x−y), may decrease arbitrarily slowly. As a result, despite their simple and natural ap- pearance, spaces of continuous functions are often not suitable for the analysis of PDEs, which is almost always based on quantitative estimates. A straightforward and useful way to strengthen the definition of continuity is α to require that the modulus of continuity is proportional to a power x−y for someexponent 0α≤ 1. Such functions are saidto be Ho¨lder continuous, or Lip- schitz continuous if α = 1. Roughly speaking, one can think of Ho¨lder continuous functions with exponent α as functions with bounded fractional derivatives of the the orderα. n Definition 1.1. Suppose that Ω is an open set inR and 0 α ≤ 1. A function u : Ω →R is uniformly Ho¨lder continuous with exponent α in Ω if the quantity u(x)−u(y) (1.1) u = sup α α,Ω x−y x,y∈ Ω x6=y isfinite. Afunctionu: Ω→RislocallyuniformlyHo¨ldercontinuouswithexponent ′ 0,α αinΩifu isfiniteforeveryΩ ⋐ Ω. WedenotebyC (Ω)thespaceoflocally ′ α,Ω uniformly Ho¨lder continuous functions with exponent α in Ω. If Ω is bounded,  0,α we denote by C Ω the space of uniformly Ho¨lder continuous functions with exponentα in Ω.p 1.4. L SPACES 3 We typically use Greek letters such as α, β both for Ho¨lder exponents and multi-indices; it should be clear from the context which they denote. When α and Ω are understood, we will abbreviate ‘u is (locally) uniformly Ho¨lder continuous with exponentα in Ω’ to ‘u is (locally) Ho¨lder continuous.’ Ifu is Ho¨lder continuous with exponent one, then we say that u is Lipschitz continu- ous. Thereis no purpose in considering Ho¨lder continuous functions with exponent greater than one, since any such function is differentiable with zero derivative and therefore is constant. The quantity u is a semi-norm, but it is not a norm since it is zero for α,Ω  0,α constant functions. The space C Ω , where Ω is bounded, is a Banach space with respect to the norm kuk = supu+u . 0,α C Ω α,Ω ( ) Ω α Example 1.2. For 0 α 1, define u(x) : (0,1)→R by u(x) = x . Then 0,α 0,β u∈C (0,1), but u∈/ C (0,1) for αβ≤ 1. Example 1.3. Thefunctionu(x) :(−1,1)→Rgivenbyu(x) =xisLipschitz 0,1 continuous, but not continuously differentiable. Thus, u ∈ C (−1,1), but u ∈/ 1 C (−1,1). We may also define spaces of continuously differentiable functions whose kth derivative is Ho¨lder continuous. n Definition 1.4. If Ω is an open set inR , k ∈N, and 0 α ≤ 1, then k,α C (Ω) consists of allfunctionsu :Ω→R with continuous partialderivativesin Ω of order less than or equal to k whose kth partial derivatives are locally uniformly Ho¨lder continuous with exponent α in Ω. If the open set Ω is bounded, then  k,α C Ω consists of functions with uniformly continuous partial derivatives in Ω of order less than or equal to k whose kth partial derivatives are uniformly Ho¨lder continuous with exponent α in Ω.  k,α The space C Ω is a Banach space with respect to the norm X X   β β kuk = sup ∂ u + ∂ u k,α C (Ω) α,Ω Ω β≤k β=k p 1.4. L spaces n Asbefore,letΩbeanopensetinR (or,moregenerally,aLebesgue-measurable set). p Definition 1.5. For 1 ≤ p ∞, the space L (Ω) consists of the Lebesgue measurable functions f : Ω→R such that Z p f dx∞, Ω ∞ andL (Ω) consists of the essentially bounded functions. These spaces are Banach spaces with respect to the norms Z  1/p p kfk = f dx , kfk = supf p ∞ Ω Ω4 1. PRELIMINARIES where sup denotes the essential supremum, supf = infM ∈R:f ≤M almost everywhere in Ω. Ω p Strictly speaking, elements of the Banach spaceL are equivalence classes of func- tions that are equal almost everywhere, but we identify a function with its equiva- lenceclassunlessweneedtorefertothepointwisevaluesofaspecificrepresentative. p For example, we say that a function f ∈ L (Ω) is continuous if it is equal almost everywhere to a continuous function, and that it has compact support if it is equal almost everywhere to a function with compact support. Next we summarize some fundamental inequalities for integrals, in addition to p Minkowski’s inequality which is implicit in the statement thatk·k is a norm for L p≥ 1. First, we recall the definition of a convex function. n Definition 1.6. AsetC ⊂R isconvexifλx+(1−λ)y∈C foreveryx,y∈C and everyλ∈0,1. A function φ:C →R is convex if its domainC is convex and φ(λx+(1−λ)y)≤λφ(x)+(1−λ)φ(y) for everyx,y∈C and everyλ∈0,1. Jensen’s inequality states that the value of a convex function at a mean is less than or equal to the mean of the values of the convex function. n Theorem 1.7. Suppose that φ :R→R is a convex function, Ω is a set inR 1 with finite Lebesgue measure, and f ∈L (Ω). Then   Z Z 1 1 φ fdx ≤ φ◦fdx. Ω Ω Ω Ω Tostatethenextinequality,wefirstdefinetheHo¨lderconjugateofanexponent ′ ∗ p. We denote it byp to distinguish it from the Sobolev conjugatep which we will introduce later on. ′ Definition 1.8. The Ho¨lder conjugate ofp∈1,∞ is the quantityp ∈ 1,∞ such that 1 1 + =1, ′ p p with the convention that 1/∞=0. 