Uniqueness properties of functionals with Lipschitzian derivative
Summary: We prove that if \(X\) is a real Hilbert space and if \(J:X\to\mathbb R\) is a \(C^1\) functional whose derivative is Lipschitzian, with Lipschitz constant \(L\), then, for every \(x_0\in X\), with \(J'(x_0)\neq 0\), the following alternative holds: either the functional \(x\to\frac12\|x-x_0\|^2- \frac1L J(x)\) has a global minimum in \(X\), or, for every \(r>J(x_0)\), there exists a unique \(y_r\in J^{-1}(r)\) such that \(\|x_0-y_r\|= \text{dist}(x_0,J^{-1}(r))\) and, for every \(r>0\), the restriction of the functional \(J\) to the sphere \(\{x\in X:\|x-x_0\|=r\}\) has a unique global maximum.
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