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A Note on Well-posed Problems

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A recent result by Ricceri [Ri] states that a \({C^{1,1}_{loc}}\) function \({f : X \to {\mathbb R}}\), where X is a Hilbert space, attains its minimum on any small closed ball around a point where its derivative does not vanish, and that the unique minimum point belongs to the boundary of the ball. The proof is based on a saddle-point theorem. We show that the result, which we extend to Banach spaces having a norm with modulus of convexity of power type 2, can be obtained by means of a purely variational argument.

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Correspondence to Roberto Lucchetti.

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The work of the second author was partially supported by the PRIN project “Variational and topological methods in the study of nonlinear phenomena”.

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Corvellec, JN., Lucchetti, R. A Note on Well-posed Problems. Mediterr. J. Math. 8, 181–189 (2011).

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