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The normality of some elliptic functional-differential operators of second order

Evgenii M Varfolomeev

2006
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Russian Mathematical Surveys
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In non-linear optical systems with a transformation of the field in two-dimensional feedback, there arise various regular periodic phenomena called "multipetal waves" [1], [2] . These light structures are used in modern computer technology for creating optical analogues of neuron networks. Such systems are modelled mathematically in terms of bifurcations of periodic solutions of quasi-linear parabolic functional-differential equations with a transformation g(x) of the space variables. This
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... ariables. This problem was considered in [3], [4] in the case when the domain Q is a disc or annulus and g is a rotation through some angle θ. The case when Q ⊂ R 2 and g are arbitrary was studied in [5], [6] . It was assumed in these papers that the linearized elliptic functional-differential operator is normal. Necessary and sufficient conditions for this operator to be normal were obtained in [7] in terms of Q ⊂ R n and g. More general cases without the assumption of normality were investigated in [8] . Here we obtain necessary and sufficient conditions for the normality of the linearized operator in the case of two transformations of the space variables. Let Q ⊂ R n be a bounded domain with boundary ∂Q ⊂ C ∞ , n 2. Let g, f be one-to-one transformations of class C 3 such that Here V is the bounded domain with Q ⊂ V , Jg(x) = [∂gi/∂xj] n i,j=1 is the Jacobian matrix of g and |Jg(x)| = |det Jg(x)|. Also let g(Q) ⊂ Q and f (Q) ⊂ Q. Consider the unbounded operator A0 : L2(Q) → L2(Q) with domain of definition D(A0) = {v ∈ W 2 2 (Q) : Bv = 0} and given by A0v = ∆v, v ∈ D(A0). Here W k 2 (Q) is the Sobolev space of complex-valued functions lying in L2(Q) along with all their generalized derivatives up through order k, and Bv = v˛∂ Q or Bv = (∂v/∂ν)˛∂ Q , where ν is the unit inner normal vector to ∂Q at the point x ∈ ∂Q. It is known that A0 is self-adjoint. Put A : L2(Q) → L2(Q), A = A0 + A1 + A2, where A1, A2 are bounded linear operators defined on the whole space L2(Q) as follows: where a1 = 0, a2 = 0 are real numbers. An operator A is said to be normal if D(AA * ) = D(A * A) and AA * v = A * Av for all v ∈ D(A * A). We put D(A) = D(A0). Consider the sets G m g = {x ∈ Q : g m (x) = x}, m = 1, 2, . . . , where g m (x) denotes the result of applying g m times. Write e G m g = Q \ G m g . We also write the superposition of transformations in the form f g(x), g −1 f (x) and so on.

doi:10.1070/rm2006v061n01abeh004303
fatcat:ivnh2lsow5dk7ddgb3tewiscci