20 2. FOUR IMPORTANT LINEAR PDE 2.2. LAPLACE’S EQUATION Among the most important of all partial differential equations are undoubt- edly Laplace’s equation (1) Δu = 0 and Poisson’s equation (2) −Δu = f. ∗ In both (1) and (2), x ∈ U and the unknown is u : ¯ U → R, u = u(x), where U ⊂ Rn is a given open set. In (2) the function f : U → R is also given. Remember from §A.3 that the Laplacian of u is Δu = ∑n i=1 uxixi . DEFINITION. A C2 function u satisfying (1) is called a harmonic func- tion. Physical interpretation. Laplace’s equation comes up in a wide variety of physical contexts. In a typical interpretation u denotes the density of some quantity (e.g. a chemical concentration) in equilibrium. Then if V is any smooth subregion within U, the net flux of u through ∂V is zero: ∂V F · ν dS = 0, F denoting the flux density and ν the unit outer normal field. In view of the Gauss–Green Theorem (§C.2), we have V div F dx = ∂V F · ν dS = 0, and so (3) div F = 0 in U, since V was arbitrary. In many instances it is physically reasonable to as- sume the flux F is proportional to the gradient Du but points in the opposite direction (since the flow is from regions of higher to lower concentration). Thus (4) F = −aDu (a 0). ∗I prefer to write (2) with the minus sign, to be consistent with the notation for general second-order elliptic operators in Chapter 6.
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