Gårding inequality for higher order elliptic operators with a non-power degeneration and its applications

For higher order elliptic operators in an arbitrary (bounded or unbouned) domain in $n$-dimensional Euclidean space $R_n$ with a non-power degeneration we prove a weighted analogue of Carding inequality. By means of this inequality we study the unique solvability of the Dirichlet variational problem, whose solution is sought in the closure of the class of infinitely differentiable compactly supported functions. The degeneration of the coefficients in different variables is characterized via different functions. The lower coefficients of the operators are assumed to belong to some weighted $L_p$-spaces. For one class of elliptic operators with a power degeneration in a half-space we study the solvability of variational Dirichlet problem with inhomogeneous boundary conditions.