Layer Potentials. Chapter Fundamental and Singular Solutions

Transcription

1 Chapter Layer Potentals We solve boundary value problems assocated wth system (1.1) (or, rather, ts homogeneous verson) by means of potental-type functons wth a sutably chosen kernel. In ths chapter we construct a matrx of fundamental solutons for the operator A ω ( x ), whch we can then use to defne generalzed sngle-layer and doublelayer plate potentals. The method used s analogous to the one employed n 14 to construct the correspondng matrx for the operator A( x ) defned by (1.11), whch occurs n the study of the equlbrum bendng of plates. Later, we wrte the matrx of fundamental solutons n a form that allows us to decompose t nto an nfnte seres of so-called wavefunctons. Ths form, constructed n Theorem.1, s smlar to the correspondng matrx n the theory of plane elastodynamcs 48 wth the added complcaton of certan computatonal constants. In Secton. we nvestgate the sngulartes of the matrx of fundamental solutons and of ts assocated matrx of sngular solutons. Thus, n Theorems.3 and.4 we fnd that these sngulartes concde wth those of the correspondng matrces from equlbrum plate theory. Therefore, the sngle-layer and double-layer potentals ntroduced n Secton.3 behave n the same way as the potentals consdered n 14. The mportant propertes of these functons, used extensvely n the subsequent analyss, are contaned n Theorems Fundamental and Sngular Solutons We construct a matrx of fundamental solutons for the operator A ω ( x ) usng the method descrbed n 13. If (ξ ) s the adjont of the matrx A ω (ξ ), then u(x) ( x )B(x), (.1) where B satsfes (deta ω )( x )B(x)H(x). (.) From (1.1) and (1.11) t follows that G.R. Thomson and C. Constanda, Statonary Oscllatons of Elastc Plates, A Boundary Integral Equaton Analyss, DOI 1.17/ _, Sprnger Scence+Busness Meda, LLC 11 7

2 8 Layer Potentals deta ω (ξ ) μξ 1 μ ξ 1 (h Δ 1)+ρω h μξ 1 + μξ μ ξ (h Δ 1)+ρω h μξ +(μδ + ρω ) h 4 μ(λ + μ)δ + h (λ + 3μ)(ρω h μ)δ +(ρω h μ) μ 3 (ξ1 + ξ )(h Δ 1)+ρω h μ (ξ1 + ξ ) + h 4 μ (λ + μ)δ 3 + h μ(λ + 3μ)(ρω h μ)+ρω h 4 μ(λ + μ) Δ + μ(ρω h μ) + ρω h (λ + 3μ)(ρω h μ) Δ + ρω (ρω h μ) h 4 μ (λ + μ)δ 3 + ρω h 4 μ(λ + 5μ) h μ (λ + μ) Δ + ρω h (ρω h μ)(λ + 4μ)Δ + ρω (ρω h μ). Factorng ths expresson, we obtan ( deta ω (ξ )h 4 μ (λ + μ) Δ + λ + 3μ λ + μ k Δ + μ ) λ + μ k k3 (Δ + k3), where and hence, where k ρω μ (.3) k 3 k 1 h ; (.4) deta ω (ξ )h 4 μ (λ + μ)(δ + k1 )(Δ + k )(Δ + k 3 ), (.5) k1 + k λ + 3μ λ + μ k, k1 k μ (.6) λ + μ k k3. Wthout loss of generalty, we assume that k1 k. Usng (.4) and (.6), we fnd that k1, k, and k 3 are connected by the equalty h (k1 k 3 )(k k 3 )h k1 k k 3 (k 1 + k )+k4 3 h ( μ λ + μ k k 3 λ + 3μ λ + μ k k 3 + k 4 3 h ( k k3 + k3) 4 ( h k3 k + k 1h ), ) from whch h (k 1 k 3)(k k 3) k 3. (.7)

