ABSTRACT
It is well known that in two-dimensional problems a minimal sequence does not necessarily converge uniformly to the solution; on the contrary, it may diverge everywhere [1]. In this paper I give same conditions sufficient for a minimal sequence to be uniformly convergent. As an example, I shall consider the Dirichlet problem
for an equation of elliptic type
where a(z,y), b(x,y), c(x ,y), f ( x , y ) are continuous in the region bounded by the contour T and a, 6 > 0, c > 0 in D. This problem is equivalent to the problem of minimizing the integral
Lemma. If rj(x, y) is a continuous function defined in D possessing con tinuous partial derivatives inside D and satisfying the following conditions:
for any £,a,/3 such that the segment with endpoints (x,a) and (x,/3) is contained in D , then the inequality
Now substituting Cartesian coordinates for the polar ones (?/ = psin0) in (10) we obtain
holds, where d is the diameter of the region and C2, C3 are constants. Proof. We set r](x,y) = 0 outside of D. From (5) we have
Now let N be an arbitrary point inside D. We take M as the origin of coordinates. Choose a positive number p > 0, (p < d) and denote by px{6) the distance between M and the first point of the intersection of the contour T with the ray 9, of this distance is > p, and px(ff) = p otherwise. Then we have
Suppose that D x is the region bounded by the lines We have
Take a positive number A. We obviously have
and according to (9)
Now, making use of (6), we get
If we now take
we come to (7). This lemma almost directly implies: T heorem 1. Let u(x,y) be a solution of equation (2) which satisfies
condition (1) and is such that I (u ) < +oo; assume
Let un(x,y) be a sequence of functions vanishing on T such that
Then un{x,y) converges uniformly to u(x,y) in D and
Let H > 0 be a positive number. We shall write (x,y) £ Dh if there exist a and (3 such that: 1) ol < y < /?, 2) j3 — a > h, 3) the segment joining the points (x ,a) and (x,/3) is contained in D. Put D'h — D — Dh and
max\i1n(x,y)\ = Mn, m ax \u(xyy)\ = M0, rjn = u - un. (23)
Now, using the Markov theorem [2] on estimation for the derivative of a polynomial, we obtain
and
The rate of growth of K n can be estimate if the nature of un is of a more special character. In Theorem 2 we consider the case in which the un are polynomial in y.
