## 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 *A* has Hölder Poisson bounds.

This is a preview of subscription content, access via your institution.

## References

- 1.
Arendt, W. and Elst, A.F.M. ter, Gaussian estimates for second order elliptic operators with boundary conditions.

*J. Oper. Theory***38**(1997), 87–130. - 2.
Arendt, W. and Elst, A.F.M. ter, Sectorial forms and degenerate differential operators.

*J. Oper. Theory***67**(2012), 33–72. - 3.
Arendt, W. and Elst, A.F.M. ter, The Dirichlet problem without the maximum principle.

*Annales de l’Institut Fourier***69**(2019), 763–782. - 4.
Arendt, W., Elst, A.F.M. ter, Kennedy, J. B. and Sauter, M., The Dirichlet-to-Neumann operator via hidden compactness.

*J. Funct. Anal.***266**(2014), 1757–1786. - 5.
Aronson, D. G., Bounds for the fundamental solution of a parabolic equation.

*Bull. Am. Math. Soc.***73**(1967), 890–896. - 6.
Auscher, P., Regularity theorems and heat kernels for elliptic operators.

*J. Lond. Math. Soc.***54**(1996), 284–296. - 7.
Auscher, P. and Tchamitchian, P., Gaussian estimates for second order elliptic divergence operators on Lipschitz and $C^1$ domains. In

*Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998)*, vol. 215 of Lecture Notes in Pure and Appl. Math., 15–32. Marcel Dekker, New York, 2001. - 8.
Behrndt, J. and Elst, A. F. M. ter, Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps.

*J. Spectr. Theory*(2019). In press. - 9.
Choi, J. and Kim, S., Green’s functions for elliptic and parabolic systems with Robin-type boundary conditions.

*J. Funct. Anal.***267**(2014), 3205–3261. - 10.
Choulli, M. and Kayser, L., Gaussian lower bound for the Neumann Green function of a general parabolic operator.

*Positivity***19**(2015), 625–646. - 11.
Daners, D., Heat kernel estimates for operators with boundary conditions.

*Math. Nachr.***217**(2000), 13–41. - 12.
Daners, D., Inverse positivity for general Robin problems on Lipschitz domains.

*Arch. Math.***92**(2009), 57–69. - 13.
Davies, E. B.,

*Heat kernels and spectral theory*. Cambridge Tracts in Mathematics 92. Cambridge University Press, Cambridge etc., 1989. - 14.
Elst, A.F.M. ter and Ouhabaz, E.-M., Partial Gaussian bounds for degenerate differential operators II.

*Ann. Sc. Norm. Super. Pisa Cl. Sci.***14**(2015), 37–81. - 15.
Elst, A.F.M. ter and Ouhabaz, E.-M., Poisson bounds for the Dirichlet-to-Neumann operator on a $C^{1+\kappa }$-domain.

*J. Differ. Equ.***267**(2019), 4224–4273. - 16.
Elst, A.F.M. ter and Rehberg, J., Hölder estimates for second-order operators on domains with rough boundary.

*Adv. Differ. Equ.***20**(2015), 299–360. - 17.
Elst, A.F.M. ter and Robinson, D.W., Local lower bounds on heat kernels.

*Positivity***2**(1998), 123–151. - 18.
Evans, L. C. and Gariepy, R. F.,

*Measure theory and fine properties of functions*. Studies in advanced mathematics. CRC Press, Boca Raton, 1992. - 19.
Gesztesy, F., Mitrea, M. and Nichols, R., Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions.

*J. Anal. Math.***122**(2014), 229–287. - 20.
Giaquinta, M.,

*Multiple integrals in the calculus of variations and nonlinear elliptic systems*. Annals of Mathematics Studies 105. Princeton University Press, Princeton, 1983. - 21.
Kato, T.,

*Perturbation theory for linear operators*. Second edition, Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag, Berlin etc., 1980. - 22.
Nittka, R., Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains.

*J. Differ. Equ.***251**(2011), 860–880. - 23.
Ouhabaz, E.-M.,

*Analysis of heat equations on domains*, vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2005.

