Homotopical properties of a class of nonsmooth functions (Q756211)

The author develops an equivariant critical point theory for a class of (non-smooth) lower semicontinuous functionals defined on an open subset of a Hilbert space. This class of functionals f is given by the condition \[ (\alpha -\beta,u-v)\geq -\chi (u,v,f(u),f(v))(1+| \alpha |^ 2+| \beta |^ 2)| u-v|^ 2, \] for any \(\alpha \in \partial^-f(u)\), \(\beta \ni \partial^-f(v)\), where \(\chi\) is a continuous function and \(\partial^-f\) is the subdifferential of f. One of the main steps is the proof of the fact that the set \(D(f):=\{u; f(u)<\infty \}\) equipped with a suitable metric is an absolute neighborhood retract. The author also shows homotopical stability of sublevels of f under \(\Gamma\)-convergence.