Lp-Mean coercivity, regularity and relaxation in the calculus of variations

Yan, Baisheng; Zhou, Zhengfang

doi:10.1016/s0362-546x(00)00150-4

Cited by 5 publications

(9 citation statements)

References 30 publications

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“…1) Existence of quasiconvex functions satisfying (2.5) follows from property (2.4) and general results established in Yan and Zhou [30] using the important technique of nonlinear Hodge decomposition discovered in Iwaniec [11] and Iwaniec and Sbordone [15]. In fact, by [30], Theorem 2.1, one can choose g to be the quasiconvexification function…”

Section: Theorem 22 ([20] [30]) Letmentioning

confidence: 79%

“…Another important property of this function F L is that it easily satisfies the so-called L n -mean coercivity condition (see, e.g., [13], [30]):…”

Section: Quasiconvex Functions Vanishing On Quasiregular Setsmentioning

confidence: 99%

“…Existence of the quasiconvex functions with growth below the natural power n and vanishing exactly on the quasiregular set is the main content of our next theorem; we refer to [20], [30] for the proof and further discussions.…”

Section: Quasiconvex Functions Vanishing On Quasiregular Setsmentioning

confidence: 99%

“…We refer to, e.g., [4], [7], [8], [9], [18], [19], [22], [29], [30], [31] for more properties of this and other generalized convex hulls in the calculus of variations. …”

Section: A General Existence Theoremmentioning

confidence: 99%

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A linear boundary value problem for weakly quasiregular mappings in space

Yan¹

2001

Calculus of Variations and Partial Differential Equations

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Abstract. Given a number L ≥ 1, a weakly L-quasiregular map on a domain Ω in space R n is a map u in a Sobolev spaceIn this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space R n with dimension n ≥ 3. It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in W 1,p (Ω; R n ) can only assume a quasiregular linear boundary value; however, for all L ≥ 1 and 1 ≤ p < nL L+1 , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in W 1,p (Ω; R n ). Classification (1991):30C65, 30C70, 35F30, 49J30 Mathematics Subject

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Section: Theorem 22 ([20] [30]) Letmentioning

confidence: 79%

“…Another important property of this function F L is that it easily satisfies the so-called L n -mean coercivity condition (see, e.g., [13], [30]):…”

Section: Quasiconvex Functions Vanishing On Quasiregular Setsmentioning

confidence: 99%

Section: Quasiconvex Functions Vanishing On Quasiregular Setsmentioning

confidence: 99%

“…We refer to, e.g., [4], [7], [8], [9], [18], [19], [22], [29], [30], [31] for more properties of this and other generalized convex hulls in the calculus of variations. …”

Section: A General Existence Theoremmentioning

confidence: 99%

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A linear boundary value problem for weakly quasiregular mappings in space

Yan¹

2001

Calculus of Variations and Partial Differential Equations

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“…In this section we investigate coercivity in the context of nonnegative positively p-homogeneous integrands F . In this case the L p coercivity, as noted in [32,38], only depends on the zero set {F = 0} and not on other features of F :…”

Section: Examples Of L P Coercive Integrandsmentioning

confidence: 99%

On coercive variational integrals

Chen

Kristensen

2017

Nonlinear Analysis: Theory, Methods & Applications

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It is well-known that sequential weak lower semicontinuity of a variational integral F(u, Ω) = ∫ Ω F (∇u(x)) dx on the Sobolev space W 1,p (Ω, R N) under a p-growth condition on the integrand F is equivalent to quasiconvexity in the sense of Morrey. We show that coercivity on Dirichlet classes likewise is equivalent to a quasiconvexity condition. We also discuss more general notions of coercivity, and in the case of positively p-homogeneous integrands F we establish the existence of minimizers for a class of non-coercive quasiconvex variational integrals.

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Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands

Guerra¹,

Kristensen

2022

Arch Rational Mech Anal

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Lp-Mean coercivity, regularity and relaxation in the calculus of variations

Cited by 5 publications

References 30 publications

A linear boundary value problem for weakly quasiregular mappings in space

A linear boundary value problem for weakly quasiregular mappings in space

On coercive variational integrals

Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands

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