Equation–Domain Duality in the Dirichlet Problem for General Differential Equations in the Space $L_2$

A development of an author's observation that led to the creation of the equation–domain duality method is presented. This method is used in the study of the Dirichlet problem for a general partial differential equation in a semialgebraic domain. The exposition involves results of the general theory of boundary value problems and is aimed at extending these results to the generalized statements of such problems in $L_2(\Omega )$. Results on the boundary properties of the $L_2$-solution of a general linear partial differential equation in a domain are employed. It is demonstrated how the general construction under consideration is used in the study of the Dirichlet problem for specific equations with constant coefficients on the basis of the equation–domain duality method. It is also shown how one can extend to the generalized statement of the Dirichlet problem the earlier obtained necessary and sufficient conditions for the existence of a nontrivial smooth solution to the homogeneous Dirichlet problem for a general second-order equation with constant complex coefficients and a homogeneous symbol in a disk, as well as for an ultrahyperbolic equation in the $n$-dimensional ball.