Abstract
In this work we prove, under symmetry and convexity assumptions on the domain \(\Omega \), the non- degeneracy at zero of the Hessian matrix of the Robin function for the spectral fractional Laplacian. This work extends to the fractional setting the results of M. Grossi concerning the classical Laplace operator.
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Notes
The difference between the sign of the results of [12] and the results proven here is due to, in [12], the normalizing constant for the fundamental solution is \(\frac{1}{N(2-N)\omega _N}=-c_{N,1}\), according to our notation, which corresponds to the fundamental solution of negative Laplace operator \(\Delta \). This translates in a positive sign in [12, Eq. 2.9] in contrast to the negative sign in below, which corresponds to the fractional analogue of the classical Green’s representation formula (cf. [10, §2.2.4 Theorem 12]).
It is worth to note the following on the change of variables in the surface integral. Let us express the surface \(\partial _L\mathcal {C}_{\Omega }\) as a level set \(\{(x,y):\Gamma (x,y)=\lambda \}\), for some \(\Gamma (x,y)\) bounding the solid \(\mathcal {C}_{\Omega }\), i.e., \(\mathcal {C}_{\Omega }=\{(x,y):\Gamma (x,y)\le \lambda \}\). By the co-area formula, the change of variables \((x,y)=\Upsilon (z,y)=(\Psi (z),y)\), with \(\Psi (z)\) defined as above, produces
$$\begin{aligned} \begin{aligned} \int \limits _{\Gamma (x,y)=\lambda }\!\!\!\!\!\! f(x,y)d\sigma _{(x,y)}&=\frac{\partial }{\partial \lambda }\int \limits _{\Gamma (x,y)\le \lambda }\!\!\!\! f(x,y)|\nabla \Gamma (x,y)|\,dxdy\\ {}&=\frac{\partial }{\partial \lambda }\int \limits _{\Gamma (\Psi (z),y)\le \lambda }\!\!\!\! f(\Psi (z),y)|\nabla ^\top \Gamma (\Psi (z),y)D\Upsilon ^{-1}(z,y)||D\Upsilon (z,y)|\,dzdy\\&=\frac{\partial }{\partial \lambda }\int \limits _{\Gamma (\Psi (z),y)\le \lambda }\!\!\!\!\! f(\Psi (z),y)|\nabla \Gamma (\Psi (z),y)|\,dzdy\\ {}&=\int \limits _{\Gamma (\Psi (z),y)=\lambda }\!\!\!\!\! f(\Psi (z),y)d\sigma _{(z,y)}=\int \limits _{\Gamma (z,y)=\lambda }\!\!\!\!\! f(\Psi (z),y)d\sigma _{(z,y)}. \end{aligned} \end{aligned}$$The third equality follows since \(|\nabla ^\top \Gamma (\Psi (z),y)D\Upsilon ^{-1}(z,y)|=|\nabla ^\top \Gamma (\Psi (z),y)\Pi ^{-1}|=|\Pi \, \nabla \Gamma (\Psi (z),y)|=|\nabla \Gamma (\Psi (z),y)|\). Finally, as the surface is invariant under the action of \(\Psi \), then \(\{\Gamma (\Psi (z),y)=\lambda \}=\{\Gamma (z,y)=\lambda \}\).
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Acknowledgements
This work has been partially supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation). The author is partially supported by the Ministry of Economy and Competitiveness of Spain, under research project PID2019-106122GB-I00. The author wishes to thank the referees for their careful reading and valuable comments which helped to improving considerably this manuscript.
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Ortega, A. On the Non-degeneracy of the Robin Function for the Fractional Laplacian on Symmetric Domains. Bull. Iran. Math. Soc. 50, 4 (2024). https://doi.org/10.1007/s41980-023-00841-0
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DOI: https://doi.org/10.1007/s41980-023-00841-0