## 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.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## The chain condition

### The chain condition

Let \(\Omega \subset \mathbb {R}^d\) be open and connected. We say that \(\Omega \) satisfies the **chain condition** if there exists a \(c > 0\) such that for all \(x,y \in \Omega \) and \(n \in \mathbb {N}\) there are \(x_0,\ldots ,x_n \in \Omega \) such that \(x_0 = x\), \(x_n = y\) and \(|x_{k+1} - x_k| \le \frac{c}{n} \, |x-y|\) for all \(k \in \{ 0,\ldots ,n-1 \} \). Obviously in general \(\Omega \) does not satisfy the chain condition.

### Proposition A.1

Let \(\Omega \subset \mathbb {R}^d\) be open bounded connected and Lipschitz. Then \(\Omega \) satisfies the chain condition.

The proof requires some preparation. Let \(\Omega \subset \mathbb {R}^d\) be open bounded connected and Lipschitz. If \(T > 0\) and \(\gamma :[0,T] \rightarrow \Omega \) is a Lipschitz curve, then \(\gamma \) is differentiable almost everywhere. We define the **length** of \(\gamma \) by \(\ell (\gamma ) = \int _0^T |\gamma '(t)| \, dt\). Define the **geometric distance** \(d :\Omega \times \Omega \rightarrow [0,\infty )\) by *d*(*x*, *y*) is the infimum of \(\ell (\gamma )\), where \(T > 0\) and \(\gamma :[0,T] \rightarrow \Omega \) is a Lipschitz curve with \(\gamma (0) = x\) and \(\gamma (T) = y\). Obviously \(|x-y| \le \ell (\gamma )\) and hence \(|x-y| \le d(x,y)\).

We first consider a special Lipschitz chart.

### Lemma A.2

Let \(U \subset \mathbb {R}^d\) be an open set and \(\Phi \) be a bi-Lipschitz map from an open neighbourhood of \(\overline{U}\) onto an open subset of \(\mathbb {R}^d\) such that \(\Phi (U) = E\) and \(\Phi (\Omega \cap U) = E^-\). Then there are \(c_1,c_2 > 0\) such that \(d(x,y) \le c_1 \, |x-y|\) and \(|x-y| \le c_2\) for all \(x,y \in \Omega \cap U\).

### Proof

Let \(L \in \mathbb {R}\) be larger than both the Lipschitz constant for \(\Phi \) and \(\Phi ^{-1}\). Further, let \(x,y \in \Omega \cap U\). Define \(\gamma :[0,1] \rightarrow \Omega \) by \(\gamma (t) = \Phi ^{-1} ( (1-t) \, \Phi (x) + t \, \Phi (y) )\). Then \(\gamma (0) = x\) and \(\gamma (1) = y\). Moreover, the curve \(\gamma \) is Lipschitz continuous and \(|\gamma '(t)| \le L \, |\Phi (y) - \Phi (x)| \le L^2 \, |y-x|\) for almost every \(t \in [0,1]\). So \(d(x,y) \le \ell (\gamma ) \le L^2 \, |x-y|\). Also \(|x-y| \le L \, |\Phi (x) - \Phi (y)| \le 2 L\). \(\square \)

We next show that the geometric distance is equivalent with the induced Euclidean distance on \(\Omega \).

### Lemma A.3

There exists a \(c > 0\) such that \(|x-y| \le d(x,y) \le c \, |x-y|\) for all \(x,y \in \Omega \).

### Proof

By a compactness argument there are \(N \in \mathbb {N}\) and for all \(k \in \{ 1,\ldots ,N \} \) there are open \(U_k \subset \mathbb {R}^d\) and a bi-Lipschitz map \(\Phi _k\) from an open neighbourhood of \(\overline{U_k}\) onto an open subset of \(\mathbb {R}^d\) such that \(\Phi _k(U_k) = E\) and \(\Phi _k(\Omega \cap U_k) = E^-\); and moreover, \(\Gamma \subset \bigcup _{k=1}^N U_k\). For all \(k \in \{ 1,\dots ,N \} \) fix \(w_k \in \Omega \cap U_k\). Again by compactness there are \(N' \in \{ N+1,N+2,\ldots \} \) and for all \(k \in \{ N+1,\ldots ,N' \} \) there are \(w_k \in \Omega \) and \(r_k > 0\) such that \(B(w_k,r_k) \subset \Omega \) and

