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October, 2018 Diffusion with nonlocal Robin boundary conditions
Wolfgang ARENDT, Stefan KUNKEL, Markus KUNZE
J. Math. Soc. Japan 70(4): 1523-1556 (October, 2018). DOI: 10.2969/jmsj/76427642


We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq \beta\in L^{\infty}(\partial\Omega)$ and $\mu\colon\partial\Omega\to{\mathscr{M}}(\overline{\Omega})$, and boundary conditions of the form $$ \partial_{\nu}^{{\mathscr{A}}}u(z)+\beta(z)u(z)=\int_{\overline{\Omega}}u(x)\mu(z)(\mathrm{d}x), \quad z\in\partial\Omega, $$ where $\partial_{\nu}^{{\mathscr{A}}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $\mu$ we show that $A_{\beta, \mu}$ generates a holomorphic semigroup $T_{\beta,\mu}$ on $L^{\infty}(\Omega)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overline{\Omega})$. Its restriction to $C(\overline{\Omega})$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.


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Wolfgang ARENDT. Stefan KUNKEL. Markus KUNZE. "Diffusion with nonlocal Robin boundary conditions." J. Math. Soc. Japan 70 (4) 1523 - 1556, October, 2018.


Received: 24 October 2016; Revised: 23 April 2017; Published: October, 2018
First available in Project Euclid: 3 October 2018

zbMATH: 07009711
MathSciNet: MR3868216
Digital Object Identifier: 10.2969/jmsj/76427642

Primary: 35B35 , 47D07 , 60J35

Keywords: diffusion process , non-local boundary condition , stability

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 4 • October, 2018
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