Equations of the Mixed Type by A. V. Bitsadze (Auth.)

By A. V. Bitsadze (Auth.)

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40) can be found in the paper of Chi Min l u (1). 33), as was shown, the curve of parabolic degeneracy y = 0 is itself a characteristic. I t is therefore natural to expect t h a t the Cauchy problem for this equation with the boundary conditions given on the curve of degeneracy is, in general, incorrect. 53) where φ and ψ are arbitrary twice differentiable functions. 54) remains bounded then the above solution becomes uniquely determined. e. , . τ (x) const. , ' τ (x) ~ const. and consequently the required solution is given by the expres­ sion u(x,y) = ^ r { x + 2^1/2) + i_ ^ _ 2^/1/2).

40). 48), the Cauchy problem for this equation belonging to boundary conditions given along the curve of parabolic degeneracy nevertheless has always a solution which is also unique and stable. A very frequent occurrence of this problem is investigated in the paper of K. I. Karapatian (1) [cf. also I. L. 40) can be found in the paper of Chi Min l u (1). 33), as was shown, the curve of parabolic degeneracy y = 0 is itself a characteristic. I t is therefore natural to expect t h a t the Cauchy problem for this equation with the boundary conditions given on the curve of degeneracy is, in general, incorrect.

Hausdorff (1), C. M. Nikolskij (1)]. 32) where A is the Laplace operator. 23). 32) in complex form d^u . du . du . „ -áF + (^-^ + CU = 0, Wdt'~^ "^'dT^'-'dC ζ= X -\- iy e D, ζ= X — iy Ε D*, in the same way as in § 2 of Ch. 33) + ψ it) dt, where φ (z) = { {z), . , , ψη {ζ) } in an arbitrary holomorphic vector. 32) Αυ~-^{υΑ) --^{vB)+vC = 0 with a logarithmic singularity at point ζ = ZQ. 1) [cf. I. N. Vekua (1), A. V. Bitsadze (3)1. I t is found that if '^(-a^ + - 8 F - 2 C ) , > 0 . 35) where G {z, z^) is the matrix Green's function; which always exists whenever the contour Γ is sufficiently smooth.

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