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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)
##### MSC:
 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
##### Keywords:
regularity at the boundary
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