1 The following result is called Ho¨lder’s inequality. The special case when p = ′ p =1/2 is the Cauchy-Schwartz inequality. ′ p p 1 Theorem 1.9. If 1 ≤ p ≤ ∞, f ∈ L (Ω), and g ∈ L (Ω), then fg ∈ L (Ω) and kfgk ≤kfk kgk . ′ 1 p p Repeated application of this inequality gives the following generalization. Theorem 1.10. If 1≤p ≤∞ for 1≤i≤N satisfy i N X 1 =1 p i i=1 1 1/p p In retrospect, it might have been better to use L spaces instead of L spaces, just as it would’ve been better to use inverse temperature instead of temperature, with absolute zero corresponding to infinite coldness.p 1.4. L SPACES 5 Q N p 1 i and f ∈L (Ω) for 1≤i≤N, then f = f ∈L (Ω) and i i i=1 N Y kfk ≤ kfk . i 1 pi i=1 p Suppose that Ω has finite measure and 1≤q≤p. Iff ∈L (Ω), an application q of Ho¨lder’s inequality to f = 1·f, shows that f ∈L (Ω) and 1/q−1/p kfk ≤Ω kfk . q p p q Thus, the embedding L (Ω)֒→L (Ω) is continuous. This result is not true if the measure of Ω is infinite, but in general we have the following interpolation result. p r q Lemma 1.11. If 1≤p≤q≤r, then L (Ω)∩L (Ω)֒→L (Ω) and θ 1−θ kfk ≤kfk kfk q p r where 0≤θ≤1 is given by 1 θ 1−θ = + . q p r Proof. Assume without loss of generalitythatf ≥ 0. Using Ho¨lder’s inequal- ity with exponents 1/σ and 1/(1−σ), we get     Z Z Z Z σ 1−σ q θq (1−θ)q θq/σ (1−θ)q/(1−σ) f dx = f f dx≤ f dx f dx . Choosingσ/θ =q/p, in which case (1−σ)/(1−θ) =q/r, we get Z Z  Z  qθ/p q(1−θ)/r q p r f dx≤ f dx f dx and the result follows.  p It is often useful to consider local L spaces consisting of functions that have finite integral on compact sets. p Definition 1.12. The space L (Ω), where 1≤p≤∞, consists of functions loc p ′ ′ f :Ω→Rsuchthatf ∈L (Ω)foreveryopensetΩ ⋐ Ω. Asequenceoffunctions p p ′ f converges to f in L (Ω) if f converges to f in L (Ω) for every open set n n loc ′ Ω ⋐ Ω. q p If p q, then L (Ω) ֒→ L (Ω) even if the measure of Ω is infinite. Thus, loc loc 1 L (Ω) is the ‘largest’ space of integrable functions on Ω. loc n Example 1.13. Considerf :R →R defined by 1 f(x) = a x 1 n where a∈R. Then f ∈L (R ) if and only if an. To prove this, let loc  f(x) ifxǫ, ǫ f (x) = 0 ifx≤ǫ.6 1. PRELIMINARIES ǫ Thenf is monotone increasing and converges pointwise almost everywhere tof + as ǫ→0 . For any R0, the monotone convergence theorem implies that Z Z ǫ fdx = lim f dx + ǫ→0 B (0) B (0) R R Z R n−a−1 = lim r dr + ǫ→0 ǫ  ∞ if n−a≤ 0, = −1 n−a (n−a) R if n−a 0, p n which proves the result. The function f does not belong to L (R ) for 1≤p∞ p for anyvalue ofa, since the integraloff divergesat infinity whenever it converges at zero. 1.5. Compactness Compactness results play a central role in the analysis of PDEs. Typically, we construct a sequence of approximate solutions of a PDE and show that they belongtoacompactset. Wethenextractaconvergentsubsequenceofapproximate solutions and attempt to show that their limit is a solution of the original PDE. There are two main types of compactness — weak and strong compactness. We begin with criteria for strong compactness. A subset F of a metric space X is precompact if the closure of F is compact; equivalently, F is precompact if every sequence in F has a subsequence that con- verges in X. The Arzel`a-Ascoli theorem gives a basic criterion for compactness in function spaces: namely, a set of continuous functions on a compact metric space is precompact if and only if it is bounded and equicontinuous. We state the result explicitly for the spaces of interest here. n Theorem 1.14. Suppose that Ω is a bounded open set inR . A subset F of  C Ω , equipped with the maximum norm, is precompact if and only if: (1) there exists a constant M such that kfk ≤M for all f ∈F; ∞ (2) for every ǫ 0 there exists δ 0 such that if x,x+h ∈ Ω and h δ then f(x+h)−f(x)ǫ for all f ∈F. Thefollowingtheorem(knownvariouslyastheRiesz-Tamarkin,orKolmogorov- Riesz, or Fr´echet-Kolmogorov theorem) gives conditions analogous to the ones in p n the Arzel`a-Ascoli theorem for a set to be precompact in L (R ), namely that the p set is