A new rearrangement inequality and its application for $$L^2$$L2-constraint minimizing problems

By Masataka Shibata. 2013

77 citations recorded by the source corpus.

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, $$H^1$$H1-precompactness of minimizing sequence of $$L^2$$L2-constraint minimizing problem is very important. Usually, by using the concentration compactness principle and some scaling argument, $$H^1$$H1-precompactness is obtained. To prove $$H^1$$H1-precompactness without scaling arguments, we introduce the rearrangement. The Steiner rearrangement is defined as a map from $$H^1$$H1 to $$H^1$$H1, whereas our rearrangement $$(\cdot \star \cdot )$$(·⋆·) is defined as a map from $$H^1 \times H^1$$H1×H1 to $$H^1$$H1. By using the rearrangement, we show a strict inequality $$\Vert \nabla (u \star v) \Vert _{L^2}^2 < \Vert \nabla u\Vert _{L^2}^2 + \Vert \nabla v\Vert _{L^2}^2$$‖∇(u⋆v)‖L22<‖∇u‖L22+‖∇v‖L22 under simple assumptions.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/458fac6953120328fb980a02fc1dd3b7842c125c

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Multiple positive normalized solutions for nonlinear Schrödinger systems (2017, methodology): “To prove the compactness of the minimizing sequences, instead of trying directly to check the strict inequalities proposed by Lions [25, 26] we make use of the rearrangement introduced by Shibata [35] as presented in [21, lemma A.”