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

John M. Ball; Christopher L. Horner

doi:10.1007/s00526-024-02719-3

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

John M. Ball^*, Christopher L. Horner

^*Corresponding author for this work

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Abstract

It is shown that a possibly infinite-valued proper lower semicontinuous convex function on Rn has an extension to a convex function on the half-space Rⁿ×[0, ∞) which is finite and smooth on the open half-space Rⁿ×(0, ∞). 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)→∞ as detA→0+.

Original language	English
Article number	107
Journal	Calculus of Variations and Partial Differential Equations
Volume	63
Issue number	4
Early online date	17 Apr 2024
DOIs	https://doi.org/10.1007/s00526-024-02719-3
Publication status	Published - May 2024

Keywords

74B20
49J45
26B25

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abstract = "It is shown that a possibly infinite-valued proper lower semicontinuous convex function on Rn has an extension to a convex function on the half-space Rn×[0, ∞) which is finite and smooth on the open half-space Rn×(0, ∞). 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)→∞ as detA→0+.",

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AU - Horner, Christopher L.

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AB - It is shown that a possibly infinite-valued proper lower semicontinuous convex function on Rn has an extension to a convex function on the half-space Rn×[0, ∞) which is finite and smooth on the open half-space Rn×(0, ∞). 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)→∞ as detA→0+.

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

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