Diffusion with nonlocal boundary conditions
Diffusion with nonlocal boundary conditions
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DOI：
10.1016/j.jfa.2016.01.025
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发表时间：
2014-09
期刊：
影响因子：
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通讯作者：
W. Arendt;Stefan Kunkel;M. Kunze
中科院分区：
文献类型：
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作者：
W. Arendt;Stefan Kunkel;M. Kunze
考虑有界Dirichlet正则集Ω <$Rd上的二阶微分算子A μ，满足非局部边界条件u（z）=<$Ω u（x）μ（z，dx），其中z∈ <$Ω.这里函数μ：<$Ω→ M+（Ω）是σ（M（Ω），B（Ω））-连续的，对所有z∈ <$Ω，0≤ μ（z，Ω）≤ 1.在对A μ中系数的适当假设下，证明了A μ在L∞（Ω）上生成一个全纯正压缩半群T μ.半群T μ从来不是强连续的，但它具有强Feller性质，因为它由核算子组成，取值于C（Ω Ω）。我们还证明了T μ是紧的，并研究了T μ（t）在t→∞时的渐近行为.
We consider second order differential operators A μ on a bounded, Dirichlet regular set Ω⊂ R d, subject to the nonlocal boundary conditions u (z)=∫ Ω u (x) μ (z, d x) for z∈∂ Ω. Here the function μ:∂ Ω→ M+(Ω) is σ (M (Ω), C b (Ω))-continuous with 0≤ μ (z, Ω)≤ 1 for all z∈∂ Ω. Under suitable assumptions on the coefficients in A μ, we prove that A μ generates a holomorphic positive contraction semigroup T μ on L∞(Ω). The semigroup T μ is never strongly continuous, but it enjoys the strong Feller property in the sense that it consists of kernel operators and takes values in C (Ω‾). We also prove that T μ is immediately compact and study the asymptotic behavior of T μ (t) as t→∞.