Read e-book online A ''sup+ c inf'' inequality for Liouville-type equations PDF

By Bartolucci D.

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Step 4: We carry out the operations P[2+1u(s+2)] and P[3+1u(-s-2)] A 4 ( s) 0 0º ª 1 « s  2 ( s  2)(2s  5) 0 » . ¬ ¼ Step 5: We carry out the operation L[2+1u(-s-2)] and P[2u1 / 2] A s (s) 0 0º ª1 « ». 3). , ir d1d 2 ... d r . 1) can be written in the form A S (s) ª d1 «0 « «# « «0 «0 « «# «0 ¬ 0 ! 0 d1d 2 ! 0 # % 0 ! d1d 2 ... d r # 0 ! 0 # % # 0 ! 0 0 0 ! 0º 0 0 ! 0 »» # # % #» » 0 0 ! 0» . 0 0 ! 0» » # # % #» 0 0 ! 2. 4) where Dk(s) is the greatest common divisor of all minors of degree k of matrix A(s) (D0(s) = 1).

0 # 0 % ! % ! # 1 # 0 % ! % ! # 0 # 0 ! % ! 0º 0 »» #» » 0» . 1) 22 Polynomial and Rational Matrices The same operations carried out on columns are equivalent to postmultiplication of the matrix A(s) by the following matrices: i -th column Pm (i, c) ª1 «0 « «# « «0 «# « «¬ 0 0 ! 0 0 ! 0º 1 ! 0 0 ! 0 »» # % # # % #»   nun , » 0 ! c 0 ! 0 » i -th row # % # # % #» » 0 ! 0 0 ! 1 »¼ i Pd (i, j , w( s )) ª1 «0 « «# « «0 «0 « «# «0 ¬ 0 ! 1 ! # % 0 ! 0 ! # % 0 ! i Pz (i, j ) ª1 «0 « «# « «0 «# « «0 «# « ¬«0 0 !

S  skq ) k mrk , qk its elementary divisors. 4) is the sum of the sets of elementary divisors of Ak(s), k = 1,2. 1. 4) for A1 ( s ) 0 º ªs  1 1 « 0 s  1 1 »» , A 2 ( s ) « «¬ 0 0 s  1»¼ 0 º ªs  1 1 « 0 s 1 0 »» . 5) are A1S ( s ) 0 º ª1 0 «0 1 » , A ( s) 0 2S « » «¬ 0 0 ( s  1)3 »¼ 0 ª1 0 º «0 1 ». 5) are thus equal (s  1)3, (s  1)2, and (s  2), respectively. 7) and its elementary divisors are (s - 1)2, (s - 1)3, (s - 2). Consider a matrix A nun and its corresponding polynomial matrix [Ins - A].

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A ''sup+ c inf'' inequality for Liouville-type equations with singular potentials by Bartolucci D.


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