Elliptic geometric variational problems

• Funding agency: Polish National Science Center (NCN)
• Grant type: SONATA BIS
• Grant number: 2022/46/E/ST1/00328
• Years: 2023 -- 2028
• Principal investigator: Sławomir Kolasiński
• Total funding amount: 1 778 760 PLN

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• Opportunities
• Team members
• Papers
• Project description
  • Introduction
  • Scientific goal of the project
  • Significance of the project
  • Bibliography

Opportunities

• Ph.D. candidates are welcome to join the project. Generous funding is available. More information via e-mail.

• Call for a Ph.D. student (deadline 21st March 2025, 23:59)

• Two year postdoc position will be open in 2026.

Team members

• Maciej Leśniak
  Ph.D. student, topic: notions of ellipticity and relations between them.
• Michał Sajkowski
  M.Sc. student, topic: anisotropic principal curvatures and curvature measures.

Papers

• M. Leśniak, On Polyconvexity and Almgren Uniform Ellipticity With Respect to Polyhedral Test Pairs, arXiv:2501.12472

Project description

Introduction

In the theory of optimisation one studies configurations of objects which are optimal in some predefined sense. If the set of competitors is closed under smooth deformations and optimality is expressed in terms of an energy functional, then one can use methods of calculus of variations to find a minimiser, or just a critical point, of the energy. I am mainly interested in optimising shapes of subsets of ${\mathbf{R}}^n$. A typical example is the Plateau problem where one is interested in finding a surface in ${\mathbf{R}}^3$ of smallest possible area among all surfaces with a given boundary curve. Another example is the isoperimetric problem which is about describing an open set in ${\mathbf{R}}^n$ which minimises the perimeter (i.e. $(n-1)$ dimensional area of the boundary) among all open sets of volume equal one. These two classical and well understood problems give rise to many others, e.g., by considering arbitrary dimension and co-dimension, modelling the notions of surface and boundary with various objects, or by changing the way area is defined. Of course, studying the structure and regularity of critical points is a lot harder than the same for minimisers. The ultimate tool, tailored to tackle these kind of problems, is the theory of varifolds (cf. [Allard1972]) which can be thought of as currents deprived of orientation.

If $T\in \mathbf{G}(n,k)$ is a $k$-dimensional linear subspace of ${\mathbf{R}}^n$, a non-zero translation invariant (inside $T$) Radon measure over $T$ must be a constant multiple of the Lebesgue measure which coincides with ${\mathcal{H}}^{k}T$, where ${\mathcal{H}}^k$ is the $k$-Hausdorff measure over ${\mathbf{R}}^n$; hence, a choice of a positive constant for each $T\in \mathbf{G}(n,k)$ gives rise to a $k$-dimensional measure over ${\mathbf{R}}^n$. More precisely, let $F:\mathbf{G}(n,k)\to (0,\mathrm{\infty })$ be continuous. The measure ${\mathrm{\Phi }}_F$ is defined first for $k$-dimensional “polyhedral surfaces” $S=\bigcup _{i=1}^{\mathrm{\infty }}{S}_i$, where each $S_i$ is a Borel subset of some $T_i\in \mathbf{G}(n,k)$, by $${\mathrm{\Phi }}_F(S)=\sum _{i=1}^{\mathrm{\infty }}F(T_i){\mathcal{H}}^k(S_i).$$ Next, one extends ${\mathrm{\Phi }}_F$ to all $k$-submanifolds of ${\mathbf{R}}^n$ by approximation and to all countably $k$-rectifiable sets by countable additivity. For purely $k$-unrectifiable sets $S$ one can set ${\mathrm{\Phi }}_F(S)=0$. This motivates the study of anisotropic energies of the form $${\mathrm{\Phi }}_F(M)={\int }_M^{}F(\mathrm{Tan}(M,x))\theta (x)\mathrm{d}{\mathcal{H}}^k(x),$$ where $M\subseteq {\mathbf{R}}^n$ is countably $k$-rectifiable and $\theta :M\to [0,\mathrm{\infty })$ is a Borel multiplicity (or density) function.

