The Łojasiewicz gradient inequality in the infinite-dimensional Hilbert space framework (Q531832)

Let \(V\) be a real Hilbert space with dual \(V'\). An analytic function \(F: U\to\mathbb{R}\), where \(U\) is an open subset of \(V\), is said to satisfy a Łojasiewicz gradient inequality if, for any \(a\in U\), there exists a \(\theta\in (0,1/2)\), a neighborhood \(W\) of \(a\) in \(U\) and \(c> 0\) such that \(\| DF(u)\|> c|F(u)- F(a)|^{1-\theta}\) for all \(u\) in \(W\). In the case of a quadratic \(F\), \(F(u)= 1/2(Au, u)\), with \(A\) linear selfadjoint, Theorem 2.1 claims that \(F\) satisfies the property at the origin for some \(\theta> 0\) if and only if it satisfies the property at any point for \(\theta= 1/2\) and if and only if \(0\) is not an accumulation point of \(\text{spec}(A)\). In the nonlinear case, it is shown that, if the second derivative is ``bad'', the functional cannot satisfy the inequality with \(\theta= 1/2\). An almost optimal result is given in Theorem 4.1, providing a sufficient condition in terms of \(D^2F(0)\) for the inequality at the origin. Finally, some applications to nonlinear partial differential equations are given.