A linear boundary value problem for weakly quasiregular mappings in space

Yan, Baisheng

doi:10.1007/s005260000074

Cited by 11 publications

(23 citation statements)

References 32 publications

(46 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then v ρ is a piece-wise affine map in ψ + W 1,n 0 (Ω; R n ) (see, e.g., [11,Lemma 1.6]) and satisfies Dv ρ (x) ∈ U for almost every x ∈ Ω. Therefore, v ρ ∈ X.…”

Section: Proof Except For the Requirementmentioning

confidence: 99%

“…In Yan [12], the proof of Theorem 1.1 has relied on an important technique developed in Yan [10,11] using the idea of convex integration motivated by the work of Müller &Šverák [7] (see also Müller & Sychev [8]). …”

Section: Theorem 14 For Any > 0 and Any Piece-wise Affine Mapmentioning

confidence: 99%

“…In Yan [10,11,12], we studied the boundary value problem for weakly Lquasiregular mappings in W 1,p (Ω; R n ) with 1 ≤ p < nL L+1 . In particular, the following result has been proved in [12] (see also [11]).…”

Section: Below)mentioning

confidence: 99%

See 2 more Smart Citations

A Baire's category method for the Dirichlet problem of quasiregular mappings

Yan

2003

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any > 0 and any piece-wise affine mapThe theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

show abstract

“…Then v ρ is a piece-wise affine map in ψ + W 1,n 0 (Ω; R n ) (see, e.g., [11,Lemma 1.6]) and satisfies Dv ρ (x) ∈ U for almost every x ∈ Ω. Therefore, v ρ ∈ X.…”

Section: Proof Except For the Requirementmentioning

confidence: 99%

Section: Theorem 14 For Any > 0 and Any Piece-wise Affine Mapmentioning

confidence: 99%

See 1 more Smart Citation

A Baire's category method for the Dirichlet problem of quasiregular mappings

Yan

2003

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…When L = 1 , it is easily seen that the set K 1 = Z 1 coincides with the set of conformal matrices, that is, K 1 = Z 1 = f R j 0 R 2 S O (n)g: Weakly L-quasiregular mappings are the mappings u 2 W 1 p loc ( R n ) satisfying Du(x) 2 K L almost everywhere in (see Iwaniec 6]). Many regularity and stability properties of weakly quasiregular mappings have been studied in connection with the certain semi-convex hulls and the attainment result of the quasiconformal sets K L in Yan 13,14,15]. In this paper we study similar problems related to the set Z L :…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…To prove Theorem 3.3, or the more general Theorem 1.4, we need some techniques in convex integration theory we refer to 10,11,14] for more references on this theory.…”

Section: L+1mentioning

confidence: 99%

Relaxation and attainment results for an integral functional with unbounded energy well

Yan

2002

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

View full text Add to dashboard Cite

Consider the functional I(u) = ∫Ω ‖Du|n − L det Du| dx, whose energy well consists of matrices satisfying |ξ|n = L det ξ. We show that the relaxations of this functional in various Sobolev spaces are significantly different. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, |Du|n = L det Du, in the so-called grand Sobolev space W1,q)(Ω; Rn), with q = nL/(L + 1).

show abstract

The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

et al. 2020

View full text Add to dashboard Cite

We consider conformally invariant energies W on the group $${{\,\mathrm{GL}\,}}^{\!+}(2)$$ GL + ( 2 ) of $$2\times 2$$ 2 × 2 -matrices with positive determinant, i.e., $$W:{{\,\mathrm{GL}\,}}^{\!+}(2)\rightarrow {\mathbb {R}}$$ W : GL + ( 2 ) → R such that $$\begin{aligned} W(A\, F\, B) = W(F) \quad \text {for all }\; A,B\in \{a\, R\in {{\,\mathrm{GL}\,}}^{\!+}(2) \,|\,a\in (0,\infty ),\; R\in {{\,\mathrm{SO}\,}}(2)\}, \end{aligned}$$ W ( A F B ) = W ( F ) for all A , B ∈ { a R ∈ GL + ( 2 ) | a ∈ ( 0 , ∞ ) , R ∈ SO ( 2 ) } , where $${{\,\mathrm{SO}\,}}(2)$$ SO ( 2 ) denotes the special orthogonal group and provides an explicit formula for the (notoriously difficult to compute) quasiconvex envelope of these functions. Our results, which are based on the representation $$W(F)=h\bigl (\frac{\lambda _1}{\lambda _2}\bigr )$$ W ( F ) = h ( λ 1 λ 2 ) of W in terms of the singular values $$\lambda _1,\lambda _2$$ λ 1 , λ 2 of F, are applied to a number of example energies in order to demonstrate the convenience of the singular-value-based expression compared to the more common representation in terms of the distortion $${\mathbb {K}}:=\frac{1}{2}\frac{\Vert F \Vert ^2}{\det F}$$ K : = 1 2 ‖ F ‖ 2 det F . Applying our results, we answer a conjecture by Adamowicz (in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol 18(2), pp 163, 2007) and discuss a connection between polyconvexity and the Grötzsch free boundary value problem. Special cases of our results can also be obtained from earlier works by Astala et al. (Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, 2008) and Yan (Trans Am Math Soc 355(12):4755–4765, 2003). Since the restricted domain of the energy functions in question poses additional difficulties with respect to the notion of quasiconvexity compared to the case of globally defined real-valued functions, we also discuss more general properties related to the $$W^{1,p}$$ W 1 , p -quasiconvex envelope on the domain $${{\,\mathrm{GL}\,}}^{\!+}(n)$$ GL + ( n ) which, in particular, ensure that a stricter version of Dacorogna’s formula is applicable to conformally invariant energies on $${{\,\mathrm{GL}\,}}^{\!+}(2)$$ GL + ( 2 ) .

show abstract

A linear boundary value problem for weakly quasiregular mappings in space

Cited by 11 publications

References 32 publications

A Baire's category method for the Dirichlet problem of quasiregular mappings

A Baire's category method for the Dirichlet problem of quasiregular mappings

Relaxation and attainment results for an integral functional with unbounded energy well

The Quasiconvex Envelope of Conformally Invariant Planar Energy Functions in Isotropic Hyperelasticity

Contact Info

Product

Resources

About