Research article Special Issues

Fine properties of functions of bounded deformation-an approach via linear PDEs

  • Received: 04 November 2019 Accepted: 06 February 2020 Published: 27 February 2020
  • In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are L1-functions with the property that the symmetric part of their distributional derivative is representable as a bounded (matrix-valued) Radon measure. It has been known for a long time that for a (matrix-valued) Radon measure the property of being a symmetrized gradient can be characterized by an under-determined second-order PDE system, the Saint-Venant compatibility conditions. This observation gives rise to a new approach to the fine properties of BD-maps via the theory of PDEs for measures, which complements and partially replaces classical arguments. Starting from elementary observations, here we elucidate the ellipticity arguments underlying this recent progress and give an overview of the state of the art. We also present some open problems.

    Citation: Guido De Philippis, Filip Rindler. Fine properties of functions of bounded deformation-an approach via linear PDEs[J]. Mathematics in Engineering, 2020, 2(3): 386-422. doi: 10.3934/mine.2020018

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  • In this survey we collect some recent results obtained by the authors and collaborators concerning the fine structure of functions of bounded deformation (BD). These maps are L1-functions with the property that the symmetric part of their distributional derivative is representable as a bounded (matrix-valued) Radon measure. It has been known for a long time that for a (matrix-valued) Radon measure the property of being a symmetrized gradient can be characterized by an under-determined second-order PDE system, the Saint-Venant compatibility conditions. This observation gives rise to a new approach to the fine properties of BD-maps via the theory of PDEs for measures, which complements and partially replaces classical arguments. Starting from elementary observations, here we elucidate the ellipticity arguments underlying this recent progress and give an overview of the state of the art. We also present some open problems.


    The Ricci curvature in Finsler geometry naturally generalizes the Ricci curvature in Riemannian geometry. However, in Finsler geometry, there are several versions of the definition of scalar curvature because the Ricci curvature tensor is defined in different forms. Here we adopt the definition of scalar curvature, which was introduced by Akbar–Zadeh [1,see (2.1)]. Tayebi [11] characterized general fourth-root metrics with isotropic scalar curvature. Moreover, he studied Bryant metrics with isotropic scalar curvature. Later, a locally conformally flat (α,β)-metric with weakly isotropic scalar curvature was studied by Chen–Xia [4]. They proved that its scalar curvature must vanish. Recently, Cheng–Gong [5] proved that if a Randers metric is of weakly isotropic scalar curvature, then it must be of isotropic S-curvature. Furthermore, they concluded that when a locally conformally flat Randers metric is of weakly isotropic scalar curvature, it is Minkowskian or Riemannian. Very recently, Ma–Zhang–Zhang [8] showed that the Kropina metric with isotropic scalar curvature is equivalent to an Einstein Kropina metric according to the navigation data.

    Shimada [9] first developed the theory of m-th root metrics as an interesting example of Finsler metrics, immediately following Matsumoto and Numata's theory of cubic metrics [7]. It is applied to biology as an ecological metric by Antonelli [2]. Later, many scholars studied these metrics ([3,6,10,11,12], etc). In [13], cubic Finsler manifolds in dimensions two or three were studied by Wegener. He only abstracted his PhD thesis and barely did all the calculations in that paper. Kim and Park [6] studied the m-th root Finsler metrics which admit (α,β)types. In [12], Tayebi–Razgordani–Najafi showed that if the locally conformally flat cubic metric is of relatively isotropic mean Landsberg curvature on a manifold M of dimension n(3), then it is a Riemannian metric or a locally Minkowski metric. Tripathia–Khanb–Chaubey [10] considered a cubic (α,β)-metric which is a special class of p-power Finsler metric, and obtained the conditions under which the Finsler space with such special metric will be projectively flat. Further, they obtained in which case this Finsler space will be a Berwald space or Douglas space.

    In this paper, we mainly focus on m-th root metrics with weakly isotropic scalar curvature and obtain the following results:

    Theorem 1.1. Let the m(3)-th root metric F be of weakly isotropic scalar curvature. Then its scalar curvature must vanish.

    Let A:=ai1i2im(x)yi1yi2yim. If A=Fm is irreducible, then the further result is obtained as follows:

    Theorem 1.2. Let F=mA be the m(3)-th root metric. Assume that A is irreducible. Then the following are equivalent: (i) F is of weakly isotropic scalar curvature; (ii) its scalar curvature vanishes; (iii) it is Ricci-flat.

