Criteria for validity of the maximum modulus principle for solutions of linear strongly elliptic second order systems (Q1803186)

The systems under consideration are of the form (E) \(\sum A_{ij} \partial^ 2_{ij} u - \sum A_ j \partial_ ju - A_ 0u = 0\), where the \(A\)'s are sufficiently smooth \(m \times m\) matrix valued functions in \(\Omega \subset \mathbb{R}^ n\). The maximum modulus principle can be stated in two ways: 1) \(\| u \|_{C (\omega)} \leq \| u \|_{C (\partial \omega)}\) for all solutions of (E) in \(\Omega\) and all sufficiently regular \(\omega \subset \Omega\). 2) (E) has no solution with modulus attaining its strict local maximum in the interior of \(\Omega\). Either formulation is equivalent to the following condition: \(A_{ij} = Aa_{ij}\), where \(A\) is an \(m \times m\) matrix, \(a_{ij}\) are scalars and \(\sum a_{jk} \xi_ j \cdot \xi_ k + \sum A^{-1} A_ j \xi_ j \cdot \zeta + A^{-1} A_ 0 \zeta \cdot \zeta \geq 0\) for all \(\xi_ j\), \(\zeta \in \mathbb{R}^ m\) such that \(\xi_ j \cdot \zeta = 0\). It is shown that the condition 1) for \(\omega = \Omega\) alone does not imply the asserted form of the matrices \(A_{ij}\). The above result is a consequence and extension of the authors' previous work for systems with constant coefficients [Math. USSR, Sb. 53, 457-479 (1986); transl. from Mat. Sb., Nov. Ser. 125, No. 4, 458-480 (1984; Zbl 0577.35003)].