bounded, ‘tight,’ andL -equicontinuous. For a proof, see 44. p n Theorem 1.15. Let 1 ≤ p ∞. A subset F of L (R ) is precompact if and only if: (1) there exists M such that p kfk ≤M for all f ∈F; L (2) for every ǫ 0 there exists R such that 1/p Z p f(x) dx ǫ for all f ∈F. xR1.5. COMPACTNESS 7 (3) for every ǫ 0 there exists δ 0 such that if hδ,   Z 1/p p f(x+h)−f(x) dx ǫ for all f ∈F. n R The ‘tightness’ condition (2) prevents the functions from escaping to infinity. Example 1.16. Define f :R→R by f =χ . The set f :n∈N is n n n (n,n+1) p bounded and equicontinuous inL (R) for any 1≤p∞, but it is not precompact sincekf −f k = 2 if m6=n, nor is it tight since m n p Z ∞ p f dx = 1 for all n≥R. n R The equicontinuity conditions in the hypotheses of these theorems for strong compactness are not always easy to verify; typically, one does so by obtaining a uniform estimate for the derivatives of the functions, as in the Sobolev-Rellich embedding theorems. As we explain next, weak compactness is easier to verify, since we only need to show that the functions themselves are bounded. On the other hand, we get subsequences that converge weakly and not necessarily strongly. This can create difficulties, especiallyfor nonlinear problems,sincenonlinear functions arenotcon- tinuous with respect to weak convergence. ∗ ′ Let X be a real Banach space and X (which we also denote by X ) the dual space of bounded linear functionals on X. We denote the duality pairing between ∗ ∗ X andX byh·,·i :X ×X →R. Definition 1.17. A sequence x in X converges weakly to x∈X, written n ∗ ∗ x ⇀ x, if hω,x i → hω,xi for every ω ∈ X . A sequence ω in X converges n n n ∗ ∗ weak-star to ω∈X , written ω ⇀ω ifhω ,xi→hω,xi for every x∈X. n n ∗∗ IfX is reflexive,meaning thatX =X, then weakand weak-starconvergence are equivalent. n p ∗ Example 1.18. If Ω ⊂R is an open set and 1 ≤ p ∞, then L (Ω) = ′ p p p L (Ω). Thus a sequence of functionsf ∈L (Ω) converges weakly tof ∈L (Ω) if n Z Z ′ p (1.2) f gdx→ fgdx for every g∈L (Ω). n Ω Ω ′ ∞ ∗ 1 ∞ 1 ∗ If p = ∞ and p = 1, then L (Ω) 6= L (Ω) but L (Ω) = L (Ω) . In that case, ∞ (1.2) defines weak-star convergence in L (Ω). AsubsetE ofaBanachspaceX is(sequentially)weakly,orweak-star,precom- pact if every sequence inE has a subsequence that converges weakly, or weak-star, in X. The following Banach-Alagolu theorem characterizes weak-star precompact subsetsofaBanachspace;itmaybethoughtofasgeneralizationoftheHeine-Borel theorem to infinite-dimensional spaces. Theorem 1.19. A subset of a Banach space is weak-star precompact if and only if it is bounded. IfX is reflexive,thenboundedsets areweaklyprecompact. Thisresultapplies, in particular, to Hilbert spaces.8 1. PRELIMINARIES Example 1.20. Let H be a separable Hilbert space with inner-product (·,·) and orthonormal basis e : n ∈N. The sequence e is bounded in H, but it n n √ has no stronglyconvergentsubsequencesinceke −e k= 2 for everyn6=m. On n m P the other hand, the sequence converges weakly in H to zero: if x = x e ∈ H n n P 2 2 then (x,e ) =x →0 as n→∞ sincekxk = x ∞. n n n 1.6. Averages n For x∈R andr 0, let n B (x) =y∈R :x−yr r denote the open ball centered at x with radius r, and n ∂B (x) =y∈R :x−y=r r the corresponding sphere. n The volume of the unit ball inR is given by n/2 2π α = n nΓ(n/2) where Γ is the Gamma function, which satisfies √ Γ(1/2)= π, Γ(1) =1, Γ(x+1)=xΓ(x). Thus, for example, α = π and α = 4π/3. An integration with respect to polar 2 3 coordinates shows that the area of the (n−1)-dimensional unit sphere is nα . n 1 We denote the averageof a functionf ∈L (Ω) over a ballB (x)⋐ Ω, or the r loc corresponding sphere ∂B (x), by r Z Z Z Z 1 1 (1.3) − fdx = fdx, − fdS = fdS. n n−1 α r nα r n n B (x) B (x) ∂B (x) ∂B (x) r r r r If f is continuous at x, then Z lim − fdx =f(x). + r→0 B (x) r The following result, called the Lebesgue differentiation theorem, implies that the averages of a locally integrable function converge pointwise almost everywhere to the function as the radiusr shrinks to zero. 1 n Theorem 1.21. If f ∈L (R ) then loc Z (1.4) lim − f(y)−f(x) dx =0 + r→0 B (x) r n pointwise almost everywhere for x∈R . n A point x ∈R for which (1.4) holds is called a Lebesgue point of f. For a proof of this theorem (using the Wiener covering lemma and the Hardy-Littlewood maximal function) see Folland 11 or Taylor 42.1.7. CONVOLUTIONS 9 1.7. Convolutions n Definition 1.22. If f,g :R →R are measurable function, we define the n convolutionf∗g :R →R by Z (f∗g)(x) = f(x−y)g(y)dy n R n provided that the integral converges for x pointwise almost everywhere inR . When defined, the convolution product is both commutative and associative, f∗g =g∗f, f∗(g∗h)= (f∗g)∗h. In many respects, the convolution of two functions inherits the best properties of both functions. n n If f,g∈C (R ), then their convolution also belongs to C (R ) and c c supp(f ∗g)⊂suppf +suppg. n n n If f ∈ C (R ) and g ∈ C(R ), then f ∗g ∈ C(R ) is defined, however rapidly c g grows at infinity, but typically it does not have compact support. If neither f nor g have compact support, then we need some conditions on their growth or decay at infinity to ensure that the convolution exists. The following result, called p Young’s inequality, givesconditions for theconvolutionofL functions to exist and estimates its norm. Theorem 1.23. Suppose that 1≤p,q,r≤∞ and 1 1 1 = + −1. r p q p n q n r n If f ∈L (R ) and g∈L (R ), then f∗g∈L (R ) and kf∗gk ≤kfk kgk . r p q L L L The following special cases are useful to keep in mind. ′ Example 1.24. If p =q = 2, or more generally if q =p , then r =∞. In this n case, the result follows from the Cauchy-Schwartz inequality, since for allx∈R Z f(x−y)g(y)dx ≤kfk 2kgk 2. L L n n Moreover, a density argument shows that f∗g∈C (R ): Choosef ,g ∈C (R ) 0 k k c 2 n n such that f → f, g → g in L (R ), then f ∗g ∈ C (R ) and f ∗g → f ∗g k k k k c k k uniformly. AsimilarargumentisusedintheproofoftheRiemann-Lebesguelemma n 1 n ˆ thatf ∈C (R ) if f ∈L (R ). 0 Example 1.25. If p =q = 1, then r = 1, and the result follows directly from Fubini’s theorem, since    Z Z Z Z Z f(x−y)g(y)dy dx≤ f(x−y)g(y)dxdy = f(x) dx g(y)dy . 1 n Thus, the space L (R ) is an algebra under the convolution product. The Fourier 1 transformmaps theconvolutionproductoftwoL -functions to thepointwiseprod- uct of their Fourier transforms. Example 1.26. If q = 1, then p = r. Thus, convolution with an integrable 1 n p n function k∈L (R ) is a bounded linear mapf 7→k∗f on L (R ).10 1. PRELIMINARIES 1.8. Derivatives and multi-index notation We define the derivative of a scalar field u :Ω→R by   ∂u ∂u ∂u Du = , ,..., . ∂x ∂x ∂x 1 2 n We will also denote the ith partial derivative by ∂ u, the ijth derivative by ∂ u, i ij n and so on. The divergence of a vector field X =(X ,X ,...,X ):Ω→R is 1 2 n ∂X ∂X ∂X 1 2 n divX = + +···+ . ∂x ∂x ∂x 1 2 n LetN = 0,1,2,... denote the non-negative integers. An n-dimensional 0 n multi-index is a vectorα∈N , meaning that 0 α= (α ,α ,...,α ), α =0,1,2,.... 1 2 n i We write α=α +α +···+α , α =α α ...α . 1 2 n 1 2 n We define derivatives and powers of orderα by ∂ ∂ ∂ α α α α α 1 2 n ∂ = ... , x =x x ...x . 1 2 n α α α 1 2 n ∂x ∂x ∂x If α = (α ,α ,...,α ) and β = (β ,β ,...,β ) are multi-indices, we define the 1 2 n 1 2 n multi-index (α+β) by α+β = (α +β ,α +β ,...,α +β ). 1 1 2 2 n n n Wedenotebyχ (k)thenumberofmulti-indicesα∈N withorder0≤α≤k, n 0 and by χ˜ (k) the number of multi-indices with orderα =k. Then n (n+k) (n+k−1) χ (k) = , χ˜ (k) = n n nk (n−1)k 1.8.1. Taylor’s theorem for functions of several variables. The multi- index notation provides a compact way to write the multinomial theorem and the Taylor expansion of a function of several variables. The multinomial expansion of a power is     X X k k k α i α (x +x +···+x ) = x = x 1 2 n i α α ...α α 1 2 n α +...α =k α=k 1 n where the multinomial coefficient of a multi-index α = (α ,α ,...,α ) of order 1 2 n α=k is given by     k k k = = . α α α ...α α α ...α 1 2 n 1 2 n k Theorem 1.27. Suppose that u∈C (B (x)) and h∈B (0). Then r r α X ∂ u(x) α u(x+h)= h +R (x,h) k α α≤k−1 where the remainder is given by α X ∂ u(x+θh) α R (x,h)= h k α α=k for some 0θ 1.1.9. MOLLIFIERS 11 Proof. Letf(t)=u(x+th) for 0≤t≤1. Taylor’s theorem for a function of a single variable implies that k−1 j k X 1 d f 1 d f f(1)= (0)+ (θ) j k j dt k dt j=0 for some 0θ 1. By the chain rule, n X df =Du·h= h ∂u, i i dt i=1 and the multinomial theorem gives k   n k X X d n α α = h ∂ = h ∂ . i i k dt α i=1 α=k Using this expression to rewrite the Taylor series for f in terms of u, we get the result.  A function u : Ω→R is real-analytic in an open set Ω if it has a power-series expansion that converges to the function in a ball of non-zero radius about every ω point of its domain. We denote by C (Ω) the space of real-analytic functions on ∞ Ω. A real-analytic function is C , since its Taylor series can be differentiated ∞ term-by-term, but aC function need not be real-analytic. For example, see (1.5) below. 