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

John M. Ball, Christopher L. Horner

Abstract

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 $ψ(Dy)$ is best expressed when $ψ(A)\to\infty$ as $\det A\to 0+$.

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

Abstract

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on has an extension to a convex function on the half-space which is finite and smooth on the open half-space . The result is applied to nonlinear elasticity, where it clarifies how the condition of polyconvexity of the free-energy density is best expressed when as .
Paper Structure (3 sections, 6 theorems, 47 equations)

This paper contains 3 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['lifting']}
  3. Polyconvexity conditions

Key Result

theorem 1

Let $\Phi:\mathbb{R}^n\to (-\infty,\infty]$ be a proper lower semicontinuous convex function. Then there exists a lower semicontinuous convex function such that $(i)$$\varphi(0,y)=\Phi(y)$ for all $y\in \mathbb{R}^n$, $(ii)$$\lim_{x\to 0+}\varphi(x,y)= \Phi(y)$ for each $y\in \mathbb{R}^n$. $(iii)$$\varphi:(0,\infty)\times\mathbb{R}^n\to \mathbb{R}$ is smooth, If $\Phi\geqslant 0$, then $\varphi$

Theorems & Definitions (13)

  • theorem 1
  • corollary 1
  • proposition 1
  • corollary 2
  • proof
  • lemma 1
  • proof
  • remark 1
  • remark 2
  • remark 3
  • ...and 3 more