ABSTRACT
Equation (20.5) in turn may be rewritten as a partial integral equation of Volterra type
(20.7) +l( m(z, (,r)w(z,r)dr
+l·l(n(z,(,t,r)w(t,r)drdt=9(Z,(), where we have put
and
c = -2lJz - 2lJ( +C, Cl = - lJz - lJ( +CIt lJ2a lJ2b lJ2a1 lJc lJcl
d = lJz2 +2lJzlJ( + lJ(2 - lJz - lJ( +>.,
Here CPOt CPl t tPot and ,p1 are arbitrary holomorphic functions. The equation (20.7) with holomorphic functions (20.6) and f is a special case of the equation
(20.10)
where K is the partial integral operator of Volterra type
+10· 10' n(zt(tttT)W(ttT)dTdt. In VEKUA [1948] it was shown that equation (20.9) has a holomorphic solution if all data 't mt nt and fare holomorphic. However, this equation may have a unique solution also for more general data. The reason for this is that t under reasonable conditions on the kernels 't mt and nt the spectral radius of the operator (20.10) is zero in many natural function spaces. In fact t let r 1 = {z : Z = 11(u) : 0 ~ u ~ a} and r 2 ={( :( =12(tJ) : o~ tJ ~ b} be two smooth simple curves in the complex z-plane and (-planet respectivelYt and let r =r l X r 2• The integrals in (20.10) are understood as contour integrals over r 1 joining 0 and z, and over r 2 joining 0 and (t respectively. If the data are not holomorphict these integralst and hence also the operator K, depend of course on the contours r l and r 2 • Neverthelesst for fixed rl and r 2 the spectral radius of K is still zero. Put
(20.11)
Wt(U, v) = w( 'Yt (u), 'Y2(v», and
gt(U, V) = g( 'Yt (U), 'Y2(V». Then the operator (20.10) takes the form
+LlI mt(u, v, v)Wt(u, v) dv +LV. J: nl(u, v,u, V)Wt(U, v) dvdu,
and the equation (20.9) becomes
(20.12)
Now we may apply the results of KALITVIN [1998J to this equation. For instance, the spectral radius of the operator (20.11) is zero in the spaces C(D) and Lp(D)(D = [O,a)x[O,b), 1 ~ p ~ 00), provided that the kernels I, m, and n are bounded and integrable. Consequently, equation (20.12) has a unique solution in C(D) or Lp(D), and thus equation (20.9) is uniquely solvable for any function 9 from C(D) or Lp(D), repsectively. Analogous results hold for systems of differential equations occuring in the modelling ofthin elastic hulls (VEKUA (1948]). Here the kernels depend only on the form of the hull, and the solution determines the stress function. A solution (in a generalized sense) also exists for nonholomorphic entries of the stress tensor and for non-smooth surfaces.