1.9. Mollifiers The function    2 Cexp −1/(1−x ) ifx 1 (1.5) η(x) = 0 ifx≥ 1 ∞ n belongs to C (R ) for any constantC. We chooseC so that c Z ηdx =1 n R and for any ǫ0 define the function   1 x ǫ (1.6) η (x) = η . n ǫ ǫ ǫ ∞ Then η is a C -function with integral equal to one whose support is the closed ballB (0). We refer to (1.6) as the ‘standard mollifier.’ ǫ We remark that η(x) in (1.5) is not real-analytic when x = 1. All of its derivatives are zero at those points, so the Taylor series converges to zero in any neighborhood, not to the original function. The only function that is real-analytic with compact support is the zero function. In rough terms, an analytic function is a single ‘organic’ entity: its values in, for example, a single open ball determine its values everywhere in a maximal domain of analyticity (which in the case of one complex variable is a Riemann surface) through analytic continuation. The ∞ behavior of a C -function at one point is, however, completely unrelated to its behavior at another point. 1 Suppose that f ∈L (Ω) is a locally integrable function. For ǫ0, let loc ǫ (1.7) Ω =x∈Ω:dist(x,∂Ω)ǫ12 1. PRELIMINARIES ǫ ǫ and define f :Ω →R by Z ǫ ǫ (1.8) f (x) = η (x−y)f(y)dy Ω ǫ ǫ ǫ where η is the mollifier in (1.6). We define f for x∈ Ω so that B (x)⊂ Ω and ǫ n ǫ n ǫ we have room to averagef. If Ω =R , we have simply Ω =R . The function f is a smooth approximation of f. p Theorem 1.28. Suppose that f ∈ L (Ω) for 1≤ p ∞, and ǫ 0. Define loc ǫ ǫ ǫ ∞ ǫ ǫ f : Ω →R by (1.8). Then: (a) f ∈ C (Ω ) is smooth; (b) f → f pointwise p + ǫ + almost everywhere in Ω as ǫ→ 0 ; (c) f →f in L (Ω) as ǫ→0 . loc ǫ Proof. Thesmoothnessoff followsbydifferentiationundertheintegralsign Z α ǫ α ǫ ∂ f (x) = ∂ η (x−y)f(y)dy Ω which may be justified by use of the dominated convergence theorem. The point- wise almost everywhere convergence (at every Lebesgue point of f) follows from p the Lebesgue differentiation theorem. The convergence in L follows by the ap- loc proximation of f by a continuous function (for which the result is easy to prove) ǫ and the use of Young’s inequality, since kη k 1 = 1 is bounded independently of L ǫ.  ∞ One consequence of this theorem is that the space of test functions C (Ω) is c p dense in L (Ω) for 1≤ p ∞. Note that this is not true when p = ∞, since the uniform limit of smooth test functions is continuous. 1.9.1. Cutoff functions. ′ n Theorem 1.29. Suppose that Ω ⋐ Ω are open sets inR . Then there is a ∞ ′ function φ∈C (Ω) such that 0≤φ≤ 1 and φ= 1 on Ω . c ′ Proof. Let δ =dist(Ω,∂Ω) and define ′′ ′ Ω =x∈ Ω:dist(x,Ω )δ/2. ′′ δ/4 ǫ Let χ be the characteristic function of Ω , and define φ =η ∗χ where η is the standard mollifier. Then one may verify that φ has the required properties.  We refer to a function with the properties in this theorem as a cutoff function. ′ n Example 1.30. If 0 r R and Ω = B (0), Ω = B (0) are balls inR , r R then the corresponding cut-off function φ satisfies C Dφ≤ R−r where C is a constant that is independent of r, R. 1.9.2. Partitions of unity. Partitions of unity allow us to piece together global results from local results. n Theorem 1.31. Suppose that K is a compact set inR which is covered by a finite collection Ω ,Ω ,...,Ω of open sets. Then there exists a collection of 1 2 N P N ∞ functions η ,η ,...,η such that 0≤η ≤ 1, η ∈C (Ω ), and η = 1 on 1 2 N i i i i c i=1 K.1.10. BOUNDARIES OF OPEN SETS 13 We callη a partition of unity subordinate to the coverΩ. To prove this i i result, we use Urysohn’s lemma to construct a collection of continuous functions with the desired properties, then use mollification to obtain a collection of smooth functions. 1.10. Boundaries of open sets WhenweanalyzesolutionsofaPDEintheinterioroftheirdomainofdefinition, wecanoftenconsiderdomainsthatarearbitraryopensetsandanalyzethesolutions in a sufficiently small ball. In order to analyze the behavior of solutions at a boundary, however, we typically need to assume that the boundary has some sort ofsmoothness. Inthissection,wedefinethesmoothnessoftheboundaryofanopen set. Wealsoexplainbrieflyhowonedefines analyticallythenormalvector-fieldand the surface area measure on a smooth boundary. In general, the boundary of an open set may be complicated. For example, it can have nonzero Lebesgue measure. Example 1.32. Let q : i ∈N be an enumeration of the rational numbers i q ∈ (0,1). For each i ∈N, choose an open interval (a ,b ) ⊂ (0,1) that contains i i i q , and let i Ω= (a ,b ). i i i∈N The Lebesgue measure of Ω 0 is positive, but we can make it as small as we −i wish; for example, choosing b −a = ǫ2 , we get Ω ≤ ǫ. One can check that i i ∂Ω= 0,1\Ω. Thus, ifΩ 1, then ∂Ω has nonzero Lebesgue measure. Moreover,an open set, or domain, need not lie on one side of its boundary (we ◦ say that Ω lies on one side of its boundary if Ω =Ω), and corners, cusps, or other singularities in the boundary cause analytical difficulties. 