# Geometric measure theory and differential inclusions

@article{Lellis2019GeometricMT, title={Geometric measure theory and differential inclusions}, author={Camillo De Lellis and Guido De Philippis and Bernd Kirchheim and Riccardo Tione}, journal={arXiv: Analysis of PDEs}, year={2019} }

In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard's theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the… Expand

#### Figures from this paper

#### 4 Citations

Minimal graphs and differential inclusions

- Mathematics
- 2020

Abstract In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in We prove regularity of solutions and a compactness result for… Expand

On the constancy theorem for anisotropic energies through differential inclusions

- Medicine, Mathematics
- Calculus of variations and partial differential equations
- 2021

It is shown that if the hypothesis of non-negativity is dropped, it is also possible to construct via convex integration a very degenerate stationary point with multiplicity. Expand

On the Rank-1 convex hull of a set arising from a hyperbolic system of Lagrangian elasticity

- Mathematics
- 2019

We address the questions (P1), (P2) asked in Kirchheim-Muller-Sverak (2003) concerning the structure of the Rank-$1$ convex hull of a submanifold $\mathcal{K}_1\subset M^{3\times 2}$ that is related… Expand

The four-state problem and convex integration for linear differential operators

- Mathematics
- 2021

We show that the four-state problem for general linear differential operators is flexible. The only flexibility result available in this context is the one for the five-state problem for the curl… Expand

#### References

SHOWING 1-10 OF 38 REFERENCES

Minimal graphs and differential inclusions

- Mathematics
- 2020

Abstract In this paper, we study the differential inclusion associated with the minimal surface system for two-dimensional graphs in We prove regularity of solutions and a compactness result for… Expand

ALLARD’S INTERIOR REGULARITY THEOREM: AN INVITATION TO STATIONARY VARIFOLDS

- 2017

This is a small set of notes, taken from the last lectures of a course given in Spring 2012 at the University of Zürich. The aim is to give a short, reader-friendly but nonetheless detailed… Expand

The Regularity of Critical Points of Polyconvex Functionals

- Mathematics
- 2004

Abstract.In this paper we are concerned with the question of regularity of critical points for functionals of the type eq1 We construct a smooth, strongly polyconvex eq2, and Lipschitzian weak… Expand

Quasiconvexity and partial regularity in the calculus of variations

- Mathematics
- 1986

We prove partial regularity of minimizers of certain functionals in the calculus of variations, under the principal assumption that the integrands be uniformly strictly quasiconvex. This is of… Expand

Studying Nonlinear pde by Geometry in Matrix Space

- Mathematics
- 2003

We outline an approach to study the properties of nonlinear partial differential equations through the geometric properties of a set in the space ofm xn matrices which is naturally associated to the… Expand

Convex integration for Lipschitz mappings and counterexamples to regularity

- Mathematics
- 2003

We study Lispchitz solutions of partial differential relations $\nabla u\in K$, where $u$ is a vector-valued function in an open subset of $R^n$. In some cases the set of solutions turns out to be… Expand

Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

- Mathematics
- 2003

Let Omega subset of R-n be a bounded starshaped domain. In this note we consider critical points (u) over bar is an element of (ξ) over bary + W-0(1,p) (Omega; R-m) of the functional F(u, Omega) :=… Expand

Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations

- Mathematics
- 2003

Abstract.Let Ω⊂ℝn be a bounded domain and F:𝕄→ℝ a given strongly quasiconvex integrand of class C2 satisfying the growth condition $${{ |F(\xi)| \le c (1 + |\xi|^p)}}$$ for some c>0 and 2≤p<∞.… Expand

Equivalence of the Ellipticity Conditions for Geometric Variational Problems

- Mathematics
- 2018

We exploit the so called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin in [Comm. Pure Appl. Math.] and proved to be necessary and sufficient for the validity of the… Expand

Tartar ’ s conjecture and localization of the quasiconvex hull in R 2 × 2

- 2006

We give a concrete and surprisingly simple characterization of compact sets K ⊂ R2×2 for which families of approximate solutions to the inclusion problem Du ∈ K are compact. In particular our… Expand