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**Unformatted text preview: **u ( x , y , t ) is a solution of the heat equation u t = Δ u in a bounded region Ω ⊂ R 2 . Here one has the initial condition u ( x , y , ) = f ( x , y ) and boundary condition u ( x , y , t ) = 0 at points (x,y) on the boundary, ∂Ω and deﬁne E ( t ) : = 1 2 ZZ Ω u 2 ( x , y , t ) dxdy . a) Generalize #3(a)(b) to this setting. b) If Ω is symmetric under refection across the y-axis, so ( x , y ) → (-x , y ) and if the initial temperature is also symmetric, f ( x , y ) = f (-x , y ) , show that u ( x , y , t ) = u (-x , y , t ) for all points in Ω and all t > 0. REMARK: If Ω is the unit disk, { x 2 + y 2 < 1 } and f ( x , y ) depends only the distance to the origin, r : = p x 2 + y 2 , the same reasoning shows that the solution u also depends only on r . [Last revised: March 9, 2011] 1...

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