On generalized solutions of boundary value problems for some general differential equations (Q2735875)

The author introduces and considers notions of a generalized solution of the Dirichlet problem, the Neumann problem, and the general boundary value problem for differential equations of the form \(L^+\cdot A\cdot Lu= f\) in an arbitrary domain \(\Omega \subset \mathbb{R}^n\), where \(L= \sum_{\alpha\leq m} a_\alpha(x) D^\alpha\) is an arbitrary differential operation with complex \(j\times k\)-matrix coefficients, all elements of which are \(C^\infty (\overline{\Omega})\)-functions depending on \(x\in \overline{\Omega}\), \(L^+\) is formally adjoint operation, \(A: L_2(\Omega)\to L_2(\Omega)\) is some continuous operator. One could write in such form the Poisson equation and generally any quasilinear divergent PDE of the second order. The function \(u\) from the definition domain \(D(L_0)\) of the minimal expansion \(L_0\) of \(L\) is called a generalized solution of the Dirichlet problem for such equation, if for each \(\varphi\in C_0^\infty (\Omega)\) the ``integral identity'' \(\langle AL_0u, L\varphi\rangle= \langle f,\varphi\rangle\) holds. The author proves that this problem is well-posed, if and only if both conditions are fulfilled: 1) the Vishik condition (there exists a continuous left inverse \(L_0^{-1}\)) holds, 2) the operator \(PA: \text{Im }L_0\to \text{Im } L_0\), where \(P: L_2(\Omega)\to \text{Im }L_0\) is the orthoprojector, is a homeomorphism. There are analogous results on the Neumann problem and other problems.