Solvability of inhomogeneous boundary-value problems for fourth-order differential equations
We consider a Cauchy-type boundary-value problem of, a problem with three boundary conditions, and the Dirichlet problem for a general fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain $\Omega \subset R^2$ with smooth boundary. Using the method of the Green formula, the theory of expansion of differential operators, and the theory of $L$-traces (i.e., traces associated with a differential operation $L$), we obtain necessary and sufficient (for elliptic operators) conditions for the solvability of each of the problems under consideration in the space $H^m(\Omega),\;\; m \geq 4$.
English version (Springer): Ukrainian Mathematical Journal 63 (2011), no. 8, pp 1165-1175.
Citation Example: Buryachenko K. O. Solvability of inhomogeneous boundary-value problems for fourth-order differential equations // Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1011-1020.