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

Abstract

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