On the Fredholm and unique solvability of nonlocal elliptic problems in multidimensional domains

The authors study nonlocal elliptic boundary value problems of the form aligned Au&=f_0quadtext{for }xin Q, B_{ij}u&=f_{ij}quadtext{for }xinGamma_i, i=1,dots,N, j=1,dots,m. endaligned Here Au=sum_{|alpha|leq 2m}a_alpha(x)D^alpha u, B_{ij}u=sum_{s=0}^{S_i}(sum_{|alpha|leq m_j}b_{ijalpha s}(x)D^alpha u)(omega_{is}(x))|_{Gamma_i} are boundary operators, Qsubset {bf R}^n is a bounded domain with boundary partial Q=bigcup_{i=1}^NGamma_i, omega_{is} is a diffeomorphism from a neighborhood of Gamma_i such that omega_{is}(Gamma_i) subset Q. The boundary conditions are nonlocal due to the mappings omega_{is}. The operator A is properly elliptic and the Lopatinskiĭ condition is imposed. The authors study solvability of such boundary value problems. A priori estimates of solutions are obtained and the Fredholm property is proved. Unique solvability of the problem with a parameter is shown.