Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions: Further Results
Acta Mathematica Sinica, English Series
 Acta Mathematica Sinica, English Series  2003, Vol. 19 Issue (4): 679-694    DOI: 10.1007/s10114-003-0257-3
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Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions: Further Results
E. M. E. ZAYED
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
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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 {λv} are the eigenvalues of the negative Laplacian $- \Delta _{n}=- {\sum\nolimits_{i=1}^n {{\left( {\frac{\partial } {{\partial x^{i} }}} \right)}^{2} } }$ in Rn (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 Γj (j=1,..., k) of $\partial \Omega _{1}$ and on the components Γj(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 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.