A new rearrangement inequality and its application for L 2 -constraint minimizing problems

In this paper, we introduce a new type rearrangement inequality based on the Steiner rearrangement. To show orbital stability of standing wave solutions with respect to a nonlinear Schrödinger equation, H 1 -precompactness of minimizing sequence of L 2 -constraint minimizing problem is very important. Usually, by using the concentration compactness principle and some scaling argument, H 1 -precompactness is obtained. To prove H 1 -precompactness without scaling arguments, we introduce the rearrangement. The Steiner rearrangement is defined as a map from H 1 to H 1 , whereas our rearrangement ( · ⋆ · ) is defined as a map from H 1 × H 1 to H 1 . By using the rearrangement, we show a strict inequality ‖ ∇ ( u ⋆ v ) ‖ L 2 2 < ‖ ∇ u ‖ L 2 2 + ‖ ∇ v ‖ L 2 2 under simple assumptions.