Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions: Further Results期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

作者姓名：	E.M.E.ZAYED

作者单位：	DepartmentofMathematics,FacultyofScience,ZagazigUniversity,Zagazig,Egypt

摘要：	The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.
关键词：	渐近展开公式罗宾边界环形域拉普拉斯算子
收稿时间：	13 August 2001
Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions: Further Results

E.M.E.ZAYED.Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions: Further Results[J].Acta Mathematica Sinica,2003,19(4):679-694.

Authors:	Email author" target="_blank">E?M?E?Zayed Email author

Institution:	(1) Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

Abstract:	Abstract The asymptotic expansions of the trace of the heat kernel $\Theta {\left( t \right)} = {\sum\nolimits_{v - 1}^\infty {{\kern 1pt} {\kern 1pt} \exp {\left( { - t\lambda _{v} } \right)}} }$ for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian $- \Delta _{n} = - {\sum\nolimits_{i = 1}^n {{\left( {\frac{\partial } {{\partial x^{i} }}} \right)}^{2} } }$ in R ⁿ(n = 2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary $\partial \Omega _{1}$ and a smooth outer boundary $\partial \Omega _{2}$ , where a finite number of piecewise smooth Robin boundary conditions ${\left( {\frac{\partial } {{\partial n_{j} }} + \gamma _{j} } \right)}\phi = 0$ on the components Γ_j (j = 1, ..., k) of $\partial \Omega _{1}$ and on the components Γ_j (j = k+1, ...,m) of $\partial \Omega _{2}$ are considered such that $\partial \Omega _{1} = \cup ^{k}_{{j = 1}} \Gamma _{j}$ and $\partial \Omega _{2} = \cup ^{m}_{{j = k + 1}} \Gamma _{j}$ and where the coeffcients γ_j(j = 1, ...,m) are piecewise smooth positive functions. Some applications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.

Keywords:	Inverse problem Heat kernel Eigenvalues Robin boundary conditions Classical ideal gas
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