| 魏 茸，张 静.2 ∆ + 12 ∆ρ的Dirichlet边界值问题[J].海南师范大学学报自科版,2019,32(3):299-304 |
| 2 ∆ + 12 ∆ρ的Dirichlet边界值问题 |
| The Dirichlet Boundary Value Problem of Operator 12 ∆ + 12 ∆ρ |
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| DOI：10.12051/j.issn.1674-4942.2019.03.009 |
| 中文关键词: Dirichlet边界值问题 概率表示 狄氏型理论 |
| 英文关键词: Dirichlet boundary value problem probabilistic representation Dirichlet forms theory |
| 基金项目:国家自然科学基金（11701127）；海南省自然科学基金（117096）；海南师范大学博士科研启动基金 |
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| 中文摘要: |
| 文章应用布朗运动的时间逆转算子和狄氏型理论，给出算子 12 ∆ + 12 ∆ρ的Dirichlet边
界值问题的概率解，并证明其在边界上连续。ρ ∈ C∞0 (D) 时，边界值问题的概率解可表示为
对任意 x ∈ D，u (x) ≔ Ex éëêe2NρτD f (XτD )ùûú。对ρ ∈ W1，2 (D)，构造一列ρn ∈ C∞0 (D)使其在W0 1，2 (D)收敛到ρ。 由u在D内局部Holder连续，证明u在边界∂D上连续。 |
| 英文摘要: |
| In this paper, the time reversal operator of Brownain motion and the theory of Dirichlet forms were applied to⁃
give the probabilistic solution of the Dirichlet boundary problem of operat or 12 ∆ + 12 ∆ρ and show that it is contiuous on
boundary. It was proved that the Dirichlet boundary problem had the probalistic solution expressed as
u (x) ≔ Ex éëêe2NρτD f (XτD )ùûú for any x ∈ D when ρ ∈ C∞0 (D). A set of ρn ∈ C∞0 (D) which converged to ρ in W 1,2 0 (D) was construct⁃
ed for any ρ ∈ W1,2 (D). Due to u is locally Holder continuous in D, it was proved that uwas continuous on ∂D. |
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