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引用次数: 0
摘要
对于形式为S (u) = A: d2 u + b (x, u, Du)的椭圆型半线性微分算子,考虑泛函E∞(u) = ess sup Ω, | S (u)|。我们研究了给定边界数据的E∞极小值。因为泛函是不可微的,这个问题不会产生传统的欧拉-拉格朗日方程。在一定条件下,我们仍然可以给出一个所有极小值都必须满足的偏微分方程组。而且,这个条件等价于变分问题的一个弱版本。因此,偏微分方程的理论第一次可以用于研究L∞上的一大类变分问题。
Variational problems in involving semilinear second order differential operators p, li { white-space: pre-wrap; }
For an elliptic, semilinear differential operator of the form S ( u ) = A : D 2 u + b ( x , u , Du ), consider the functional E ∞ ( u ) = ess sup Ω , | S ( u )|. We study minimisers of E ∞ for prescribed boundary data. Because the functional is not differentiable, this problem does not give rise to a conventional Euler-Lagrange equation. Under certain conditions, we can nevertheless give a system of partial differential equations that all minimisers must satisfy. Moreover, the condition is equivalent to a weaker version of the variational problem. The theory of partial differential equations therefore becomes available for the study of a large class of variational problems in L ∞ for the first time.
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