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arXiv 1911.04166math.FA
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希尔伯特空间紧子集上$1$ -喷注的凸$C^1$扩张

Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert spaces

Daniel Azagra, Carlos Mudarra

备注：
Final version (with an improvement in the proof suggested by a referee)
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摘要：

令$X$表示Hilbert空间。给定$X$的紧子集$K$和两个连续函数$f:K\to\mathbb{R}$，$G:K\to X$，我们证明了存在凸函数$F\in C^1(X)$使得$F=f$在$K$上，$\nabla F=G$在$K$上的充要条件是$1$ -jet $(f, G)$满足（1）对所有$x, y\in K$都满足$f(x)\geq f(y)+ \langle G(y), x-y\rangle$，（2）如果$x, y\in K$和$f(x)= f(y)+ \langle G(y), x-y\rangle$则满足$G(x)=G(y)$ .我们还解决了类似的问题，$K$替换为任意有界子集的$X$，并为$C^1(X)$替换为类$C^{1,u}_{b}(X)$的可微函数与一致连续的衍生物上的有界子集的$X$。

英文摘要：

Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f:K\to\mathbb{R}$, $G:K\to X$, we show that a necessary and sufficient condition for the existence of a convex function $F\in C^1(X)$ such that $F=f$ on $K$ and $\nabla F=G$ on $K$ is that the $1$-jet $(f, G)$ satisfies (1) $f(x)\geq f(y)+ \langle G(y), x-y\rangle$ for all $x, y\in K$, and (2) if $x, y\in K$ and $f(x)= f(y)+ \langle G(y), x-y\rangle$ then $G(x)=G(y)$. We also solve a similar problem for $K$ replaced with an arbitrary bounded subset of $X$, and for $C^1(X)$ replaced with the class $C^{1,u}_{b}(X)$ of differentiable functions with uniformly continuous derivatives on bounded subsets of $X$.

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