3 .1 Fundamental and Sngular Solutons 9 We clam that, under assumptons (1.14) and (1.15), k 1, k, and k 3 are real, strctly postve, and dstnct. Frst, k1 and k are the two roots of the equaton x + λ + 3μ λ + μ k x + μ λ + μ k k3. The dscrmnant of ths quadratc s (λ + 3μ) (λ + μ) k4 4μ λ + μ k k 3 k (λ + μ) k (λ + μ) k (λ + μ) k (λ + 6λμ+ 9μ ) 4μ(λ + μ) k (λ + λμ+ μ )+ k (λ + μ) + 4μ(λ + μ) h 4μ(λ + μ) h >. (k 1h ) Consequently, k1 and k are real and dstnct. Also, by (.4), assumpton (1.15) mples that k3 >. By (.6) and (1.14), ths means that k 1 + k >, k 1 k >. Hence, k 1 and k are strctly postve. Fnally, from (.7) and the fact that k 3 t follows that k 1 k 3 and k k 3. Replacng, n turn, each component of H by δ( x y ), where δ s the Drac delta dstrbuton, and settng the other two equal to zero, from (.1) and (.) we obtan the matrx of fundamental solutons where, by (.) and (.5), t(x,y) s a soluton of D ω (x,y) ( x ) t(x,y)e 3, (.8) h 4 μ (λ + μ)(δ + k1 )(Δ + k )(Δ + k 3 )t(x,y) δ( x y ). (.9) We seek t(x,y) n the form t(x,y) 3 j1 b j (k j x y ), (.1) where s the Hankel functon of the frst knd of order zero and the b j are constants to be determned from (.9). Ths Hankel functon s a fundamental soluton of the Helmholtz operator and satsfes 75 (Δ + k j) (k j x y )4δ( x y ). (.11)

4 1 Layer Potentals From (.1) and (.11) we fnd that (Δ + k3 )t(x,y)b 1 4δ( x y )+(k 3 k1 )H(1) (k 1 x y ) + b 4δ( x y )+(k 3 k )H(1) (k x y ) + 4b 3 δ( x y ). To elmnate the Drac dstrbuton n ths equaton, we requre that Now, f (.1) s satsfed, we see that b 1 + b + b 3. (.1) (Δ + k )(Δ + k3)t(x,y) b 1 (k3 k 1 ) 4δ( x y )+(k k 1 )H(1) (k 1 x y ) + 4b (k 3 k )δ( x y ), so we must have (k 3 k 1)b 1 +(k 3 k )b. (.13) Snce t(x, y) satsfes (.9), we obtan 1 h 4 μ (λ + μ) δ( x y )(Δ + k 1 )(Δ + k )(Δ + k 3 )t(x,y) 4b 1 (k1 k )(k 1 k 3 )δ( x y ), from whch we deduce that b 1 4h 4 μ (λ + μ)(k1 k )(k 1 (.14) k 3 ). Substtutng ths nto (.1) and (.13) yelds b b 3 4h 4 μ (λ + μ)(k k 1 )(k k 3 ), 4h 4 μ (λ + μ)(k 3 k 1 )(k 3 k ). (.15) The formula for b 3 can be smplfed by means of (.7): b 3 4h μ (λ + μ)k3. (.16) These constants are well defned snce the k j are dstnct. To calculate the matrx of fundamental solutons D ω (x,y) usng (.8), we frst need to compute ( x ). From (1.1) and (1.11) t follows that