2 Example 1.33. The unit disc inR with the nonnegativex-axis removed,   2 2 2 2 Ω= (x,y)∈R :x +y 1 \ (x,0)∈R : 0≤x 1 , does not lie on one side of its boundary. In rough terms, the boundary of an open set is smooth if it can be ‘flattened out’ locally by a smooth map. Definition 1.34. Suppose that k ∈N. A map φ :U →V between open sets n k −1 U, V inR is a C -diffeomorphism if it one-to-one, onto, and φ and φ have continuous derivatives of order less than or equal to k. n n Note that the derivative Dφ(x) :R →R of a diffeomorphism φ : U → V is −1 −1 an invertible linear map for every x∈U, with Dφ(x) = (Dφ )(φ(x)). n Definition 1.35. Let Ω be a bounded open set inR andk∈N. We say that k k the boundary ∂Ω is C , or that Ω is C for short, if for every x ∈ Ω there is an n n k open neighborhood U ⊂R of x, an open set V ⊂R , and a C -diffeomorphism φ:U →V such that φ(U ∩Ω)=V ∩y 0, φ(U ∩∂Ω)=V ∩y = 0 n n n where (y ,...,y ) are coordinates in the image spaceR . 1 n14 1. PRELIMINARIES ∞ ∞ If φ is a C -diffeomorphism, then we say that the boundary is C , with an analogous definition of a Lipschitz or analytic boundary. k n Inotherwords,thedefinitionsaysthataC opensetinR is ann-dimensional k C -manifoldwithboundary. ThemapsφinDefinition1.35arecoordinatechartsfor the manifold. It follows from the definition that Ω lies on one side of its boundary n andthat∂Ωisanoriented(n−1)-dimensionalsubmanifoldofR withoutboundary. The standard orientation is given by the outward-pointing normal (see below). Example 1.36. The open set  2 Ω= (x,y)∈R :x 0,ysin(1/x) 1 lies on one side of its boundary, but the boundary is not C since there is no coordinatechartoftherequiredformfortheboundarypoints(x,0):−1≤x≤1. 1.10.1. Open sets in the plane. A simple closed curve, or Jordan curve, Γ is a set in the plane that is homeomorphic to a circle. That is, Γ = γ(T) 2 is the image of a one-to-one continuous map γ :T →R with continuous inverse −1 γ : Γ→T. (Therequirementthattheinverseiscontinuousfollowsfromtheother assumptions.) According to the Jordan curve theorem, a Jordan curve divides the 2 plane into two disjoint connected open sets, so thatR \ Γ = Ω ∪ Ω . One of 1 2 the sets (the ‘interior’) is bounded and simply connected. The interior region of a Jordan curve is called a Jordan domain. Example 1.37. The slit disc Ω in Example 1.33 is not a Jordan domain. For example, its boundary separates into two nonempty connected components when the point (1,0) is removed, but the circle remains connected when any point is removed, so ∂Ω cannot be homeomorphic to the circle. Example 1.38. The interior Ω of the Koch, or ‘snowflake,’ curve is a Jordan domain. The Hausdorff dimension of its boundaryis strictly greaterthan one. It is interesting to note that, despite the irregular nature of its boundary, this domain k,p has the property that every function in W (Ω) with k ∈N and 1 ≤ p ∞ can k,p 2 be extended to a function in W (R ). 2 1 If γ :T →R is one-to-one, C , and Dγ 6= 0, then the image of γ is the 1 C boundary of the open set which it encloses. The condition that γ is one-to- one is necessary to avoid self-intersections (for example, a figure-eight curve), and the condition that Dγ 6= 0 is necessary in order to ensure that the image is a 1 2 C -submanifold ofR .  2 3 1 Example 1.39. The curveγ :t7→ t ,t is notC att= 0 whereDγ(0)= 0. 