    Based on Theorem 1.1, we obtain the result for locally conformally flat cubic Finsler metrics as following:

    Theorem 1.3. Let F be a locally conformally flat cubic Finsler metric on a manifold M of dimension n(3). If F is of weakly isotropic scalar curvature, then F must be locally Minkowskian.

    In this section, we mainly introduce several geometric quantities in Finsler geometry and several results that will be used later.

    Let M be an n(3)-dimensional smooth manifold. The points in the tangent bundle TM are denoted by (x,y), where xM and yTxM. Let (xi,yi) be the local coordinates of TM with y=yixi. A Finsler metric on M is a function F:TM[0,+) such that

    (1) F is smooth in TM{0};

    (2) F(x,λy)=λF(x,y) for any λ>0;

    (3) The fundamental quadratic form g=gij(x,y)dxidxj, where

    gij(x,y)=[12F2(x,y)]yiyj

    is positively definite. We use the notations: Fyi:=Fyi, Fxi:=Fxi,F2yiyj:=2F2yiyj.

    Let F be a Finsler metric on an n-dimensional manifold M, and let Gi be the geodesic coefficients of F, which are defined by

    Gi:=14gij(F2xkyjykF2xj),

    where (gij)=(gij)1. For any xM and yTxM{0}, the Riemann curvature Ry:=Rik(x,y)xidxk is defined by

    Rik:=2GixkGixjykyj+2GjGiyjykGiyjGjyk.

    The Ricci curvature Ric is the trace of the Riemann curvature defined by

    Ric:=Rkk.

    The Ricci tensor is

    Ricij:=12Ricyiyj.

    By the homogeneity of Ric, we have Ric=Ricijyiyj. The scalar curvature r of F is defined as

    r:=gijRicij. (2.1)

    A Finsler metric is said to be of weakly isotropic scalar curvature if there exists a 1-form θ=θi(x)yi and a scalar function χ=χ(x) such that

    r=n(n1)(θF+χ). (2.2)

    An (α,β)-metric is a Finsler metric of the form

    F=αϕ(s),

    where α=aij(x)yiyj is a Riemannian metric, β=bi(x)yi is a 1-form, s:=βα and b:=∥βα<b0. It has been proved that F=αϕ(s) is a positive definite Finsler metric if and only if ϕ=ϕ(s) is a positive C function on (b0,b0) satisfying the following condition:

    ϕ(s)sϕ(s)+(Bs2)ϕ(s)>0,|s|b<b0, (2.3)

    where B:=b2.

    Let F=3aijk(x)yiyjyk be a cubic metric on a manifold M of dimension n3. By choosing a suitable non-degenerate quadratic form α=aij(x)yiyj and one-form β=bi(x)yi, it can be written in the form

    F=3pβα2+qβ3,

    where p and q are real constants such that p+qB0 (see [6]). The above equation can be rewritten as

    F=α(ps+qs3)13,

    which means that F is also an (α,β)-metric with ϕ(s)=(ps+qs3)13. Then, by (2.3), we obtain

    p2B+p(4p+3qB)s2>0. (2.4)

    Two Finsler metrics F and ˜F on a manifold M are said to be conformally related if there is a scalar function κ=κ(x) on M such that F=eκ(x)˜F. Particularly, an (α,β)-metric F=αϕ(βα) is said to be conformally related to a Finsler metric ˜F if F=eκ(x)˜F with ˜F=˜αϕ(˜s)=˜αϕ(˜β˜α). In the following, we always use symbols with a tilde to denote the corresponding quantities of the metric ˜F. Note that α=eκ(x)˜α, β=eκ(x)˜β, thus ˜s=s.

    A Finsler metric that is conformally related to a locally Minkowski metric is said to be locally conformally flat. Thus, a locally conformally flat (α,β)-metric F has the form F=eκ(x)˜F, where ˜F=˜αϕ(˜β˜α) is a locally Minkowski metric.

    Denoting

    rij:=12(bi|j+bj|i),sij:=12(bi|jbj|i),rij:=ailrlj,sij:=ailslj,rj:=birij,r:=biri,sj:=bisij,r00:=rijyiyj,si0:=sijyj,s0:=siyi,

    where bi:=aijbj, bi|j denotes the covariant differentiation with respect to α.