5 .1 Fundamental and Sngular Solutons (ξ ) h μδ + h (λ + μ)ξ μ + ρω h μξ μξ μδ + ρω h μ ΔΔ + h μ(λ + μ)δξ +(ρω h μ)μδ + ρω h μδ In the same way, + ρω h (λ + μ)ξ + ρω (ρω h μ)+μ ξ h μ ΔΔ + h μ(λ + μ)δδ h μ(λ + μ)δξ 1 μ ξ 1 + ρω h μδ + ρω h (λ + μ)δ ρω h (λ + μ)ξ1 + ρω (ρω h μ) h μ(λ + μ)δδ h μ(λ + μ)δξ1 + ρω h (λ + 3μ)Δ ( μ + ρω h (λ + μ) ) ξ 1 + ρω (ρω h μ). (ξ )h μ(λ + μ)δδ h μ(λ + μ)δξ + ρω h (λ + 3μ)Δ Next, 1 (ξ ) h (λ + μ)ξ 1 ξ μξ 1 μξ μδ + ρω h (λ + μ)ξ 1 ξ μξ μξ 1 μδ + ρω Aω 1 (ξ ) ( μ + ρω h (λ + μ) ) ξ + ρω (ρω h μ). h μ(λ + μ)δξ 1 ξ ( μ + ρω h (λ + μ) ) ξ 1 ξ, 13 (ξ ) h (λ + μ)ξ 1 ξ μξ 1 h μδ + h (λ + μ)ξ μ + ρω h μξ h (λ + μ)ξ 1 ξ h μδ + h (λ + μ)ξ μ + ρω h μξ 1 μξ Aω 31 (ξ ) h μ(λ + μ)ξ 1 ξ + h μ Δξ 1 + h μ(λ + μ)ξ 1 ξ +(ρω h μ)μξ 1 h μ Δξ 1 + μ(ρω h μ)ξ 1, 3 (ξ ) h μδ + h (λ + μ)ξ1 μ + ρω h μξ 1 h (λ + 1 ξ μξ Aω 3 (ξ ) h μ Δξ + μ(ρω h μ)ξ. Fnally, 33 (ξ ) h μδ + h (λ + μ)ξ1 μ + ρω h h (λ + μ)ξ 1 ξ h (λ + 1 ξ h μδ + h (λ + μ)ξ μ + ρω h

6 1 Layer Potentals h 4 μ ΔΔ + h 4 μ(λ + μ)δξ + h μ(ρω h μ)δ + h 4 μ(λ + μ)δξ 1 + h 4 (λ + μ) ξ 1 ξ + h (λ + μ)(ρω h μ)ξ 1 + h μ(ρω h μ)δ + h (λ + μ)(ρω h μ)ξ +(ρω h μ) h 4 (λ + μ) ξ 1 ξ h 4 μ(λ + μ)δδ + h (λ + 3μ)(ρω h μ)δ +(ρω h μ). Therefore, the elements of the adjont of the matrx A ω (ξ ) are α (ξ )h μ(λ + μ)δ α ΔΔ h μ(λ + μ)δξ α ξ + ρω h (λ + 3μ)δ α Δ ( ρω h (λ + μ)+μ ) ξ α ξ + ρω (ρω h μ)δ α, (.17) α3 (ξ ) Aω 3α (ξ )h μ Δξ α + μ(ρω h μ)ξ α, (.18) 33 (ξ )h4 μ(λ + μ)δδ + h (λ + 3μ)(ρω h μ)δ +(ρω h μ). (.19) Theorem.1. The elements of the matrx of fundamental solutons D ω (x,y) are Dα ω (x,y) 4h μk3 α1 x α x (k 1 x y ) α x α x (k x y ) + x α x (k 3 x y )+δ α k3 H(1) (k 3 x y ), (.) Dα3 ω (x,y) Dω 3α (x,y) 4h μk3 γ x (k 1 x y )+γ α x (k x y ), (.1) α D33 ω (x,y) 4h μk3 where 1 (k 1 x y )+ H(1) (k x y ), (.) α 1 k μ k 3 k k 1 1 h k 3 (k 1 μ k 3 ) k 1 k, α k 1 μ k3 k1, (.3) k γ μ k3 k1, (.4) k, h k3 (k μ k3 ) k, (.5) k 1 wth μ μ λ + μ. (.6)

