On boundary value problems for first-order elliptic pseudosymmetric systems in \(\mathbb R^4\). (English. Russian original) Zbl 1070.35010

Differ. Equ. 39, No. 3, 444-446 (2003); translation from Differ. Uravn. 39, No. 3, 410-412 (2003).
From the introduction: In a bounded domain \(\Omega\subset\mathbb{R}^4\) whose boundary is a sufficiently smooth three-dimensional manifold \(\partial\Omega\), we consider the boundary value problem of finding a solution \(U=U(x)\) of the elliptic system of differential equations \[ \sum^4_{j=1}A_j\frac{\partial U} {\partial x_j}=f(x),\quad x\in\Omega,\tag{1} \] satisfying the boundary conditions \[ {\mathcal B}(y, \partial/\partial x)|_{x\to y}U=g(y),\quad y\in\partial\Omega. \tag{2} \] Here the \(A_j\) \((j=1,2,3,4)\) are constant real matrices of the fourth order; moreover, \(A_1\) is the identity matrix, and the remaining matrices are skew-symmetric, and \({\mathcal B}\) is a \(2\times 4\) matrix boundary operator consisting of scalar linear sufficiently smooth pseudodifferential operators “polynomial” in the normal to \(\partial\Omega\). Theorem. For an arbitrary boundary operator \({\mathcal B}\), the boundary value problem (1), (2) is not regularizable.


35F15 Boundary value problems for linear first-order PDEs
Full Text: DOI