Estimates for solutions of the Dirichlet problem for Poisson’s equation

Posted on September 23, 2014 by fmartin

Recall that a function f:\Omega\subseteq\mathbb{R}^n\rightarrow \mathbb{R} is superharmonic if it’s twice differentiable and satisfies the inequality \Delta f\leq 0. Any superharmonic function verifies the weak minimum principle: that is, it attains its minimum on the boundary of the domain \Omega.

Suppose u is a solution for the problem

\left\{ \begin{array}{rl} \Delta u(x)= f(x) &:x\in B_1(0)\\ u(x)= g(x) &:x \in\partial B_1(0)\\ \end{array} \right.

where f and g are continuous functions. Let M_1, m_1 be the maximum (minimum) of f on the closure of the ball and let M_2, m_2 be the maximum (minimum) value of the boundary function g. Then, we can define v(x) = \frac{m_1}{2n} (|x|^2 -1) + M_2, which is a solution for the equation \Delta v = m_1 on the unit ball, with boundary value identically M_2. It follows that v-u is a superharmonic function, with positive boundary values; then, by the minimum principle we have that v-u\geq 0, or equivalently u\leq v, on the whole ball.

We thus arrive at the following estimates for u (the lower bound is obtained by an analogous procedure using the maximum principle for subharmonic functions):

\frac{M_1}{2n} (|x|^2-1)+ m_2 \leq u(x) \leq \frac{m_1}{2n} (|x|^2-1)+ M_2

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