On Equivariant Boundary Value Problems

smooth coefficients aα(x), L+ be a formally adjoint differential operation. Let L0, L0 be minimal operators (i.e., for example, D(L0) is the clozure of C∞ 0 (Ω) in the norm of the graph ‖u‖L = ‖u‖L2(Ω)+‖Lu‖L2(Ω)), and L, L+ be maximal expansions of L,L+ in the space L2(Ω) respectively (i.e. L = (L0 ) ∗, L+ = (L0)∗), L̃ = L|D(L̃) where D(L̃) is the clozure of C∞(Ω̄) in the norm of the graph ‖u‖L and it is analogous for L̃+. M.Yo. Vishik introduced the conditions: V1) the operator L0: D(L0) → L2(Ω) has a continuous left-inverse, V2) the operator L0 : D(L + 0 ) → L2(Ω) has a continuous left-inverse and proved that 1) these conditions are necessary and sufficient for the existence of a solvable expansion LB: D(LB) → L2(Ω), (that is D(L0) ⊂ D(LB), ∃ L−1 B : L2(Ω) → D(LB)); 2) under conditions V1), V2) for any solvable expansion LB the following decomposition of the domain D(L) is valid: D(L) = D(L0) + kerL+B, and L: B → kerL+ is an isomorphism. Let G be some Lie group, smoothly acting in the closed domain Ω̄. It means, that there is a group of diffeomorphisms Ug: Ω̄ x → g · x = Ug(x) ∈ Ω̄ of domain Ω̄ onto itself, group, smoothly depending on an element of G, and mapping g → Ug is a homomorphism of groups. Thus the contraction of diffeomorphisms Ug on boundary ∂Ω induces a smooth action of group G on boundary ∂Ω. The action of group G on domain Ω̄ generates a representation of the group G in function spaces: (gu)(x) = u(g−1x) (homomorphism of group G into group of converted operators). Such representation is induced on spaces C∞ 0 (Ω), C∞(Ω), Hm(Ω), H−m(Ω), D′(Ω), H(m)(Ω), H(−m)(Ω) and others. Let the differential operation L be invariant with respect to the action of group G, that is g(Lu) = L(gu). Then spaces D(L), D(L0), C(L), kerL are invariant with respect to the action of the group. If the action of group preserves the volume of the domain Ω then the scalar product in the space L2(Ω) is invariant with respect to the action of group G, and consequently the representation of the group G is unitary in this space. In this case the operation L+ is also invariant with respect to an action of group G, the spaces D(L+), D(L0 ), C(L +), kerL+ are invariant. Boundary value problem