1.10.2. Parametric representation of a boundary. If Ω is an open set in n k R with C -boundary and φ is a chart on a neighborhoodU of a boundary point, as in Definition 1.35, then we can define a local chart n n−1 Φ=(Φ ,Φ ,...,Φ ):U∩∂Ω⊂R →W ⊂R 1 2 n−1 forthe boundary∂ΩbyΦ=(φ ,φ ,...,φ ). Thus,∂Ωis an(n−1)-dimensional 1 2 n−1 n submanifold ofR . The boundary is parameterized locally byx = Ψ (y ,y ,...,y ) where 1≤ i i 1 2 n−1 −1 i≤n and Ψ = Φ :W →U ∩∂Ω. The (n−1)-dimensional tangent space of ∂Ω is spanned by the vectors ∂Ψ ∂Ψ ∂Ψ , ,..., . ∂y ∂y ∂y 1 2 n−11.10. BOUNDARIES OF OPEN SETS 15 n−1 n The outward unit normalν :∂Ω→S ⊂R is orthogonal to this tangent space, and it is given locally by ν˜ ∂Ψ ∂Ψ ∂Ψ ν = , ν˜= ∧ ∧···∧ , ν˜ ∂y ∂y ∂y 1 2 n−1 ∂Ψ /∂y ∂Ψ /∂Ψ ... ∂Ψ /∂y 1 1 1 2 1 n−1 ... ... ... ... ∂Ψ /∂y ∂Ψ /∂y ... ∂Ψ /∂y i−1 1 i−1 2 i−1 n−1 ν˜ = . i ∂Ψ /∂y ∂Ψ /∂y ... ∂Ψ /∂y i+1 1 i+1 2 i+1 n−1 ... ... ... ... ∂Ψ /∂y ∂Ψ /∂y ... ∂Ψ /∂y n 1 n 2 n n−1 Example1.40. Forathree-dimensionalregionwithtwo-dimensionalboundary, the outward unit normal is (∂Ψ/∂y )×(∂Ψ/∂y ) 1 2 ν = . (∂Ψ/∂y )×(∂Ψ/∂y ) 1 2 n The restriction of the Euclidean metric onR to the tangent space of the boundary gives a Riemannian metric on the boundary whose volume form defines the surface measure dS. Explicitly, the pull-back of the Euclidean metric n X 2 dx i i=1 to the boundary under the mapping x= Ψ(y) is the metric n n−1 X X ∂Ψ ∂Ψ i i dy dy . p q ∂y ∂y p q i=1 p,q=1 P The volume form associated with a Riemannian metric h dy dy is pq p q √ dethdy dy ...dy . 1 2 n−1 Thus the surface measure on ∂Ω is given locally by p t dS = det(DΨ DΨ)dy dy ...dy 1 2 n−1 where DΨ is the derivative of the parametrization,   ∂Ψ /∂y ∂Ψ /∂y ... ∂Ψ /∂y 1 1 1 2 1 n−1   ∂Ψ /∂y ∂Ψ /∂y ... ∂Ψ /∂y 2 1 2 2 2 n−1   DΨ = .   ... ... ... ... ∂Ψ /∂y ∂Ψ /∂y ... ∂Ψ /∂y n 1 n 2 n n−1 These local expressions may be combined to give a global definition of the surface integral by means of a partition of unity. Example 1.41. In the case of a two-dimensional surface with metric 2 2 2 ds =Edy +2Fdy dy +Gdy , 1 2 1 2 the element of surface area is p 2 dS = EG−F dy dy . 1 216 1. PRELIMINARIES Example 1.42. The two-dimensional sphere  2 3 2 2 2 S = (x,y,z)∈R :x +y +z =1 ∞ 3 ∞ is a C submanifold ofR . A localC -parametrization of  2 3 U =S \ (x,0,z)∈R :x≥0 2 2 is given by Ψ :W ⊂R →U ⊂S where Ψ(θ,φ) = (cosθsinφ,sinθsinφ,cosφ)  3 W = (θ,φ)∈R :0θ2π, 0φπ . The metric on the sphere is  ∗ 2 2 2 2 2 2 Ψ dx +dy +dz = sin φdθ +dφ and the corresponding surface area measure is dS =sinφdθdφ. The integralof a continuous functionf(x,y,z) over the sphere that is supported in U is then given by Z Z fdS = f (cosθsinφ,sinθsinφ,cosφ)sinφdθdφ. 2 S W We may use similar rotated charts to cover the points with x≥ 0 and y = 0. 1.10.3. Representation of a boundary as a graph. An alternative, and computationally simpler, way to represent the boundary of a smooth open set is as a graph. After rotating coordinates, if necessary, we may assume that the nth component of the normal vector to the boundary is nonzero. If k≥1, the implicit k function theorem implies that we may represent a C -boundary as a graph x =h(x ,x ,...,x ) n 1 2 n−1 n−1 k whereh:W ⊂R →RisinC (W)andΩisgivenlocallybyx h(x ,...,x ). n 1 n−1 If the boundary is only Lipschitz, then the implicit function theorem does not ap- ply, and it is not always possible to represent a Lipschitz boundary locally as the region lying below the graph of a Lipschitz continuous function. 1 If ∂Ω is C , then the outward normalν is given in terms of h by   1 ∂h ∂h ∂h ν =p − ,− ,...,− ,1 2 ∂x ∂x ∂x 1+Dh 1 2 n−1 and the surface area measure on ∂Ω is given by p 2 dS = 1+Dh dx dx ...dx . 1 2 n−1 n Example 1.43. Let Ω=B (0) be the unit ball inR and∂Ω the unit sphere. 1 The upper hemisphere H =x∈∂Ω:x 0 n ′ is the graph ofx =h(x ) where h:D→R is given by n q  2 ′ ′ n−1 ′ ′ h(x )= 1−x , D = x ∈R :x 11.12. DIVERGENCE THEOREM 17 ′ ′ n−1 and we write x = (x,x ) with x = (x ,...,x ) ∈R . The surface measure n 1 n−1 onH is 1 ′ q dS = dx 2 ′ 1−x and the surface integral of a function f(x) overH is given by Z Z ′ ′ f (x,h(x)) ′ q fdS = dx. 2 H D ′ 1−x The integral of a function over∂Ω may be computed in terms of such integrals by use of a partition of unity subordinate to an atlas of hemispherical charts. 1.11. Change of variables 1 We state a theorem for a C change of variables in the Lebesgue integral. A special case is the change of variables from Cartesian to polar coordinates. For proofs, see 11, 42. n n Theorem 1.44. Suppose that Ω is an open set inR and φ : Ω →R is a 1 C diffeomorphism of Ω onto its image φ(Ω). If f : φ(Ω) →R is a nonnegative Lebesgue measurable function or an integrable function, then Z Z f(y)dy = f◦φ(x)detDφ(x) dx. φ(Ω) Ω n We define polar coordinates inR \0 by x = ry, where r = x 0 and y∈∂B (0) is a point on the unit sphere. In these coordinates, Lebesgue measure 1 has the representation n−1 dx =r drdS(y) wheredS(y) is the surface area measure on the unit sphere. We have the following result for integration in polar coordinates. n Proposition 1.45. If f :R →R is integrable, then " Z Z Z ∞ n−1 fdx = f(x+ry) dS(y) r