Solution of the problem of Plateau
Solution of the problem of Plateau
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DOI：
10.1090/s0002-9947-1931-1501590-9
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发表时间：
1931
影响因子：
1.3
通讯作者：
J. Douglas
中科院分区：
文献类型：
--
作者：
J. Douglas
在表示g是第一类不适当的情况下，使得0的值至多为无穷多个，其中gi（0）是不连续的。（19.9）依赖于这样一个事实（其证明是平凡的）：如果fi（m）（t）在m-* oo时一致趋于连续fi（t），则如果tm -t as m-> oo，我们有lim fi（m）（t）=fi（t）for m-> oo。该断言现在很容易被证明，如果（19.10）Xi= =WFi（w）该内容于2016年7月18日星期一05：43：48 UTC从 157.55.39.128下载http://about.jstor.org/terms约束的304 JESSE道格拉斯[January]是由gi（0）确定的调和函数，则（19.11）Z1 F 2（w）= 0，t= 1，使得曲面（19.10）是最小的。对于没有因子w的公式（18.2）：（in）1 2 eiG（19.12）F1（w）= 2 i9 2gi（m）6）d 0。由于所有的多边形P（m）都包含在一个有限的空间区域中，函数gi（m）（0）是一致有界的;如果w是单位圆内部的任何不动点，则当ei 0描述C时，分母（ei 0-W）2在绝对值上仍然上级于一个固定的正量。因此，（19 - 12）中的被积函数在极限过程（19 - 9）中保持一致有界;因此积分的极限等于极限的积分：（19 - 13）lim Fi（w）= Fi（w）。很明显，在g是第一类不适当的情况下，这个结果不受在极限关系式（19. 9）中不考虑g，（G）的不连续点的情况的影响，因为这些点，至多是可数无穷多的，形成一个零测度集。结果（19.11）由（19.13）和（19.5）对每m的存在性推出。20.最小曲面以r为界。为了证明在前一节中已证明存在的极小曲面以r为界，我们必须证明r的表示（19. 8）是适当的。第二种不可能是不适当的，这一点在什么地方得到了证明？18，这是基于关系（19.11），适用于这里与充分有效。然而，我们不能应用的论点？17证明（19.8）式不能是第一类不适当的。因为尽管对于这类g我们仍然会有A（g）= + so，但在一般约当围线的情况下，A（g）有时取有限值就不是真的了。因此，我们根据关系式（19 - 11）使用下面的论证来获得所需的结果。假设在g下C的点P对应于r的弧Q 'Q”。由于r是一条约当曲线，Q'和Q”是不同的：如果ai表示Q'的坐标，bi表示Q”的坐标，则距离Q 'Q”或I与（20.1）12 =（b.ai）2 i-a1不等于零。本内容于2016年7月18日星期一05：43：48 UTC从 157.55.39.128下载所有使用http://about.jstor.org/terms 1931]高原305的问题假设P在w = 1时不失一般性，因为这可以通过旋转单位圆来实现，这不会改变任何本质。
ion being made, in case the representation g is improper of the first kind, of the values of 0, at most denumerably infinite in number, where gi(0) is discontinuous. (19.9) rests on the fact (whose proof is trivial) that if fi(m)(t) tends uniformly to the continuous fi(t) when m-* oo, then if tm -t as m-> oo, we have lim fi(m) (t) =fi(t) for m-> oo . The assertion is now easily proved that if (19.10) Xi= =WFi(w) This content downloaded from 157.55.39.128 on Mon, 18 Jul 2016 05:43:48 UTC All use subject to http://about.jstor.org/terms 304 JESSE DOUGLAS [January are the harmonic functions determined by gi(O), then (19.1 1) Z1F 2 (w) = 0, t= 1 so that the surface (19.10) is minimal. For consider (18.2) without the factor w: (in)' 1 2eiG (19.12) F1 (w) = 2 i9 2gi(m) 6)dO. Since all the polygons P(m) are contained in a finite region of space, the functions gi(m)(0) are uniformly bounded; and if w is any fixed point interior to the unit circle, the denominator (eiO -W)2 remains superior in absolute value to a fixed positive quantity when ei0 describes C. Therefore the integrand in (19.12) remains uniformly bounded during the limit process (19.9); consequently the limit of the integral is equal to the integral of the limit: (19 .13) lim Fi (w) = Fi (w) . M-n+00 It is evident that in case g is improper of the first kind this result is not affected by the circumstance that the points of discontinuity of g,(G) are not considered in the limit relation (19.9), since these points, being at most denumerably infinite in number, form a set of zero measure. The result (19.11) now follows from (19.13) and the subsistence of (19.5) for every m. 20. The minimal surface is bounded by r. To show that the minimal surface whose existence is proved in the preceding section is bounded by r, we must prove that the representation (19.8) of r is proper. That it cannot be improper of the second kind is proved in ?18, which, being based on the relation (19.11), applies here with full validity. We cannot however apply the argument of ?17 to prove that (19.8) cannot be improper of the first kind. For although we would still have for a g of this kind A (g) = + so, it would not be true in the case of a general Jordan contour that A (g) sometimes takes finite values. We therefore use the following argument, based on the relation (19.11), to obtain the desired result. Suppose that under g the point P of C corresponds to the arc Q'Q" of r. Since r is a Jordan curve, Q' and Q" are distinct: and if ai denote the co6rdinates of Q', bi of Q", the distance Q'Q" or I with (20.1) 12 = (b.ai)2 i-a1 is not equal to zero. This content downloaded from 157.55.39.128 on Mon, 18 Jul 2016 05:43:48 UTC All use subject to http://about.jstor.org/terms 1931] THE PROBLEM OF PLATEAU 305 There is no loss of generality in supposing P to be at w = 1, for this may be achieved by a rotation of the unit circle, which changes nothing essential.