Heat Flow from Polygons
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Heat Flow from Polygons M. van den Berg1 · P. B. Gilkey2 · K. Gittins3 Received: 15 February 2019 / Accepted: 6 August 2019 / © The Author(s) 2019
Abstract We study the heat flow from an open, bounded set D in R2 with a polygonal boundary ∂D. The initial condition is the indicator function of D. A Dirichlet 0 boundary condition has been imposed on some but not all of the edges of ∂D. We calculate the heat content of D in R2 at t up to an exponentially small remainder as t ↓ 0. Keywords Heat content · Polygon Mathematics Subject Classification (2010) 35K05 · 35K20
1 Introduction Let D be an open, bounded set in Rm with finite Lebesgue measure |D|, and with boundary ∂D. We consider the heat equation ∂u , u = ∂t and impose a Dirichlet 0 boundary condition on ∂D. That is u(x; t) = 0, x ∈ ∂D, t > 0. We denote the (weak) solution corresponding to the initial datum lim u(x; t) = 1, x ∈ D, t↓0
M. van den Berg
[email protected] P. B. Gilkey [email protected] K. Gittins [email protected] 1
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK
2
Mathematics Department, University of Oregon, Eugene, OR 97403, USA
3
Institut de Math´ematiques, Universit´e de Neuchˆatel, Rue Emile-Argand 11, CH-2000 Neuchˆatel, Switzerland
M. van den Berg et al.
by uD . Then uD (x; t) represents the temperature at x ∈ D at time t when D has initial temperature 1, and its boundary is kept at fixed temperature 0. The heat content of D at t is denoted by QD (t) =
dx uD (x; t). D
Both uD and QD (t) have been the subjects of a thorough investigation going back to the treatise by Carslaw and Jaeger, [9]. For a more recent account we refer to [2, 13]. Many different versions and extensions have already been considered. For example, the case where ∂D is smooth, and A is an open subset of ∂D on which a Neumann (insulating) boundary condition has been imposed, while the temperature 0 Dirichlet condition has been maintained on ∂D − A. This Zaremba boundary condition for the heat equation has been considered in [4], for example. Even in the case where no boundary condition has been imposed on ∂D, the corresponding heat content, denoted by HD (t), has (if ∂D is smooth) an asymptotic series as t ↓ 0 similar to the one for QD (t), see [3], for example. In this paper we consider the heat flow out of D into Rm , where a Dirichlet 0 boundary condition has been imposed on a closed subset ∂D− ⊂ ∂D, and where no boundary condition has been imposed on ∂D+ := ∂D − ∂D− . That is ∂u , (1.1) u = ∂t with boundary condition (1.2) u(x; t) = 0, x ∈ ∂D− , t > 0. We denote the solution corresponding to the initial datum lim u(x; t) = 1D (x), almost everywhere, t↓0
(1.3)
by uD,∂D− . Here 1D is the indicator function of D. Then uD,∂D− is the weak solution of Eqs. 1.1, 1.2 and 1.3, where Eq. 1.2 holds at all regular points of ∂D− . The open set D looses heat via two mechanisms: (i) part of the boundary, ∂D− , is at fixed temperature 0, and cools the interior of D; (ii) since the complement of D is at initial t
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