52 2. FOUR IMPORTANT LINEAR PDE The region UT DEFINITIONS. (i) We define the parabolic cylinder UT := U × (0,T ]. (ii) The parabolic boundary of UT is ΓT := ¯ U T − UT . We interpret UT as being the parabolic interior of ¯ U × [0,T ]: note care- fully that UT includes the top U × {t = T }. The parabolic boundary ΓT comprises the bottom and vertical sides of U × [0,T ], but not the top. We want next to derive a kind of analogue to the mean-value property for harmonic functions, as discussed in §2.2.2. There is no such simple formula. However let us observe that for fixed x the spheres ∂B(x, r) are level sets of the fundamental solution Φ(x−y) for Laplace’s equation. This suggests that perhaps for fixed (x, t) the level sets of fundamental solution Φ(x − y, t − s) for the heat equation may be relevant. DEFINITION. For fixed x ∈ Rn, t ∈ R, r 0, we define E(x, t r) := (y, s) ∈ Rn+1 | s ≤ t, Φ(x − y, t − s) ≥ 1 rn .
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