## Journal of the Mathematical Society of Japan

### Diffusion with nonlocal Robin boundary conditions

#### Abstract

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.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 4 (2018), 1523-1556.

Dates
Revised: 23 April 2017
First available in Project Euclid: 3 October 2018

https://projecteuclid.org/euclid.jmsj/1538553644

Digital Object Identifier
doi:10.2969/jmsj/76427642

Mathematical Reviews number (MathSciNet)
MR3868216

Zentralblatt MATH identifier
07009711

#### Citation

ARENDT, Wolfgang; KUNKEL, Stefan; KUNZE, Markus. Diffusion with nonlocal Robin boundary conditions. J. Math. Soc. Japan 70 (2018), no. 4, 1523--1556. doi:10.2969/jmsj/76427642. https://projecteuclid.org/euclid.jmsj/1538553644

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