7 .1 Fundamental and Sngular Solutons 13 Proof. Frst, we rewrte the elements of the adjont matrx n a more convenent form. Usng (.3) and (.4), from (.17) we obtan α (ξ )δ α h μ(λ + μ)δδ + ρω h (λ + 3μ)Δ + ρω (ρω h μ) h μ(λ + μ)δ + ρω h (λ + μ)+μ ξ α ξ δ α h μ(λ + μ) ΔΔ + λ + 3μ λ + μ k Δ + μ λ + μ k k3 so, usng (.4) and (.6), we arrve at α (ξ )δ αh μ(λ + μ)(δ + k 1)(Δ + k ) Smlarly, from (.18) and (.19), h μ(λ + μ)(δ + k )ξ α ξ μ ξ α ξ, h μ(λ + μ)(δ + k 3)ξ α ξ μ(λ + μ)ξ α ξ. α3 (ξ ) Aω 3α (ξ ) h μ Δξ α + μ(ρω h μ)ξ α ( h μ Δ + ρω h ) μ h ξ α h μ (Δ + k μ 3)ξ α, 33 (ξ )h4 μ(λ + μ)δδ + h (λ + 3μ)(ρω h μ)δ +(ρω h μ) h 4 μ(λ + μ)(δ + μ k 3)(Δ + k 3). By the defnton of D ω (x,y), Dα ω (x,y)aω α ( x)t(x,y) δ α h μ(λ + μ)(k1 k3)(k k3)b 3 (k 3 x y ) h μ(λ + μ) (k3 k 1 )b 1 x α x (k 1 x y ) +(k3 k)b x α x (k x y ) μ(λ + μ) b 1 x α x (k 1 x y )+b x α x (k x y ) Takng (.14) (.16) nto account, we fnd that + b 3 x α x (k 3 x y ).

8 14 Layer Potentals Dα ω (x,y)δ α 4h μ H(1) (k 3 x y )+ 4h μk3 x α x (k 3 x y ) + h μ(λ + μ)(k1 k 3 ) μ(λ + μ) 4h 4 μ (λ + μ)(k1 k )(k 1 k 3 ) x α x (k 1 x y ) + h μ(λ + μ)(k k 3 ) μ(λ + μ) 4h 4 μ (λ + μ)(k k 1 )(k k 3 ) x α x (k x y ) 4h μk3 α1 x α x (k 1 x y ) α x α x (k x y ) + x α x (k 3 x y )+δ α k3h (1) (k 3 x y ), where α1 k 3 λ + μ k k 1 α k 3 k 1 k λ + μ 1 h (k1 k 3 ) λ + μ λ + μ 1 h (k k 3 ),. Now, usng (.7) and (.6), we deduce that α1 k 3 k 1 μ + k k 3 k 1 k3 1 k (k k 3 μ k3 + k k3) k μ k3 1 k, k 1 as requred. Smlarly, t can be shown that Next, α k 1 μ k3 k1. k D ω α3 (x,y) Dω 3α (x,y)aω α3 ( x)t(x,y) h μ (k3 k 1 ) 4h 4 μ (λ + μ)(k1 k )(k 1 k 3 ) x (k 1 x y ) α h μ (k3 + k ) 4h 4 μ (λ + μ)(k k 1 )(k k 3 ) x (k x y ) α 4h μk3 γ x (k 1 x y )+γ α x (k x y ), α where γ s defned by (.4).

9 .1 Fundamental and Sngular Solutons 15 Fnally, D ω 33 (x,y)aω 33 ( x)t(x,y) h4 μ(λ + μ)(μ k 3 k 1 )(k 3 k 1 ) 4h 4 μ (λ + μ)(k 1 k )(k 1 k 3 ) H(1) (k 1 x y ) + h4 μ(λ + μ)(μ k3 k )(k 3 k ) 4h 4 μ (λ + μ)(k k 1 )(k k 3 ) H(1) (k x y ) 4h μk3 1 (k 1 x y )+ (k x y ), where 1 and are defned by (.5). Ths completes the proof. The representaton of D ω (x,y) gven n Theorem.1 s suffcent for our purposes. There s no need to calculate explctly the frst-order and second-order dervatves of the Hankel functons n (.) (.). To smplfy the notaton, we wrte r x y. Snce and f (r) f (r) x α y α f (r) f (r), x α x y α y from (.) (.) t s easly seen that We ntroduce the matrx of sngular solutons D ω (x,y) D ω (y,x) T. (.7) P ω (x,y) T ( y )D ω (y,x) T. (.8) Theorem.. Each column of D ω (x,y) and P ω (x,y) satsfes the homogeneous system A ω ( x )u(x) at all ponts x R,x y. Proof. From (.8), (.5), and (.9) we see that for x y, A ω ( x )D ω (x,y)a ω ( x ) ( x ) t(x,y)e 3 Usng (.8) and (.7), we fnd that for x y, (deta ω )( x )t(x,y) E 3. Ak ω ( x)pkj ω (x,y)aω k ( x) T ( y )D ω (y,x) jk Ak ω ( x)t jm ( y )Dmk ω (y,x) T jm ( y )Ak ω ( x)dkm ω (x,y), whch proves the asserton.