dr 0 ∂B (0) 1   Z Z ∞ n−1 = f (x+ry) r dr dS(y). ∂B (0) 0 1 1.12. Divergence theorem We state the divergence (or Gauss-Green) theorem. n 1 n Theorem 1.46. Let X : Ω →R be a C (Ω)-vector field, and Ω ⊂R a 1 bounded open set with C -boundary ∂Ω. Then Z Z divXdx = X·νdS. Ω ∂Ω To prove the theorem, we prove it for functions that are compactly supported 1 in a half-space, show that it remains valid under a C change of coordinates with the divergence defined in an appropriately invariant way, and then use a partition of unity to add the results together.18 1. PRELIMINARIES 1 Ω), then an application of the divergence theorem In particular, if u,v ∈ C ( to the vector field X = (0,0,...,uv,...,0), with ith component uv, gives the integration by parts formula Z Z Z u(∂v) dx =− (∂ u)vdx+ uvν dS. i i i Ω Ω ∂Ω The statement in Theorem 1.46 is, perhaps, the natural one from the perspec- tive of smooth differential geometry. The divergence theorem, however, remains valid under weaker assumptions than the ones in Theorem 1.46. For example, it 1 applies to a cube, whoseboundaryis notC , as well as to other sets with piecewise smooth boundaries. From the perspective of geometric measure theory, a general form of the diver- gence theorem holds for Lipschitz vector fields (vector fields whose weak derivative ∞ belongs to L ) and sets of finite perimeter (sets whose characteristic function has bounded variation). The surface integral is taken over a measure-theoretic bound- arywithrespectto(n−1)-dimensionalHausdorffmeasure,andameasure-theoretic normal exists almost everywhere on the boundary with respect to this measure 10, 45. 1.13. Gronwall’s inequality InestimatingsomenormofasolutionofaPDE,weareoftenledtoadifferential inequality for the norm from which we want to deduce an inequality for the norm itself. Gronwall’sinequalityallowsonetodo this: roughlyspeaking,itstatesthata solution of a differential inequality is bounded by the solution of the corresponding differential equality. There are both linear and nonlinear versions of Gronwall’s inequality. We state only the simplest version of the linear inequality. Lemma 1.47. Suppose that u : 0,T → 0,∞) is a nonnegative, absolutely continuous function such that du (1.9) ≤Cu, u(0)=u . 0 dt for some constants C, u ≥0. Then 0 Ct u(t)≤u e for 0≤t≤T. 0 −Ct Proof. Let v(t)=e u(t). Then   dv du −Ct =e −Cu(t) ≤ 0. dt dt If follows that Z t dv v(t)−u = ds≤ 0, 0 ds 0 −Ct ore u(t)≤u , which proves the result.  0 In particular, if u = 0, it follows that u(t) = 0. We can alternatively write 0 (1.9) in the integral form Z t u(t)≤u +C u(s)ds. 0 0CHAPTER 2 Laplace’s equation There can be but one option as to the beauty and utility of this analysisbyLaplace;butthemannerinwhichithashithertobeen presented has seemed repulsive to the ablest mathematicians, 1 and difficult to ordinary mathematical students. Laplace’s equation is Δu= 0 where the Laplacian Δ is defined in Cartesian coordinates by 2 2 2 ∂ ∂ ∂ Δ= + +···+ . 2 2 2 ∂x ∂x ∂x 1 2 n We may also write Δ = divD. The Laplacian Δ is invariant under translations n (it has constant coefficients) and orthogonal transformations ofR . A solution of Laplace’s equation is called a harmonic function. Laplace’s equation is a linear, scalar equation. It is the prototype of an elliptic partial differential equation, and many of its qualitative properties are shared by more general elliptic PDEs. The non-homogeneous version of Laplace’s equation −Δu =f is called Poisson’s equation. It is convenient to include a minus sign here because Δ is a negative definite operator. The Laplace and Poisson equations, and their generalizations, arise in many different contexts. (1) Potential theory e.g. in the Newtonian theory of gravity, electrostatics, heat flow, and potential flows in fluid mechanics. (2) Riemannian geometry e.g. the Laplace-Beltrami operator. (3) Stochastic processes e.g. the stationary Kolmogorovequation for Brown- ian motion. (4) Complexanalysise.g. therealandimaginarypartsofananalyticfunction of a single complex variable are harmonic. As with any PDE,we typically wantto find solutions of the Laplaceor Poisson equation that satisfy additional conditions. For example, if Ω is a bounded domain n inR ,thentheclassicalDirichletproblemforPoisson’sequationistofindafunction  2 u:Ω→R such that u∈C (Ω)∩C Ω and −Δu =f in Ω, (2.1) u =g on∂Ω. 1 Kelvin and Tait, Treatise on Natural Philosophy, 1879 19

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.