2.2. LAPLACE’S EQUATION 27 a. Strong maximum principle, uniqueness. We begin with the asser- tion that a harmonic function must attain its maximum on the boundary and cannot attain its maximum in the interior of a connected region unless it is constant. THEOREM 4 (Strong maximum principle). Suppose u C2(U) C( ¯ U ) is harmonic within U. (i) Then max ¯ U u = max ∂U u. (ii) Furthermore, if U is connected and there exists a point x0 U such that u(x0) = max ¯ U u, then u is constant within U. Assertion (i) is the maximum principle for Laplace’s equation and (ii) is the strong maximum principle. Replacing u by −u, we recover also similar assertions with “min” replacing “max”. Proof. Suppose there exists a point x0 U with u(x0) = M := max ¯ U u. Then for 0 r dist(x0,∂U), the mean-value property asserts M = u(x0) = B(x0,r) u dy M. As equality holds only if u M within B(x0,r), we see u(y) = M for all y B(x0,r). Hence the set {x U | u(x) = M} is both open and relatively closed in U and thus equals U if U is connected. This proves assertion (ii), from which (i) follows. Positivity. The strong maximum principle asserts in particular that if U is connected and u C2(U) C( ¯ U ) satisfies Δu = 0 in U u = g on ∂U, where g 0, then u is positive everywhere in U if g is positive somewhere on ∂U. An important application of the maximum principle is establishing the uniqueness of solutions to certain boundary-value problems for Poisson’s equation.
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