10 16 Layer Potentals. Order of Sngularty The nvestgaton of the sngulartes of D ω and P ω plays an mportant role n the study of the behavor of the sngle-layer and double-layer plate potentals. It s known 1 that, as ξ, (ξ ) π so, as r, from (.1) we see that t(x, y)t(r) 3 j1 π π b j (k jr) 3 j1 3 j1 ( ξ ξ 4 ) lnξ ; b j (1 1 4 k j r + 1 ) 64 k4 j r4 ln(k j r) b j ( k j r k4 j r4 )lnr + d + O(r 6 lnr), where d s a constant. From (.14) and (.15) t follows that 3 j1 b j 3 j1 b j k j, 3 j1 b j k 4 j 4h 4 μ (λ + μ), whch leads to where t(r)c 1 r 4 lnr + d + O(r 6 lnr), (.9) 1 c 1 18πh 4 μ (λ + μ). (.3) Theorem.3. As r, ( D ω (x,y)lnr d 1 E γγ 1 ) πμ E (x α y α )(x y ) 33 + d r E α where C s a constant (3 3)-matrx, +C + D ω (x,y), (.31) λ + 3μ d 1 4πh μ(λ + μ), d λ + μ 4πh μ(λ + μ), (.3) and D ω (x,y) s a (3 3)-matrx whose elements are O(r lnr) as r.

11 . Order of Sngularty 17 Proof. By (.8), (.17), (.9), and (.3), we fnd that D ω α (x,y)aω α ( x)t(x,y) h μ(λ + μ)δ α ΔΔ(c 1 r 4 lnr) h μ(λ + μ) Δ(c 1 r 4 lnr) x α x hence, + ρω h (λ + 3μ)δ α Δ(c 1 r 4 lnr) ρω h (λ + μ)+μ (c 1 r 4 lnr) x α x + ρω (ρω h μ)δ α c 1 r 4 lnr +C α + O(r lnr) 64c 1 h μ(λ + μ)δ α lnr c 1 h μ(λ + μ) (16r lnr + 8r ) x α x +C α + O(r lnr); Dα ω (x,y)64c 1h μ(λ + μ)δ α lnr c 1 h μ(λ + μ) 3 (x α y α )(x y ) r + 3δ α lnr +C α + O(r lnr) λ + 3μ 4πh μ(λ + μ) δ λ + μ (x α y α )(x y ) α lnr + 4πh μ(λ + μ) r Also, usng (.18) and (.19), we see that and +C α + O(r lnr). D ω α3 (x,y) Dω 3α (x,y)aω α3 ( x)t(x,y) h μ x α Δ(c 1 r 4 lnr)+μ(ρω h μ) x α (c 1 r 4 lnr) C α3 + O(r lnr) +C α3 + O(r3 lnr) D33 ω (x,y)aω 33 ( x)t(x,y) h 4 μ(λ + μ)δδ(c 1 r 4 lnr)+h (λ + 3μ)(ρω h μ)δ(c 1 r 4 lnr) +(ρω h μ) c 1 r 4 lnr +C 33 + O(r lnr) 64c 1 h 4 μ(λ + μ)lnr +C 33 + O(r lnr) 1 πμ lnr +C 33 + O(r lnr). The asserton s proved.