    Let Gi and Giα denote the geodesic coefficients of F and α, respectively. The geodesic coefficients Gi of F=αϕ(βα) are related to Giα by

    Gi=Giα+αQsi0+(2Qαs0+r00)(Ψbi+Θα1yi),

    where

    Q:=ϕϕsϕ,Θ:=ϕϕs(ϕϕ+ϕϕ)2ϕ[(ϕsϕ)+(Bs2)ϕ],Ψ:=ϕ2[(ϕsϕ)+(Bs2)ϕ].

    Assume that F=αϕ(βα) is conformally related to a Finsler metric ˜F=˜αϕ(˜β˜α) on M, i.e., F=eκ(x)˜F. Then

    aij=e2κ(x)˜aij,bi=eκ(x)˜bi,˜b:=∥˜β˜α=˜aij˜bi˜bj=b.

    Further, we have

    bi|j=eκ(x)(˜bij˜bjκi+˜blκl˜aij),αΓlij=˜α˜Γlij+κjδli+κiδljκl˜aij,rij=eκ(x)˜rij+12eκ(x)(˜bjκi˜biκj+2˜blκl˜aij),sij=eκ(x)˜sij+12eκ(x)(˜biκj˜bjκi),ri=˜ri+12(˜blκl˜bib2κi),r=eκ(x)˜r,si=˜si+12(b2κi˜blκl˜bi),rii=eκ(x)˜rii+(n1)eκ(x)˜biκi,sji=eκ(x)˜sji+12eκ(x)(˜bjκi˜biκj).

    Here ˜bij denotes the covariant derivatives of ˜bi with respect to ˜α, αΓmij and ˜α˜Γmij denote Levi–Civita connections with respect to α and ˜α, respectively. In the following, we adopt the notations κi:=κ(x)xi, κij:=2κ(x)xixj, κi:=˜aijκj, ˜bi:=˜aij˜bj, f:=˜biκi, f1:=κij˜biyj, f2:=κij˜bi˜bj, κ0:=κiyi, κ00:=κijyiyj and κ2˜α:=˜aijκiκj.

    Lemma 2.1. ([4]) Let F=eκ(x)˜F, where ˜F=˜αϕ(˜β˜α) is locally Minkowskian. Then the Ricci curvature of F is determined by

    Ric=D1κ2˜α˜α+D2κ20+D3κ0f˜α+D4f2˜α2+D5f1˜α+D6˜α2+D7κ00,

    where Dk(k=1,...,7) is listed in Lemma 3.2 in [4].

    Lemma 2.2. ([4]) Let F=eκ(x)˜F, where ˜F=˜αϕ(˜β˜α) is locally Minkowskian. Then the scalar curvature of F is determined by

    r=12e2κ(x)ρ1[Σ1(τ+ηλ2)Σ2λη˜αΣ3η˜α2Σ4],

    where

    τ:=δ1+δB,η:=μ1+Y2μ,λ:=εδs1+δB,δ:=ρ0ε2ρ2ρ,ε:=ρ1ρ2,μ:=ρ2ρ,Y:=AijYiYj,Aij:=aij+δbibj,ρ:=ϕ(ϕsϕ),ρ0:=ϕϕ+ϕϕ,ρ1:=s(ϕϕ+ϕϕ)+ϕϕ,ρ2:=s[s(ϕϕ+ϕϕ)ϕϕ],

    and Σi(i=1,...,4) are listed in the proof of Lemma 3.3 in [4].

    Lemma 2.3. ([14]) Let m-th root metric F=mai1i2im(x)yi1yi2yim be a Finsler metric on a manifold of dimension n. Then the Ricci curvature of F is a rational function in y.

    In this section, we will prove the main theorems. Firstly, we give the proof of Theorem 1.1.

    The proof of Theorem 1.1. For an m-th root metric F=mai1i2im(x)yi1yi2yim on a manifold M, the inverse of the fundamental tensor of F is given by (see [14])

    gij=1(m1)F2(AAij+(m2)yiyj), (3.1)

    where Aij=1m(m1)2Ayiyj and (Aij)=(Aij)1. Thus, F2gij are rational functions in y.

