TABLE OF CONTENTS Function u( x) satisfying ? u = 0 in a domain ? ? R n is said to be harmonic there. It is a fundamental problem in mathematical physics to solve
where ? is a bounded domain and f( ?) is a continuous function on its boundary ??. Actually, we have the following theorem for the Dirichlet problem to harmonic function, (5.47).
Problem (5.47) admits the solution 
for arbitrary given f ? C( ??) if and only if any point on ?? is regular.
Here, the boundary point ? ? ?? is said to be regular if it has a barrier w( x). This means that w( x) is continuous on 
, super-harmonic there, positive in 
, and is equal to zero at x = ?. Furthermore, super-harmonicity of the continuous function w( x) is defined through the mean value property,
It is actually equivalent to ? w ? 0 in ? if w( x) is twice differentiable. Remember that B( x 0, r) denotes the open ball with the center x 0 and the radius r and
If 
satisfies (5.47), then it is called the classical solution. A sufficient condition for the regularity of boundary point is the outer circumscribing ball condition.
Theorem 5.7 is proven by
