On some definitions and properties of generalized convex sets arising in the calculus of variations

Bernard Dacorogna; Ana Margarida Fernandes Ribeiro

doi:10.1142/9789812774170_0005

On some definitions and properties of generalized convex sets arising in the calculus of variations

Bernard Dacorogna, Ana Margarida Fernandes Ribeiro

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We deal with generalized notions of convexity for sets. Namely, thepolyconvexity, quasiconvexity, rank one convexity and separate convexity. The question has its origin in the calculus of variations. We try tosystematize the results concerning these generalized notions imitating asmuch as possible the classical approach of convex analysis. Throughoutthe article, we will discuss the relations between the different convexities,separation and Carath´eodory type theorems, the notion of hull of a setand extremal points.

Original language	English
Title of host publication	Recent Advances on Elliptic and Parabolic Issues
Subtitle of host publication	Proceedings of the 2004 Swiss-Japanese Seminar, Zurich, Switzerland, 6 – 10 December 2004
Editors	Michel Chipot, Hirokazu Ninomiya
Place of Publication	Singapore
Publisher	World Scientific
Pages	103-128
Number of pages	26
ISBN (Electronic)	978-981-4472-99-9
ISBN (Print)	978-981-256-675-1
DOIs	https://doi.org/10.1142/9789812774170_0005
Publication status	Published - 2006
Event	2004 Swiss-Japanese Seminar - Zurique, Switzerland Duration: 6 Dec 2004 → 10 Dec 2004

Seminar

Seminar	2004 Swiss-Japanese Seminar
Country/Territory	Switzerland
City	Zurique
Period	6/12/04 → 10/12/04

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Dacorogna, B., & Ribeiro, A. M. F. (2006). On some definitions and properties of generalized convex sets arising in the calculus of variations. In M. Chipot, & H. Ninomiya (Eds.), Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, Zurich, Switzerland, 6 – 10 December 2004 (pp. 103-128). World Scientific. https://doi.org/10.1142/9789812774170_0005

Dacorogna, Bernard ; Ribeiro, Ana Margarida Fernandes. / On some definitions and properties of generalized convex sets arising in the calculus of variations. Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, Zurich, Switzerland, 6 – 10 December 2004. editor / Michel Chipot ; Hirokazu Ninomiya. Singapore : World Scientific, 2006. pp. 103-128

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title = "On some definitions and properties of generalized convex sets arising in the calculus of variations",

abstract = "We deal with generalized notions of convexity for sets. Namely, thepolyconvexity, quasiconvexity, rank one convexity and separate convexity. The question has its origin in the calculus of variations. We try tosystematize the results concerning these generalized notions imitating asmuch as possible the classical approach of convex analysis. Throughoutthe article, we will discuss the relations between the different convexities,separation and Carath´eodory type theorems, the notion of hull of a setand extremal points.",

author = "Bernard Dacorogna and Ribeiro, {Ana Margarida Fernandes}",

year = "2006",

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language = "English",

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editor = "Michel Chipot and Hirokazu Ninomiya",

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note = "2004 Swiss-Japanese Seminar ; Conference date: 06-12-2004 Through 10-12-2004",

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Dacorogna, B & Ribeiro, AMF 2006, On some definitions and properties of generalized convex sets arising in the calculus of variations. in M Chipot & H Ninomiya (eds), Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, Zurich, Switzerland, 6 – 10 December 2004. World Scientific, Singapore, pp. 103-128, 2004 Swiss-Japanese Seminar, Zurique, Switzerland, 6/12/04. https://doi.org/10.1142/9789812774170_0005

On some definitions and properties of generalized convex sets arising in the calculus of variations. / Dacorogna, Bernard; Ribeiro, Ana Margarida Fernandes.
Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, Zurich, Switzerland, 6 – 10 December 2004. ed. / Michel Chipot; Hirokazu Ninomiya. Singapore: World Scientific, 2006. p. 103-128.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - We deal with generalized notions of convexity for sets. Namely, thepolyconvexity, quasiconvexity, rank one convexity and separate convexity. The question has its origin in the calculus of variations. We try tosystematize the results concerning these generalized notions imitating asmuch as possible the classical approach of convex analysis. Throughoutthe article, we will discuss the relations between the different convexities,separation and Carath´eodory type theorems, the notion of hull of a setand extremal points.

AB - We deal with generalized notions of convexity for sets. Namely, thepolyconvexity, quasiconvexity, rank one convexity and separate convexity. The question has its origin in the calculus of variations. We try tosystematize the results concerning these generalized notions imitating asmuch as possible the classical approach of convex analysis. Throughoutthe article, we will discuss the relations between the different convexities,separation and Carath´eodory type theorems, the notion of hull of a setand extremal points.

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BT - Recent Advances on Elliptic and Parabolic Issues

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Dacorogna B, Ribeiro AMF. On some definitions and properties of generalized convex sets arising in the calculus of variations. In Chipot M, Ninomiya H, editors, Recent Advances on Elliptic and Parabolic Issues: Proceedings of the 2004 Swiss-Japanese Seminar, Zurich, Switzerland, 6 – 10 December 2004. Singapore: World Scientific. 2006. p. 103-128 doi: 10.1142/9789812774170_0005

On some definitions and properties of generalized convex sets arising in the calculus of variations

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