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Existence theorems for some elliptic systems. (English) Zbl 0826.35033
The following theorem is proved: Consider the elliptic system \[ - \Delta u_j = f_j (x, u_1, \ldots, u_m), \;j = 1, \ldots, m, \text{ in } \Omega, \quad u_j = \psi_j, \;j = 1, \ldots, m, \text{ on } \partial \Omega, \] where \(\Omega\) is a bounded Hölder smooth domain in \(\mathbb{R}^n\), \(n \geq 1\) and \(f_j\) is locally Hölder continuous with exponent \(\alpha\). Assume that for some suitable constants \(a_{jk}\), \(c_j\), \(|f_j (x,u_1, \ldots, u_m) |\leq \sum_k a_{jk} |u_k |+ c_j\), and that the lowest eigenvalue of the Laplacian on \(\Omega\) is greater or equal than the spectral radius of the matrix \(a_{jk}\). Then if the Dirichlet data are in \(C^{2, \alpha} (\partial \Omega)\) the solution is also in \(C^{2, \alpha} (\partial \Omega)^m\).
Reviewer: A.Bove (Bologna)
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: EuDML