An intrinsic approach to manifold constrained variational problems

Focardi, Matteo; Spadaro, Emanuele

doi:10.1007/s10231-011-0216-z

Cited by 7 publications

(9 citation statements)

References 34 publications

(28 reference statements)

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“…Moreover, by Lemma A.2 in [23] and the equi-integrability of dw k in B l , which easily follows from (4.10) and the definition of w k itself, we have that…”

Section: Truncationmentioning

confidence: 80%

“…Therefore, the Lipschitz continuity of r k , the locality of the approximate differentials and the growth hypothesis on e M , together with (4.10), (4.20) and Lemma A.2 in [23] imply that…”

mentioning

confidence: 87%

“…In addition, for Q > 1 the local structure of the given affine Q-valued map has to be taken into account as summation is not defined for Q-points in general. Let us also emphasize that the definition above is stated in intrinsic terms following the contributions in [14] and [23] for similar problems. Alternatively, one can use local charts (cp.…”

Section: Definition 41 (Quasi-convexity) Let E M : H(r M T M)mentioning

confidence: 99%

“…For it, we follow the intrinsic approach developed in [14] and [23]. We avoid any embedding of the manifold M into a linear space for the reasons underlined in the introduction.…”

Section: Theorem 43 Let P ∈ [1 ∞[ and E M : B× H(r M T M)mentioning

confidence: 99%

“…According to Lemma A.1 in [23], without relabeling the subsequence, there exist sets B l , l ∈ N, such that…”

Section: Sufficiency Of Quasiconvexitymentioning

confidence: 99%

See 4 more Smart Citations

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

Focardi

Mariano

Spadaro

2015

Arch Rational Mech Anal

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“…Moreover, by Lemma A.2 in [23] and the equi-integrability of dw k in B l , which easily follows from (4.10) and the definition of w k itself, we have that…”

Section: Truncationmentioning

confidence: 80%

“…Therefore, the Lipschitz continuity of r k , the locality of the approximate differentials and the growth hypothesis on e M , together with (4.10), (4.20) and Lemma A.2 in [23] imply that…”

mentioning

confidence: 87%

Section: Definition 41 (Quasi-convexity) Let E M : H(r M T M)mentioning

confidence: 99%

“…For it, we follow the intrinsic approach developed in [14] and [23]. We avoid any embedding of the manifold M into a linear space for the reasons underlined in the introduction.…”

Section: Theorem 43 Let P ∈ [1 ∞[ and E M : B× H(r M T M)mentioning

confidence: 99%

“…According to Lemma A.1 in [23], without relabeling the subsequence, there exist sets B l , l ∈ N, such that…”

Section: Sufficiency Of Quasiconvexitymentioning

confidence: 99%

See 3 more Smart Citations

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

Focardi

Mariano

Spadaro

2015

Arch Rational Mech Anal

Self Cite

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Equilibrium configurations of mixture thin films undergoing large strains

Mariano

2017

Math Methods in App Sciences

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We describe a mixture thin film as a membrane endowed with multiple out-of-tangent-plane vectors at each point, with vector sequence defined to within a permutation to account for the mixing of the mixture components.Such a description is motivated by a proposal for an atomistic-to-continuum derivation of a representation of multiatomic layer thin films, a view not requiring the introduction of phenomenological parameters. Differences between that proposal and the model discussed here are the definition of the values of the layer-descriptor map to within permutations and the explicit introduction of a bending-like term in the energy. The out-of-tangent-plane vectors satisfy a condition forbidding them to fall within the tangent plane after deformation.We consider a generic weakly surface-polyconvex membrane energy and a quadratic bending term involving the out-of-tangent-plane multiple vectors, a term which is also quasiconvex. Under appropriate energy growth assumptions and Dirichlet-type boundary conditions, we prove existence of ground states, i.e., equilibrium configurations described by the solutions to balance equations.An obvious corollary is the existence of equilibrium configurations of single out-of-tangent-plane vector Cosserat surfaces, a natural scheme for plates of simple materials.

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Geometric Finite Elements

Hardering

Sander

2020

Handbook of Variational Methods for Nonlinear Geometric Data

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An intrinsic approach to manifold constrained variational problems

Cited by 7 publications

References 34 publications

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

Multi-Value Microstructural Descriptors for Complex Materials: Analysis of Ground States

Equilibrium configurations of mixture thin films undergoing large strains

Geometric Finite Elements

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