# Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators

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## Abstract

Consider the elliptic operator on a bounded connected open set \(\Omega \subset \mathbb {R}^d\) with Lipschitz boundary conditions, where \(c_{kl} \in L_\infty (\Omega ,\mathbb {R})\) and \(a_k,b_k,a_0 \in L_\infty (\Omega ,\mathbb {C})\), subject to Robin boundary conditions \(\partial _\nu u + \beta \, {\mathop {\mathrm{Tr \,}}}u = 0\), where \(\beta \in L_\infty (\partial \Omega , \mathbb {C})\) is complex valued. Then we show that the kernel of the semigroup generated by \(-A\) satisfies Gaussian estimates and Hölder Gaussian estimates. If all coefficients and the function \(\beta \) are real valued, then we prove Gaussian lower bounds. Finally, if \(\Omega \) is of class \(C^{1+\kappa }\) with \(\kappa > 0\), \(c_{kl} = c_{lk}\) is Hölder continuous, \(a_k = b_k = 0\) and \(a_0\) is real valued, then we show that the kernel of the semigroup associated to the Dirichlet-to-Neumann operator corresponding to

$$\begin{aligned} A = - \sum _{k,l=1}^d \partial _k \, c_{kl} \, \partial _l + \sum _{k=1}^d a_k \, \partial _k - \sum _{k=1}^d \partial _k \, b_k + a_0 \end{aligned}$$

*A*has Hölder Poisson bounds.## Keywords

Robin boundary conditions Dirichlet-to-Neumann operator Heat kernel estimates## Mathematics Subject Classification

35K05 35B45 35J25## Notes

### Acknowledgements

The authors wish to thank Wolfgang Arendt for many discussions at various stages of this project. The authors wish to thank Mourad Choulli for helpful comments regarding the chain condition. Moreover, the authors wish to thank the referees for their comments. The first-named author is most grateful for the hospitality extended to him during a fruitful stay at the University of Ulm. He wishes to thank the University of Ulm for financial support. Part of this work is supported by an NZ-EU IRSES counterpart fund and the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. Part of this work is supported by the EU Marie Curie IRSES Program, Project ‘AOS’, No. 318910.

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