标题： 从超平面到半空间的凸函数扩展 显示英文标题

标题： Extension of convex functions from a hyperplane to a half-space

摘要： 证明了一个可能为无限值的在${\mathbb R}^n$上的正常下半连续凸函数可以扩展为在半空间${\mathbb R}^n\times[0,\infty)$上的凸函数，该函数在开半空间${\mathbb R}^n\times(0,\infty)$上是有限且光滑的。 该结果应用于非线性弹性，其中当$\psi(A)\to\infty$时，它阐明了自由能密度$\psi(Dy)$的多凸性条件如何最佳地表示为$\det A\to 0+$。 显示英文摘要

摘要： It is shown that a possibly infinite-valued proper lower semicontinuous convex function on ${\mathbb R}^n$ has an extension to a convex function on the half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the open half-space ${\mathbb R}^n\times(0,\infty)$. The result is applied to nonlinear elasticity, where it clarifies how the condition of polyconvexity of the free-energy density $\psi(Dy)$ is best expressed when $\psi(A)\to\infty$ as $\det A\to 0+$.