A Boundary Value Problem for a Second-Order Singular by Larin A. A.

By Larin A. A.

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Extra info for A Boundary Value Problem for a Second-Order Singular Elliptic Equation in a Sector on the Plane

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Thus, the following theorem is proved. 2. (x) E s T), k ~ n - x). 2. C a s e 2. WM. 4. 1) from the W F ' s {ffElk(A)}~=2, n. r ~ of the W F ' s is a P~-system for x = { 2 , . . ,n}, and At # 0, A~ # 0, l > 1. 1 we obtain the following theorem. 3. 4 has a unique solution in the class p~(x) E PA[O, T), k = O , n - 2. 1. T h e counterexample from Sec. 3 shows that there are no P~-systems for ~ = { 2 , . . , n} with the exeption of the first row of the WM. Remark. 1) w i t h the conditions y(k)(0) = y(T) .

For the matrix g2*(A) de_f (9)t(A))-X to have a simple pole at ko it 'is necessary and suflficient that 91(ko)gt(ko) = 0. Proof. The necessity part of the lemma is obvious. We prove the sufficiency. Let 91(Ao)~(Ao) = 0. Denote by Xp the set of matrices A = [A,j],,j=~,~ such that A,j = 0 for j - z~ < n - p. It is clear that if A ~ Xp, B ~ Xq, then A B ~ Xp+q_n. Since 9)l(A)9)I*(A) = E, it follows that oo 9)I*(A)= n--k-I (A-Ao)/~9")I~>(Ao), ~ E ffR~_j_~>(Ao)~(Ao)=0, k=l,n-1. j=--i k=l-n From this, in view of the relation 9~(ko)~R(ko) = 0, we ~_~>(~o)=-~_~>(~o)m(~o)- n-k obtain (~(~0)) -~, ~_~_~>(~o)~<~>(~o) \ ~=~ / m~<_j_~+~>(ao)~<~>(ao)(~(ao))-M(~o), ~m<~_~>(ao)m(ao) = - ~= ~ , .

The eigenvalues {A~}~_>l and {7~}n_>~ of Q1 and Q2 coincide with the zeros of A(A) and 5(A) respectively. Let f(x) e C2[0,77], f ( 0 ) = f ( T r ) = 0 , and let FN = {A: IAI = RN} be circumferences in the A-plane with radii RN -+ ec such that FN are e > 0 distant from the spectra {A~} and {Tn}. ,)+O2n2a~(X,~n), yn(x) = (A(A~)X(A~))-~ ( ( b 3 ( x , ? ) - b 3 ( x , An)) - kn(a3(Z, An) - a3(x,A~))), e~_ ~,1(77,~,) (a~(77,~)-~(~,~)) end= ~(77'~')6(~') and the series converge absolutely and uniformly for [0, 7r].

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