Boundary Value Problems for Partial Differential Equations by N. E. Tovmasyan, L. Z. Gevorkyan, G. V. Zakaryan

By N. E. Tovmasyan, L. Z. Gevorkyan, G. V. Zakaryan

This article is dedicated to boundary price difficulties for basic partial differential equations. It develops effective tools of answer of boundary price difficulties for elliptic equations, in accordance with the speculation of analytic services, having nice theoretical and sensible significance. a brand new method of the research of electromagnetic fields is sketched, allowing legislation of propagation of electromagnetic strength at an exceptional distance to be defined and asymptotic formulae for recommendations of Maxwell's equation to be got. those equations also are utilized to the effective answer of difficulties.

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We as f u n c t i o n a l s over are shall linearly prove t h a t the class independent as functionals they are also l i n e a r l y of analytic boundary v a l u e s b e l o n g i n g t o t h e Holder over independent vector-functions i n D* with class. IB) i n t o | 0 " G ' V f - (2-19) 0 ( 2 . 21) o k k u ( t ) ,. . , u ( t ) also h C u ( t ) , H=m' . the are the solutions solution of this of equation equation. 22) obtain: " * V •lb J 34 " S 3 ? t a . 24) and i(t ). 25) one *(t ) - u(t ) o Hence, ( J ( t ) finds: , t er.

32) condition which and functions teD variables and on D*ur, t o be f o u n d , a n a l y t i c on D*ur, i s analytic In(1-2) with with respect satisfying is to Holder's f(z) is a given i n d o m a i n D* a n d that respect t o z branch of ( i n domain t e r and e q u a l t o z e r o a t z=0. 1 Since class. 1) t o s i n g u l a r inteqral t o zero. 54): Substituting J — r t=ct(z) Integrating both f f sides | e D ( 2 . 3 5 ) , we («(z»- 35! 35) g- with ' C e D ' J = 1 2 ' - 2 < - 3 5 ) obtain: .

9) , t h e n cj(z)=0. n with are simple analytic , . . ,ej (z) ) s a t i s f i e s Let u(z) the theorems. t h e number non-homogeneous I f of numbers. 9). assumed o v e r t h e f i e l d The are g homogeneous e q u a t i o n complex of lemma 2 . 1 . 16) r w h e r e D~ i s t h e c o m p l e m e n t o f D*u Proceeding to the limit at T t o t h e whole z-»t (zeD~) 33 complex p l a n e . 66) we obtain D(t)K(t,t ) d t + - ^ ' . 17), we find t d ( t ) =0, t e r . 1 is proved. 14)): | o(t)f(t)dt, j j = l m;.

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