Classical solutions for a class of degenerate elliptic operators with a parameter
Reiko Sakamoto · Kyoto journal of mathematics · 1988
Boundary value problems are generally investigated fo r elliptic differential operators, degenerating on the boundary o f a domain ( [ 1 ] , [ 2 ] , [ 3 ] , etc.), which are well posed in appropriate Hilbert spaces.On the other hand, classical solutions seem little investigated except fo r equations o f second order ( [ 4 ] , [ 5 ] , etc.).In this paper, we consider a class o f degenerate elliptic differential operators with a positive parameter, and we seek classical solutions, restricting the parameter small enough.In §1, the regularity of solutions are considered for F -ty p e operators, analogously in [ 6 ] .In § 2 , two types of half space problems are set for F -ty p e operators corresponding to the location of the invertible zon e.In § 3 , the existence of solutions fo r half space problems is considered fo r F -ty p e elliptic operators with a parameter, using th e energy estimates for adjoint operators ( [ 7 ] ) .In § 4 , some examples of 4 th order operators are given.§ 1 .R e g u la r it y fo r F-type o p e r a to r s. .. F -ty p eo p e r a t o r s .Let x = (x i , x2,..., xn ) = (x i , x') R n , and let A (x ; D )= E a " (x )D '-= E a i(x ; D ') where D = (D 1 , D2,..•, D n) = (D 1 , D '), D i= D x j =-i a a x i , a"(x)EM -(Rn), ani (x ; e ')* 0 near x1 = 0 , and ai,(0, x '; e ')0 0 for some j o (0 ,j0 m ).Let / i be an integer such that D fai(O, x'; for 1= 0 , / i-1, and