Assume $U\subseteq {\mathbf{R}}^n$ is open and $M\subseteq U$ is as above and has locally finite ${\mathcal{H}}^k$-measure. Adopting the viewpoint that ${\mathrm{\Phi }}_F$ defines an action of the Radon measure $V$ over $U\times \mathbf{G}(n,k)$, given for Borel sets $A\subseteq U\times \mathbf{G}(n,k)$ by $$V(A)={\int }_{}^{}{\mathbf{1}}_{\{x:(x,T)\in A,T=\mathrm{Tan}(M,x)\}}\cdot \theta \mathrm{d}{\mathcal{H}}^k,$$ on the function $[U\times \mathbf{G}(n,k)\ni (x,T)\mapsto F(T)]$, we arrive at the definition of a rectifiable $k$-varifold in $U$. If $\mathrm{im}\theta \in \mathbf{Z}$, we say that $V$ is integral. The weight measure, denoted $\Vert V\Vert$, is the projection of $V$ onto $U$.

Let us emphasise that some very classical measures may be expressed as ${\mathrm{\Phi }}_F$ for an appropriate choice of $F$, e.g., given a non-euclidean norm on ${\mathbf{R}}^n$ the corresponding anisotropic $k$-Hausdorff measure coincides (on countably $k$-rectifiable sets) with ${\mathrm{\Phi }}_F$ provided $F$ is the Busemann-Hausdorff integrand and the anisotropic symplectic measure is produced by the Holmes-Thompson integrand; see [APT2004].

Scientific goal of the project

Almost everywhere regularity of minima of ${\mathrm{\Phi }}_F$ has been proven by Almgren under the assumption that $F$ is elliptic; cf. [Almgren1968 §1.6(2) and §1.4]. His ellipticity condition is a natural counterpart of quasi-convexity of Morrey; cf. [Morrey1966 p. 18] and [DRT2022 Proposition 3.13]. However, it is rather hard to check so that no examples of elliptic integrands in high co-dimension are known apart from the area integrand and its class $2$ neighbourhood. Regularity of critical points is still an open problem. Recently the atomic condition (abbreviated AC) was introduced in [DPDRG2018] as a necessary and sufficient condition for rectifiability of critical points. It is stronger than Almgren’s ellipticity as we showed in [DRK2022]. Some partial results in high co-dimension concerning existence of non-trivial AC integrands and regularity of critical points were obtained by De Rosa and Tione [DT2022] but the most important questions remain still open.

The main goal of the project is to expand the regularity theory for critical points of ${\mathrm{\Phi }}_F$ in any co-dimension by exploiting the atomic condition. In co-dimension one (uniform) ellipticity is the same as strict (uniform) convexity (see [DPDRG2018 Theorem 1.3]) but focusing on convexity distracts attention from the true nature of ellipticity, which is a more subtle notion. I am strongly convinced that AC reveals the hidden geometric structure of ellipticity and deep understanding of this condition will lead to solving (at least partially) some of the big questions mentioned in the next section.

Significance of the project

The regularity theory for critical points of the area integrand (i.e. $F\equiv 1$) rests on the monotonicity formula which implies Ahlfors regularity; cf. [Allard1972 5.1]. For anisotropic integrands this formula is inaccessible; cf. [Allard1974]. Ellipticity for $F$ was defined by Almgren [Almgren1968] so to prove partial regularity for minimisers of ${\mathrm{\Phi }}_F$. His proof employs a form of the isoperimetric inequality, which is provable for minimisers and acts as replacement for monotonicity. Allard [Allard1986] showed partial regularity for critical points $V$ assuming $n-k=1$, the integrand is uniformly convex, and the set of points of positive $\Vert V\Vert$-density is closed. The last condition is crucial and would follow from monotonicity.

The essential question, that is now open for more than 50 years and that prevents any substantial progress, is how to derive density ratio bounds for $F$-stationary varifolds under an ellipticity condition on $F$? The most promising route seems to lead via the isoperimetric inequality.

According to Almgren’s definition, $F$ is elliptic if any flat $k$-disc $D$ minimises ${\mathrm{\Phi }}_F$ among all countably $k$-rectifiable surfaces that cannot be retracted onto the relative boundary of $D$. Another, great problem is the very existence of elliptic integrands which are not just perturbations of the area integrand. In particular, it is not known whether the anisotropic $k$-Hausdorff measure constructed from some non-euclidean norm is elliptic. In case $k=2$ and surfaces are understood as Lipschitz integral chains, this has been resolved by Burago and Ivanov [BI2012]. Note that ellipticity heavily depends on the precise definition of a surface. In case we are considering only integral currents, then uniform convexity of the integrand (now being a norm on ${\bigwedge }_k{\mathbf{R}}^n$) is sufficient for uniform ellipticity in any co-dimension but an analogous fact is not yet proven in case surfaces are allowed to be more general objects; cf. [Federer1969 §5.1.2].