12 18 Layer Potentals We ntroduce the alternatng symbol ε α α for α, {1,}. In what follows, ν(y) and denote the dervatves n the normal and tangental drectons, respectvely. s(y) Theorem.4. As r, P ω (x,y) 1 { μ ε π α s(y) lnr E α + ν(y) lnr E 3 (λ + μ (x α y α )(x y ) )ε αγ s(y) r E γ 1 (xα y α )lnr E 3α ν(y) + 1 ε α (xα y α )lnr ( λ E s(y) ) h E } 1 ν α(y) h E α3 + (λ + μ )E 3α where D s a constant (3 3)-matrx, λ λ (λ + μ), + D + P ω (x,y), (.33) μ s defned by (.6), and P ω (x,y) s a (3 3)-matrx whose elements are O(r lnr) as r. Proof. From (.8) (.3) we obtan so, P11 ω (x,y)t 11( y )D11 ω (y,x)+t 1( y )D1 ω (y,x)+t 13( y )D31 ω (y,x) h (λ + μ)ν 1 A ω 11 ( y)t(y,x), 1 +h μν A ω 11 ( y)t(y,x), + h μν A ω 1 ( y)t(y,x), 1 +h λν 1 A ω 1 ( y)t(y,x), h (λ + μ)ν 1 h μ(λ + μ)δδt, 1 h μ(λ + μ)δt, h μν h μ(λ + μ)δδt, h μ(λ + μ)δt, 11 + h μν h μ(λ + μ)δt, 11 + h λν 1 h μ(λ + μ)δt, 1 + D 11 + O(r lnr) h 4 μ(λ + μ)(λ + μ)ν 1 ΔΔt, 1 +h 4 μ (λ + μ)ν 1 ΔΔt, 1 + h 4 μ (λ + μ)ν ΔΔt, h 4 μ (λ + μ)ν 1 Δt, 111 h 4 μ (λ + μ)ν Δt, 11 h 4 λμ(λ + μ)ν 1 Δt, 111 h 4 λμ(λ + μ)ν 1 Δt, 1 +D 11 + O(r lnr);

13 . Order of Sngularty 19 P11 ω (x,y) h 4 μ (λ + μ) ν(y) ΔΔt + h4 μ (λ + μ)ν 1 Δt, 111 h 4 μ (λ + μ)ν 1 Δt, 111 +h 4 μ (λ + μ)ν 1 Δt, 1 h 4 μ (λ + μ)ν Δt, 11 + D 11 + O(r lnr) h 4 μ (λ + μ) ν(y) ΔΔt + h4 μ (λ + μ) s(y) Δt, 1 +D 11 + O(r lnr) 64h 4 μ (λ + μ)c 1 ν(y) lnr + 64h4 μ (λ + μ)c 1 s(y) whch means that P ω 11 (x,y) 1 π ν(y) lnr 1 π (λ + μ ) s(y) (x 1 y 1 )(x y ) r + D 11 + O(r lnr), (x 1 y 1 )(x y ) r + D 11 + O(r lnr). Analogous manpulaton yelds P1 ω (x,y) h4 λμ s(y) Δt, 11 +h 4 μ (λ + μ) s(y) Δt, +D 1 + O(r lnr); hence, P ω 1 (x,y) λ 4π(λ + μ) 1 Smlarly, we fnd that P ω π μ (x y ) s(y) r (x y ) 1 4π s(y) r + lnr + lnr + D 1 + O(r lnr) s(y) lnr 1 π (λ + μ (x y ) ) s(y) r + D 1 + O(r lnr). 13 (x,y)h μ (λ + μ) s(y) Δt, +D 13 + O(r lnr) 1 (x 4πh y )lnr 1 s(y) 4πh ν 1(y)+D 13 + O(r lnr), 1 (x,y) h4 μ (λ + μ) s(y) Δt, 11 +h 4 λμ s(y) Δt, +D 1 + O(r lnr) 1 π μ s(y) lnr + 1 π (λ + μ (x 1 y 1 ) ) s(y) r + D 1 + O(r lnr), P ω

