Higher dimensional inverse problem for a multi-connected bounded domain with piecewise smooth Robin boundary conditions and its physical applications

The asymptotic expansions of the trace of the heat kernel @Q(t)[email protected]?"j"="1^~exp([email protected]"j) for short-time t, have been derived for a variety of bounded domains, where {@l"j}"j"="1^~ are the eigenvalues of the negative Laplace operator [email protected]?^[email protected]?"i"="1^3(@?/@?x^i)^2 in the (x^1,x^2,x^3)-space. The dependence of @Q(t) on the connectivity of bounded domains and the boundary conditions is analyzed. Particular attention is given for an arbitrary multiply-connected bounded domain @W in R^3 together with piecewise smooth Robin boundary conditions, where the coefficients in these conditions are assumed to be piecewise smooth positive functions. Some applications of an ideal gas enclosed in the multiply-connected bounded domain @W are given. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it.

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