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

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 $(-\Delta {)}^s\varphi =\lambda \varphi$ in $\Omega$, $\varphi \equiv 0$ in ${\mathbb{R}}^n\setminus \Omega$.

The sign of the boundary term in Theorem 1.5 has been corrected

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• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results
• Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics

Cited by in corpus (2)

• Spectral results for mixed problems and fractional elliptic operators
• The Nirenberg problem and its generalizations: A unified approach