On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain

Bakytbek Danebekovich Koshanov, Alexander P. Soldatov

Contemporary Mathematics Fundamental Directions, 2021

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DOI 10.22363/2413-3639-2021-67-3-564-575

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Abstract

We consider the generalized Neumann problem for a 2lth-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the (kj-1){(k_j - 1)}th-order normal derivatives where 1≤k1...kl≤2l{1\le k_1 < ... < k_l \le 2l}; for kj=j{k_j = j} it turns into the Dirichlet problem, and for kj=j+1{k_j = j +1} into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Holder spaces is given.