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

Abstract

Consider the elliptic operator

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

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.

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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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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(xy) 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

$$\begin{aligned} \overline{\Omega } \subset \bigcup _{k=1}^N U_k \cup \bigcup _{k=N+1}^{N'} B(w_k,r_k) . \end{aligned}$$

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

$$\begin{aligned} n \, |x_n - y_n| < d(x_n,y_n) \le c_1 \, |x_n - y_n| \le n \, |x_n - y_n| . \end{aligned}$$

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

$$\begin{aligned} t_k = \min \{ t \in [0,T] : \ell (\gamma |_{[0,t]}) = \frac{k \, \ell (\gamma )}{n} \} , \end{aligned}$$

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

$$\begin{aligned} |x_{k+1} - x_k| \le d(x_{k+1},x_k) \le \frac{\ell (\gamma )}{n} \le \frac{2 d(x,y)}{n} \le \frac{2 c}{n} \, |x-y| \end{aligned}$$

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

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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

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Keywords

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

Mathematics Subject Classification

  • 35K05
  • 35B45
  • 35J25