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A symmetry theorem for variational problems. (English) Zbl 0725.47013
In the present paper nonlinear equations of the form \[ (1)\quad Au=F(u),\quad u\in H, \] are considered. Here H is a Hilbert space, A: \(H\to H\) is a self-adjoint operator and F: \(H\to H\) is a potential operator, i.e. \(F=\text{grad} G\) with G: \(H\to R\). The author provides some assertions regarding the solvability of the system and the existence of special solutions respectively. Obviously (1) can be regarded as optimality condition of the variational problem \[ (2)\quad \min \{f(u)=<Au,u>-G(u)| \quad u\in H\}. \] Accordingly the proof of the theorems are constructed in such a way that by a suitable decomposition of H in orthogonal subspaces (this is guaranteed by assumptions to the spectrum of A) and by monotony demands of F convexity properties in (2) and thus the existence of solutions of (1) can be pointed out.
Specifying A and F finally the author formulates special results for ordinary differential equation systems, for the Laplace equation, for discrete dynamical systems and for systems of nonlinear wave equations respectively.

MSC:
47A50 Equations and inequalities involving linear operators, with vector unknowns
47B25 Linear symmetric and selfadjoint operators (unbounded)
49R50 Variational methods for eigenvalues of operators (MSC2000)
47N20 Applications of operator theory to differential and integral equations
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