ON CERTAIN FUNCTIONS WITH POSITIVE DEFINITE HESSIAN
ON CERTAIN FUNCTIONS WITH POSITIVE DEFINITE HESSIAN
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DOI：
10.2307/1969882
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发表时间：
1960
影响因子：
4.9
通讯作者：
H. Flanders
中科院分区：
文献类型：
--
作者：
H. Flanders
如果u = u（x，y）是rt-82 = 1在整个E2中的C "解，则u是一个二次多项式.事实证明，Nitsche的简洁证明可以扩展到证明这个结果的推广到多个变量。定理设u = u（x1，-**，xn）是E1上的C "函数.设Hessian矩阵R = 11 82 ulfxt，II是半正定的。最后，假设满足方程tr（I + R）-1 = B，B为常数。那么u是一个二次多项式。让我们使用列向量记法，Xi~（dxj~dx =~ix
If u = u(x, y) is a C" solution of rt - 82 = 1 in the whole E2, then u is a quadratic polynomial. It turns out that the neat proof of Nitsche can be extended to prove a generalization of this result to several variables. THEOREM. Let u = u(x1, - * *, xn) be a C" function on El. Suppose that the Hessian matrix R = 11 82ulfxt, II is positive semi-definite. Finally, suppose that the equation tr (I + R)-1 = b, b constant, is satisfied. Then u is a quadratic polynomial. Let us use the column vector notation, Xi ~~~(dxj ~ dx=~~~~~~~~~~~~ix