By Lemma A.2 there are \(c_1,c_2 \ge 1\) such that \(d(x,y) \le c_1 \, |x-y|\) and \(|x-y| \le c_2\) for all \(k \in \{ 1,\ldots ,N \} \) and \(x,y \in \Omega \cap U_k\). Without loss of generality we may assume that \(2 r_k \le c_2\) for all \(k \in \{ N+1,\ldots ,N' \} \). For simplicity write \(U_k = B(w_k,r_k)\) for all \(k \in \{ N+1,\ldots ,N' \} \). Then \(d(x,y) \le c_1 \, |x-y|\) and \(|x-y| \le c_2\) for all \(k \in \{ N+1,\ldots ,N' \} \) and \(x,y \in U_k\).

We next prove that the geometric distance *d* is bounded on \(\Omega \). Define \(M = 2 c_2 + \max \{ d(w_k,w_l) : k,l \in \{ 1,\ldots ,N' \} \} \). Let \(x,y \in \Omega \). Then there are \(k,l \in \{ 1,\ldots ,N' \} \) such that \(x \in U_k\) and \(y \in U_l\). Hence, \(d(x,y) \le d(x,w_k) + d(w_k,w_l) + d(w_l,y) \le M\). Therefore, *d* is bounded by *M*.

Finally, suppose that there is no \(c > 0\) such that \(d(x,y) \le c \, |x-y|\) for all \(x,y \in \Omega \). Then for all \(n \in \mathbb {N}\) there are \(x_n,y_n \in \Omega \) such that \(d(x_n,y_n) > n \, |x_n - y_n|\). It follows that \(|x_n - y_n| \le \frac{M}{n}\) for all \(n \in \mathbb {N}\). The sequence \((x_n)_{n \in \mathbb {N}}\) is bounded since \(\Omega \) is bounded. Passing to a subsequence if necessary, we may assume that the sequence \((x_n)_{n \in \mathbb {N}}\) is convergent. Let \(x = \lim _{n \rightarrow \infty } x_n\). Then \(\lim _{n \rightarrow \infty } y_n = x\) and \(x \in \overline{\Omega }\). Since \(\overline{\Omega } \subset \bigcup _{k=1}^N U_k \cup \bigcup _{k=N+1}^{N'} B(w_k,r_k)\), there exists a \(k \in \{ 1,\ldots ,N' \} \) such that \(x \in U_k\). Because \(U_k\) is open there exists an \(N_0 \in \mathbb {N}\) such that \(x_n \in U_k\) and \(y_n \in U_k\) for all \(n \in \mathbb {N}\) with \(n \ge N_0\). Finally, choose \(n \in \mathbb {N}\) such that \(n \ge \max \{ N_0,c_1 \} \). Then

This is a contradiction. \(\square \)

Now we are able to prove the proposition.

### Proof of Proposition A.1

Let \(c > 0\) be as in Lemma A.3. Let \(x,y \in \Omega \) and \(n \in \mathbb {N}\). Since the case \(x=y\) is trivial, we may assume that \(x \ne y\). There exist \(T > 0\) and a Lipschitz curve \(\gamma :[0,T] \rightarrow \Omega \) such that \(\gamma (0) = x\), \(\gamma (1) = y\) and \(\ell (\gamma ) \le 2 d(x,y)\). For all \(k \in \{ 1,\ldots ,n-1 \} \) let

which exists by continuity. Set \(x_k = \gamma (t_k)\). Further define \(x_0 = x\) and \(x_n = y\). Then

for all \(k \in \{ 0,\ldots ,n-1 \} \), as required. \(\square \)

## Rights and permissions

## About this article

### Cite this article

ter Elst, A.F.M., Wong, M.F. Hölder kernel estimates for Robin operators and Dirichlet-to-Neumann operators.
*J. Evol. Equ.* **20, **1195–1225 (2020). https://doi.org/10.1007/s00028-019-00552-2

Published:

Issue Date:

### Keywords

- Robin boundary conditions
- Dirichlet-to-Neumann operator
- Heat kernel estimates

### Mathematics Subject Classification

- 35K05
- 35B45
- 35J25