    By Lemma 2.3, the Ricci curvature Ric of m-th root metric is a rational function in y. Thus, Ricij:=Ricyiyj are rational functions. According to (2.1), we have

    F2r=F2gijRicij. (3.2)

    This means that F2r is a rational function in y.

    On the other hand, if F is of weakly isotropic scalar curvature, according to (2.2), we obtain

    F2r=n(n1)(θF+χF2),

    where θ is a 1-form and χ is a scalar function. The right side of the above equation is an irrational function in y. Comparing it with (3.2), we have r=0.

    In the following, the proof of Theorem 1.2 is given.

    The proof of Theorem 1.2. By Theorem 1.1, we conclude that F is of weakly isotropic scalar curvature if and only if its scalar curvature vanishes. So we just need to prove that (ii) is equivalent to (iii). Assume that the scalar curvature vanishes. Hence, by (3.1), 0=r=gijRicij=1m1F2(AAij+(m2)yiyj)Ricij holds. It means that

    0=(AAij+(m2)yiyj)Ricij=AAijRicij+(m2)Ric.

    Since A is irreducible, Ric must be divided by A. Thus, Ric=0.

    Conversely, if Ric=0, then by the definition of r we have r=0.

    Based on Theorem 1.1, we can prove Theorem 1.3 for locally conformally flat cubic metrics.

    The proof of Theorem 1.3. Assume that the locally conformally flat cubic metric F is of weakly isotropic scalar curvature. Then, by Lemma 2.2 and Theorem 1.1, we obtain the scalar curvature vanishes, i.e.,

    Σ1(τ+ηλ2)Σ2λη˜αΣ3η˜α2Σ4=0.

    Further, by detailed expressions of Σi(i=1,4), the above equation can be rewritten as

    B(4p+3qB)κ204(4p+3qB)˜βκ0f+4p˜α2f2(4p+3qB)8˜α2s2γ7+Tγ6=0, (3.3)

    where γ:=pB˜α2(4p+3qB)˜β2 and T has no γ1.

    Thus, the first term of (3.3) can be divided by γ. It means that there is a function h(x) on M such that

    B(4p+3qB)κ204(4p+3qB)˜βκ0f+4p˜α2f2=h(x)γ.

    The above equation can be rewritten as

    B(4p+3qB)κ204(4p+3qB)˜βκ0f+4p˜α2f2=h(x)[pB˜α2(4p+3qB)˜β2]. (3.4)

    Differentiating (3.4) with yi yields

    B(4p+3qB)κ0κi2(4p+3qB)(˜biκ0+˜βκi)f+4p˜ailylf2=h(x)[pB˜ailyl(4p+3qB)˜β˜bi]. (3.5)

    Differentiating (3.5) with yj yields

    B(4p+3qB)κiκj2(4p+3qB)(˜biκj+˜bjκi)f+4p˜aijf2=h(x)[pB˜aij(4p+3qB)˜bi˜bj].

    Contracting the above with ˜bi˜bj yields

    Bf2(8p+9qB)=3B2h(x)(p+qB).

    Thus, we have

    h(x)=(8p+9qB)f23B(p+qB). (3.6)

    Substituting (3.6) into (3.5) and contracting (3.5) with ˜bi yield

    (4p+3qB)f(f˜βBκ0)=0. (3.7)

    Furthermore, by (2.4) and 4p+3qB0, we have f(f˜βBκ0)=0.

    Case I: f=0. It means h(x)=0 by (3.6). Thus, one has that κi=0 by (3.4), which means

    κ=constant.

    Case II: f0. It implies that f˜βBκ0=0. Substituting it into (3.4), we obtain

    ˜β2=B˜α2,

    which does not exist.

    Above all, we have κ=constant. Thus we conclude that the conformal transformation must be homothetic.

    X. Zhang: Conceptualization, Validation, Formal analysis, Resources, Software, Investigation, Methodology, Supervision, Writing-original draft, Writing-review and editing, Project administration, Funding acquisition; C. Ma: Formal analysis, Software, Investigation, Supervision, Writing-review and editing; L. Zhao: Resources, Investigation, Supervision. All authors have read and agreed to the published version of the manuscript.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Xiaoling Zhang's research is supported by National Natural Science Foundation of China (No. 11961061, 11761069).

    The authors declare that they have no conflicts of interest.



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