MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

Transcription

1 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further detils see ssocited texts on function spces. You my hve lso seen some of this mteril in MATH20122 Metric Spces. Look bck t those notes if you need to do so. However I stress tht this course is methods course nd therefore this mteril is minly bckground mteril necessry to put lter (more pplied) mteril on sound bsis. Note however tht this mteril is exminble - so mke sure you understnd it, especilly prts relting to opertors from pge 4. onwrds. Definition 1.1: A set S of functions forms liner function (vector) spce if 1. f, g S nd α, β R (or C) then αf + βg S, 2. f + g = g + f nd f + (g + h) = (f + g) + h, f, g, h S, 3. zero element 0 such tht f + 0 = f, f S, 4. f S, n element ( f) such tht f + ( f) = 0, 5. (αβ)f = α(βf), f S nd α, β R (or C), 6. (α + β)f = αf + βf nd α(f + g) = αf + αg, f, g S nd α, β R (or C), 7. n element (identity) I S such tht I f = f, f S, with 1. being key property in this course. Exmples: C (, b) - set of ll continuous functions defined on [, b]. C n (, b) - set of ll functions with continuous nth derivtives defined on [, b]. L 2 (, b) - set of ll (Lebesgue) squre integrble functions ( f (x) 2 dx is bounded). Definition 1.2: An inner product spce is liner function spce on which there is defined n inner (sclr) product f, g R (or C) such tht (i) g, f = f, g - rel, (or g, f = f, g - complex - br denotes complex conjugte,) (ii) g, αf 1 + βf 2 = α g, f 1 + β g, f 2, (iii) f, f 0 with equlity f = 0. Notes: () Definition (1.2) extends notion of orthogonlity (but not ngle) to functions, (b) f, f is rel, (c) αf 1 + βf 2, g = ᾱ f 1, g + β f 2, g, (d) If g, f = 0 for ll g S then f = 0, (proof: choose g = f nd use (iii) bove).

2 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 2 Definition 1.3: A normed spce is liner function spce on which there is defined norm f R such tht (i) f 0, (ii) f = 0 f = 0, (iii) αf = α f, (iv) f + g f + g, (tringle inequlity). Note: the norm extends the notion of length, distnce or closeness to functions. It cn be defined in wide vriety of wys. For exmple, we could hve the supremum or uniform norm: f = mx f(x). x [,b] Proposition 1.4: A norm f for function f (x) my be defined to be the non-negtive rel number f = + f, f i.e. we get norm from n inner product. (Note: Cn check tht this is norm.) In this wy it is cler tht we cn lwys define norm from n inner product. However we cnnot lwys define n inner product given norm. In fct it trnspires tht necessry nd sufficient condition tht norm gives rise to n inner product is tht the so-clled Prllelogrm identity must hold, i.e. u + v 2 + u v 2 = 2( u 2 + v 2 ). For finite dimensionl vector spce tke two elements v = (v 1, v 2,...,v n ), w = (w 1, w 2,...,w n ), then the inner product is just the nturl extension of the simple sclr or dot product you hve seen in simple 2-D nd 3-D vector theory to n dimensions: v,w = v w =v 1 w 1 + v 2 w v n w n. Functions re like vectors with infinitely mny components ech component being the function s vlue t distinct point in the dependent vrible. So, imgine tking n intervl of the rel line [, b], nd divide it up into n equl pieces so tht x 0 (= ), x 1, x 2,...,x n (= b) represent the eqully spced points t the ends of ll the line segments. The function f(x) is thus pproximted on [, b] by the vector (f(x 1 ), f(x 2 ), f(x 3 ),...,f(x n )) nd s n the pproximtion gets better. In the simplest cses for function spces we define the inner product s g, f = g, f = f (x) g (x) dx, - relf, g, (1.1) f (x) g (x)dx, - complexf, g, Cre must be tken with regrd to representing functions s infinite vector spces the former hs dependent vrible with n uncountbly infinite domin whilst the ltter hs countbly infinite components. This difference cn be significnt if the function is rther exotic!

