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

A. F. M. ter Elst; M. F. Wong

doi:10.1007/s00028-019-00552-2

Journal of Evolution Equations

pp 1–31 | Cite as

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

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A. F. M. ter Elst
M. F. Wong

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First Online: 27 November 2019

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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.

Keywords

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

Mathematics Subject Classification

35K05 35B45 35J25

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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.

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

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

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 $

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.MathSciNetzbMATHGoogle Scholar
2.
Arendt, W. and Elst, A.F.M. ter, Sectorial forms and degenerate differential operators. J. Oper. Theory 67 (2012), 33–72.MathSciNetzbMATHGoogle Scholar
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.MathSciNetCrossRefGoogle Scholar
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.MathSciNetCrossRefGoogle Scholar
5.
Aronson, D. G., Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73 (1967), 890–896.MathSciNetCrossRefGoogle Scholar
6.
Auscher, P., Regularity theorems and heat kernels for elliptic operators. J. Lond. Math. Soc. 54 (1996), 284–296.MathSciNetCrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.MathSciNetCrossRefGoogle Scholar
10.
Choulli, M. and Kayser, L., Gaussian lower bound for the Neumann Green function of a general parabolic operator. Positivity 19 (2015), 625–646.MathSciNetCrossRefGoogle Scholar
11.
Daners, D., Heat kernel estimates for operators with boundary conditions. Math. Nachr. 217 (2000), 13–41.MathSciNetCrossRefGoogle Scholar
12.
Daners, D., Inverse positivity for general Robin problems on Lipschitz domains. Arch. Math. 92 (2009), 57–69.MathSciNetCrossRefGoogle Scholar
13.
Davies, E. B., Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92. Cambridge University Press, Cambridge etc., 1989.Google Scholar
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.MathSciNetzbMATHGoogle Scholar
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.CrossRefGoogle Scholar
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.zbMATHGoogle Scholar
17.
Elst, A.F.M. ter and Robinson, D.W., Local lower bounds on heat kernels. Positivity 2 (1998), 123–151.MathSciNetCrossRefGoogle Scholar
18.
Evans, L. C. and Gariepy, R. F., Measure theory and fine properties of functions. Studies in advanced mathematics. CRC Press, Boca Raton, 1992.zbMATHGoogle Scholar
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.MathSciNetCrossRefGoogle Scholar
20.
Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies 105. Princeton University Press, Princeton, 1983.Google Scholar
21.
Kato, T., Perturbation theory for linear operators. Second edition, Grundlehren der mathematischen Wissenschaften 132. Springer-Verlag, Berlin etc., 1980.Google Scholar
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.MathSciNetCrossRefGoogle Scholar
23.
Ouhabaz, E.-M., Analysis of heat equations on domains, vol. 31 of London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2005.Google Scholar

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A. F. M. ter Elst
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M. F. Wong
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1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgements

References

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