14 Layer Potentals (x,y)h4 μ (λ + μ) ν(y) ΔΔt h4 μ (λ + μ) s(y) Δt, 1 P ω 1 π ν(y) lnr + 1 π (λ + μ ) s(y) + D + O(r lnr) (x 1 y 1 )(x y ) r + D + O(r lnr), P3 ω (x,y) h μ (λ + μ) s(y) Δt, 1 +D 3 + O(r lnr) 1 (x1 4πh y 1 )lnr 1 s(y) 4πh ν (y)+d 3 + O(r lnr), P31 ω (x,y)h4 μ (λ + μ) ν(y) Δt, 1 +h 4 λμ s(y) Δt, +D 31 + O(r lnr) 1 (x1 y 1 )lnr 1 4π ν(y) 4π λ (x y )lnr s(y) 1 π (λ + μ )ν 1 (y)+d 31 + O(r lnr), P3 ω (x,y) h4 λμ s(y) Δt, 1 +h 4 μ (λ + μ) ν(y) Δt, +D 3 + O(r lnr) 1 4π λ (x1 y 1 )lnr 1 (x y )lnr s(y) 4π ν(y) 1 π (λ + μ )ν (y)+d 3 + O(r lnr), P33 ω (x,y)h4 μ (λ + μ) ν(y) ΔΔt + D 33 + O(r lnr) 1 π ν(y) lnr + D 33 + O(r lnr), as requred. Remark.1. Expansons (.31) and (.33) of D ω (x,y) and P ω (x,y), respectvely, for y close to x concde wth those of the correspondng matrces arsng n the equlbrum bendng of plates, whch can be found n Propertes of the Potentals We defne the sngle-layer potental (V ω ϕ)(x) D ω (x,y)ϕ(y)ds(y) (.34) S

15 .3 Propertes of the Potentals 1 and the double-layer potental (W ω ϕ)(x) P ω (x,y)ϕ(y)ds(y), (.35) S where ϕ s a (3 1)-vector functon known as the densty. The propertes of these potentals play an mportant role n formulatng sutable ntegral equatons for varous boundary value problems. We are especally concerned wth the behavor of (V ω ϕ)(x) and (W ω ϕ)(x) as x approaches S. Ths requres a very detaled analyss based on the expansons (.31) and (.33). As was noted n Remark.1, the sngulartes of D ω and P ω are exactly the same as those occurrng n the correspondng matrces n the equlbrum bendng of plates. The behavor of the correspondng potentals s thoroughly nvestgated n 14, and many relevant propertes are smply quoted from there. We denote by C,α (X) the vector space of Hölder contnuous functons (wth ndex α) onx, and by C 1,α (X) the subspace of C 1 (X) of functons whose frstorder dervatves belong to C,α (X). In the sequel, when we menton that a functon ϕ C,α (X), α (,1), satsfes a certan equaton or condton, we mean that ϕ C,α (X) for all α n the nterval (,1). The next three assertons follow from Theorem. and the results n 14. Theorem.5. () If ϕ C(S), then V ω ϕ and W ω ϕ are analytc and satsfy A ω ( x )u(x) n S + S. () If ϕ C,α (S), α (,1), then the drect values V ωϕ and W ωϕ of V ω ϕ and W ω ϕ on S exst (the latter n the sense of prncpal value). Theorem.6. If ϕ C,α (S), α (,1), then the functons V ω+ (ϕ)(v ω ϕ) S +, V ω (ϕ)(v ω ϕ) S (.36) are of class C (S + ) C 1,α ( S + ) and C (S ) C 1,α ( S ), respectvely, and T V ω+ (ϕ) ( W ω + 1 I ) ϕ on S, (.37) T V ω (ϕ) ( W ω 1 I ) ϕ on S, (.38) where W ω s the adjont of W ω and I s the dentty operator. Theorem.7. If ϕ C 1,α (S), α (,1), then the functons { W ω+ (W ω ϕ) (ϕ) S + n S +, ( W ω 1 I ) ϕ on S, { W ω (W ω ϕ) (ϕ) S n S, ( W ω + 1 I ) ϕ on S, (.39) (.4) are of class C (S + ) C 1,α ( S + ) and C (S ) C 1,α ( S ), respectvely, and TW ω+ (ϕ)t W ω (ϕ) on S. (.41)

16 Layer Potentals Remark.. Theorems.5.7 are used n Chapters 6 11 to rgorously justfy the constructon of regular solutons of the fundamental boundary value problems n S + and S n the form of layer potentals.

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