3 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 3 or, for more generl situtions, f, g = where the weight function w (x) > 0. w (x) f (x) g (x) dx In mny cses we will consider rel functions nd the inner product, defined bove, i.e. g, f = f (x) g (x) dx nd we note tht the norm induced by this inner product is ( 1/2 f = f (x)dx) 2. Theorem 1.5: In n inner product spce we hve the following sttements which prove very useful (see Sheet 1, Q. 3) () Cuchy-Schwrz inequlity f, g f g (b) Tringle inequlity f + g f + g Convergence of sequences of functions Definition 1.6: A sequence of functions f 1, f 2,...,f n,... is (strongly) convergent, or convergent in norm or uniformly convergent to limit f if f n f 0 s n. The ε definition of uniform convergence is: x (, b) nd ε > 0, N such tht N n, f n f < ε. In prticulr N is dependent only on ε nd NOT x. Exmple: in L 2 (, b) f n (x) f (x) 2 dx 0 s n (convergent in men, men squre convergence). We must distinguish this from pointwise convergence Definition 1.7: A sequence of functions f n (x) converges pointwise to f(x) if f n (x) f (x) 0 s n for ll x [, b]. Pointwise convergence sys tht the N in the definition of uniform convergence bove is dependent not only on ε but lso x. Definition 1.8: A Cuchy sequence is such tht ε > 0, N N such tht for ll m, n > N. f n f m < ε If sequence of functions converges then it is Cuchy sequence. Thus if sequence is NOT Cuchy sequence it will not converge.

4 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 4 Hilbert nd Bnch Spces Note the following distinction () A complete inner product spce is one in which every Cuchy sequence hs strong limit (in tht spce). It is clled Hilbert spce. (b) A complete normed liner spce is clled Bnch spce. It must be stressed tht not ll normed liner function spces re complete. For exmple, the set of continuous functions is complete with respect to the uniform norm, but not with respect to the L 2 norm. Every inner product gives rise to norm, i.e. u = u, u. Thus every Hilbert spce (complete inner product spce) is Bnch spce (complete normed spce) by definition. However, not every Bnch spce cn be Hilbert spce since s we described bove, only those norms which stisfy the prllelogrm equlity cn yield n inner product. Orthogonlity Definition 1.9: (i) f nd g re sid to be orthogonl if f, g = 0. (ii) A sequence of functions φ 1, φ 2,...,φ n,... is sid to be orthogonl if φ i, φ j = 0. for ll i, j N, i j. (iii) An orthogonl sequence φ 1, φ 2,...,φ n,... is sid to be orthonorml if Note: Write φ i, φ i = φ i 2 = 1. φ i, φ j = δ ij = where δ ij is clled the Kronecker delt. Exmple: For n N, the sequence 1 2π, 1 π cosnx, { 1 if i = j 0 if i j 1 π sin nx is orthonorml on [ π, π] nd is fmilir from Fourier series. Definition 1.10: A sequence of functions φ 1, φ 2,...,φ n,... is complete if, for ny f S, f (x) = i φ i (x) i=1 for some sequence of sclrs i, i N.

5 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 5 Theorem 1.11: If the sequence φ 1, φ 2,...,φ n,... is orthonorml then the coefficients i in Definition 1.9 re given by i = φ i, f. Proof: Suppose tht the expnsion in Definition 1.10 holds, then tking the inner product with φ i : φ i, f = φ i, j φ j = = j=1 j δ ij = i. j=1 j φ i, φ j j=1 Opertors Definition 1.12: Let S, T be two liner spces nd let the mpping L : S T be such tht L (αf + βg) = αl (f) + βl (g), then L is clled liner opertor. Exmples: () In finite dimensionl vector spce R n, liner opertors my be represented by mtrices. (b) In function spce in which the functions re sufficiently differentible, the differentil opertor L = n i=0 i (x) di dx i is defined nd is liner opertor. (c) An integrl opertor K defined on L 2 (, b) is given by Kf (x) = K (x, y)f (y)dy where K (x, y) is function of two vribles clled the kernel. Adjoint nd self-djoint opertors Definition 1.13: Let S, T be inner product spces, S, T the inner products on S, T respectively, nd let L : S T be liner opertor. The djoint L : T S of L is defined by g, Lf T = L g, f S, f S nd g T. Theorem 1.14: L exists, is unique nd is liner.

6 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 6 Proof: None for existence nd uniqueness. Linerity: for ny f L (αg 1 + βg 2 ),f S = αg 1 + βg 2, Lf T by Definition 1.13 = ᾱ g 1, Lf T + β g 2, Lf T by conjugte linerity = ᾱ L g 1, f S + β L g 2, f S by Definition 1.13 = αl g 1 + βl g 2, f S by conjugte linerity. So L (αg 1 + βg 2 ) = αl g 1 + βl g 2. Definition 1.15: If S = T then (i) L is clled self-djoint if L = L, (ii) L is clled skew-djoint if L = L. Theorem 1.16: (i) (L ) = L, (ii) 1 2 (L + L ) is self-djoint, (iii) 1 2 (L L ) is skew-djoint, (iv) Any liner opertor L my be expressed s the sum of self- nd skew-djoint opertor. Proof: (i) By definition of (L ) : S T, f S, g T, (L ) f, g T = f, L g S pplying 1.13 to L = L g, f S by 1.2(i) = g, Lf T by 1.13 = Lf, g T by 1.2(i) So (L ) f = Lf, f S nd so (L ) = L. (ii) ( ) 1 2 (L + L ) = 1 2 (L + (L ) ) = 1 2 (L + L). (iii) (iv) ( ) 1 2 (L L ) = 1 2 (L (L ) ) = 1 2 (L L) = 1 2 (L L ). L = 1 2 (L + L ) (L L ). Why study the djoint opertor? Well, it trnspires tht it is importnt in vrious theorems regrding the existence nd uniqueness of solutions to boundry vlue problems (BVPs). In prticulr Self-Adjoint opertors possess very nice properties s we shll see lter. Exmple 1: Suppose tht f () = f (b) = g () = g (b) = 0 nd let L = d dx.