In co-dimension one (uniform) ellipticity is equivalent to (uniform) strict convexity of the norm naturally associate with F; cf. [DPDRG2018 Theorem 1.3] and [Almgren1968 §5.7]. In case $F$ is of class $2$ critical points of the associated isoperimetric functional are then classified as finite sums of Wulff shapes; cf. [HLMG2009] and also our recent work [DRKS2020]. In this context, however, the most interesting are crystalline energies, where the integrand is only Lipschitz continuous; cf. [Taylor2002]. Characterisation of critical points of these kind of energies is yet another outstanding problem; cf. [DMv1]. We call it the anisotropic Alexandrov’s problem.

Assume $F$ is of class $2$. In this case one can define the principal $F$-curvatures of a hypersurface; cf. [DRKS2020 Definition 2.26]. If the total $F$-variation measure of a varifold $V$ is Radon, then one defines the mean $F$-curvature vector of $V$ using the Radon-Nikodym and a Riesz-type representation theorems. Assume $U$ is an open set with smooth boundary $M$. We say that a maximum principle holds if any $k$-varifold in $U$ that touches $M$ at a point $a$ has mean $F$-curvature at $a$ bigger than the sum of $k$ greatest principle $F$-curvatures of $M$ at $a$. This holds in co-dimension one given $F$ is smooth and uniformly convex; cf. [SW1989]. In higher co-dimension this is resolved only for the area integrand; cf. [White2010]. Maximum principle for other integrands and high co-dimension is still an open problem. Proving it could potentially lead to a generalisation of Allard’s co-dimension one theory [Allard1986]; in particular, the part where he is using barriers to prove a local height bound.

Bibliography

• [Allard1972] Allard, William K., On the first variation of a varifold, 1972, Ann. of Math. (2) , Vol. 95, p. 417-491

• [Allard1974] William K. Allard, A characterization of the area integrand, In Symposia Mathematica, Vol. XIV (Convegno di Teoria Geometrica dell’Integrazione e Varietà Minimali, INDAM, Rome, 1973), pages 429–444. Academic Press, London, 1974

• [Allard1986] William K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, In Geometric measure theory and the calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 1–28. Amer. Math. Soc., Providence, RI, 1986

• [Almgren1968] Almgren Jr., F. J., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, 1968, Annals of Mathematics. Second Series , Vol. 87, p. 321-391

• [APT2004] Álvarez Paiva, J. C. and Thompson, A. C., Volumes on normed and Finsler spaces, 2004, A sampler of Riemann-Finsler geometry, Vol. 50,Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge p. 1-48

• [BI2012] Dmitri Burago and Sergei Ivanov. Minimality of planes in normed spaces. Geom. Funct. Anal., 22(3):627–638, 2012.

• [DMv1] M. G. Delgadino and F. Maggi, Alexandrov’s theorem revisited, arXiv:1711.07690v1, 2017

• [DPDRG2018] Philippis, Guido De, De Rosa, Antonio, and Ghiraldin, Francesco, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, 2018, Communications on Pure and Applied Mathematics, Vol. 71, No. 6 Wiley, p. 1123-1148

• [DRKS2020] De Rosa, Antonio, Kolasiński, Sławomir, and Santilli, Mario, Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets, 2020, Archive for Rational Mechanics and Analysis, Vol. 238, No. 3, p. 1157-1198

• [DRK2022] De Rosa, Antonio and Tione, Riccardo, Regularity for graphs with bounded anisotropic mean curvature, Invent. Math. 230 (2022), no. 2, 463–507.

• [DRT2022] De Rosa, Antonio and Tione, Riccardo, Regularity for graphs with bounded anisotropic mean curvature, 2022, Inventiones mathematicae , Vol. 230, No. 2, p. 463-507

• [Federer1969] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969

• [HLMG2009] Yijun He, Haizhong Li, Hui Ma, and Jianquan Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, Indiana Univ. Math. J., 58(2):853–868, 2009.

• [Morrey1966] Morrey Jr., Charles B., Multiple integrals in the calculus of variations, 2008 Classics in Mathematics Springer-Verlag, Berlin, p. x+506

• [SW1989] Bruce Solomon and Brian White, A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J., 38(3):683–691, 1989

• [Taylor2002] Taylor, Jean E., Crystalline variational methods, 2002, Proceedings of the National Academy of Sciences, Vol. 99, No. 24, p. 15277-15280

• [White2010] Brian White, The maximum principle for minimal varieties of arbitrary codimension, Comm. Anal. Geom., 18(3):421–432, 2010