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Heat equation for a region in R 2 with a polygonal boundary. (English) Zbl 0609.35003

Let D be an open, bounded and connected set in \(R^ 2\) with a polygonal boundary \(\partial D\) and \(\Delta\) be the Dirichlet Laplacian for D. Denote the area of D by \(| D|\), length of \(\partial D\) by \(| \partial D|\) and the inward pointing angles by \(\gamma _ 1,...,\gamma _ n\). The authors prove that for all \(t>0\) \[ | Trace(e^{t\Delta})-\frac{| D|}{4\pi t}+\frac{| \partial D|}{8(\pi t)^{1/2}}-\sum ^{n}_{i=1}\frac{\pi ^ 2-\gamma ^ 2_ i}{24\pi \gamma _ i}| \leq (5n+ \]
\[ +\frac{20| D|}{R^ 2})\cdot \frac{1}{\gamma ^ 2}\cdot e^{-(R \sin \gamma /2)^ 2/(16t)} \] where \(\gamma =\min _{i} \gamma _ i\) and R is a geometrical constant depending upon D. This is an improvement on the work of M. Kac, D. B. Ray, P. B. Bailey and F. H. Brownell because it proves ”Brownell’s conjecture on the \(O(e^{-c/t})\), \(t\downarrow 0\) behaviour of the remainder” with precise geometrical constants.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B35 Stability in context of PDEs
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