Local integration by parts and Pohozaev identities for higher order fractional Laplacians

Xavier Ros-Oton; Joaquim Serra

doi:10.3934/dcds.2015.35.2131

Article Contents

2015, Volume 35, Issue 5: 2131-2150. Doi: 10.3934/dcds.2015.35.2131

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Local integration by parts and Pohozaev identities for higher order fractional Laplacians

Xavier Ros-Oton^1, and
Joaquim Serra^2,

1.
The University of Texas at Austin, Department of Mathematics, 2515 Speedway, Austin, TX 78751
2.
Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Avda. Diagonal 647, 08028 Barcelona

Received: May 31, 2014

Revised: August 31, 2014

Published: May 2017

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Abstract

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\Delta)^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$.
As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-\Delta)^s\phi=\lambda\phi$ in $\Omega$, $\phi\equiv0$ in $\mathbb{R}^n\setminus\Omega$.

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Mathematics Subject Classification: Primary: 47G30, 35S15.

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