7 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 7 Show tht i.e. L is skew-djoint. L = d dx, See Exmples Sheet 1. NOTE: If ssumptions bout boundry conditions re required to prove self-djointness, the opertor is sid to be formlly self-djoint. Exmple 2: Given homogeneous boundry conditions (e.g. f () = 0 or f () = 0 nd similrly for g) show tht the Sturm-Liouville opertor L is formlly self-djoint, where L = 1 { [ d p (x) d ] } + q (x), r (x) dx dx r (x) > 0 nd See Exmples Sheet 1. g, f = r (x) f (x) g (x) dx. Definition 1.17: A non-trivil function φ S is clled n eigenfunction of L if there exists λ R (or C) such tht Lφ = λφ. The prmeter λ is clled n eigenvlue of L corresponding to φ. Notes: (i) φ 0 is lwys solution of this homogeneous eqution, for ny λ. Some vlues of λ my llow non-trivil φ. (ii) In finite dimensionl vector spce L my be represented by mtrix (φ is clled n eigenvector). When the mtrix is symmetric the following theorem looks very fmilir. Theorem 1.18: Let L be self-djoint opertor then (i) The eigenvlues of L re rel, (ii) The eigenfunctions φ corresponding to distinct eigenvlues re orthogonl. Proof: (i) Since L is self-djoint, by 1.13, φ, Lφ = Lφ, φ = φ, Lφ by definition 1.2(i). Thus the complex number φ, Lφ is equl to its complex conjugte nd hence is rel. Also, by Definition 1.17, Lφ = λφ so tking the inner product of both sides gives φ, Lφ = φ, λφ = λ φ, φ = λ φ 2. (1.2)

8 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 8 Now, φ 0 so λ = φ, Lφ φ 2 which is rel. (ii) Let φ 1, φ 2 be eigenfunctions corresponding to distinct eigenvlues λ, µ, i.e. λ µ, Lφ 1 = λφ 1, Lφ 2 = µφ 2 nd µ φ 1, φ 2 = φ 1, µφ 2 linerity in 2nd rgument of, = φ 1, Lφ 2 eigenvlue definition 1.17 = Lφ 1, φ 2 L is self-djoint = λφ 1, φ 2 eigenvlue definition 1.17 = λ φ 1, φ 2 conjugte linerity in 1st rgument of, = λ φ 1, φ 2 λ is rel. But µ λ so φ 1, φ 2 = 0. Note: More thn one linerly independent eigenfunction my correspond to the sme eigenvlue - these eigenfunctions spn subspce S of S. The dimension of S is clled the multiplicity of the eigenvlue. An orthogonl bsis for S exists (Grm-Schmidt orthogonlistion). So, we my ssume ll linerly independent eigenfunctions re orthogonl. L my be such tht there exists denumerble sequence λ 1, λ 2,...,λ n,... of eigenvlues nd corresponding sequence of orthonorml eigenfunctions φ 1, φ 2,...φ n,.... If this set is complete then ny function f (x) S my be expnded in n infinite series s in Definition Definition 1.19: An opertor L is sid to be positive if f, Lf is rel nd positive for ll f S. Proposition 1.20: A positive opertor (i) is self-djoint, (ii) hs rel nd positive eigenvlues. No proof, see Exmples Sheet 1. Exmple: Show tht the generl solution of the inhomogeneous eqution Lu = f where L is self-djoint liner opertor nd L possesses complete orthonorml set of eigenfunctions φ 1, φ 2, φ 3,... with corresponding eigenvlues λ 1, λ 2, λ 3,..., is given by u = φ n, f λ n φ n.

9 MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 9 Solution: Since the φ n form complete set, expnd the solution in the form u = n φ n, where the coefficients n re to be found. Then ( ) f = Lu = L n φ n = = n Lφ n by linerity n λ n φ n, by Definition 1.17 of eigenvlue/eigenfunction. Thus, tking the inner product with φ m, m = 1, 2,... φ m, f = φ m, n λ n φ n = n λ n φ m, φ n = n λ n δ mn = m λ m. Thus, provided λ m 0, Hence u = m = φ m, f λ m. φ n, f λ n φ n.