Multivalued fixed point index and nonlocal problems involving relative rearrangement

We apply the fixed point index theory for multivalued problems to nonlocal problems involving relative rearrangement mappings. These applications need an extension of the Almgren–Lieb theorem and an improvement of the strong continuity of the relative rearrangement mapping on arbitrary open sets (bounded or not). The proofs of these results are based on some pointwise equations relating the relative rearrangement of the gradient and the weighted De Giorgi perimeter.