On solvability of variational Dirichlet problem for a class of degenerate elliptic operators

Abstract

The paper is devoted to investigation of unique solvability of the Dirichlet variational
problem associated with integro-differential sesquilinear form
????[????, ????] =
Σ︁
????∈????
???????? [????, ????], (*)
where
???????? [????, ????] =
Σ︁
|????|=|????|=????
∫︁
Ω
????(????)2???????? ????????????(????)????(????)(????) ????(????)(????)????????,
Ω — a bounded domain in the euclidian space ???????? with a closed (???? − 1)-dimensional boundary
????Ω, ????(????), ???? ∈ Ω, — a regularized distance from a point ???? ∈ Ω to ????Ω, ???? — a multi-index,
????(????)(????) — a generalized derivative of multi-index ???? of a function ????(????), ???? ∈ Ω, ????????????(????) — bounded
in Ω complex-valued functions, ???? ⊂ {1, 2, . . . , ????} and ???????? , ???? ∈ ????, — real numbers. It is assumed
that ???? ∈ ????. A degeneracy of coefficients of the differential operator associated with the form
(*), is said to be coordinated if there exist a number ???? such that ???????? = ???? + ???? − ???? for all ???? ∈ ????.
Otherwise it is called uncoordinated.
The variational Dirichlet problem associated with the form (*) in the case of coordinated
degeneracy of coefficients is well studied in many papers, where it is also assumed that the form
(*) satisfies a coercivness condition. It should be mentioned that the case of uncoordinated
degeneracy of the coefficients is fraught with some technical complexities and it was only
considered in some separate papers. In this case with the aid of embedding theorems for spaces
of differentiable functions with power weights leading forms ???????? [????, ????], ???? ∈ ????2 ⊂ ????, are separated
and it is proved that solvability of the variational Dirichlet problem is generally depends on the
leading forms.
We consider the case of uncoordinated degeneracy of coefficients of the operator under
investigation and, in contrast to previously published works on this direction, it is allowed that
the main form (*) does not obey coerciveness condition.