(in Polish) Geometryczna teoria miary i zagadnienia wariacyjne 1000-1M23TMW

The lecture is a natural continuation of the course "Measure Theory", the material of which will be the entry point for further considerations. The end point is intended to be the current state of knowledge of geometric variational problems and knowledge of the main open problems in this field. In particular, we will focus on the hitherto poorly understood, but crucial, notion of ellipticity.

Inevitably, a lot of material will be presented in an illustrative way, without giving detailed proofs, although always with reference to specific scientific papers. The aim is to present the current state of knowledge at a level sufficient to undertake independent research.

Lecture

1. Crash-course of multilinear algebra [1, §1] (1 lecture)




1. Tensor product of linear spaces


2. Tensor algebra and exterior algebra


3. The isomorphism: Hom(A,B) ≃ A* ⊗ B


4. Norm and scalar product on the exterior power


5. Wedge product and contraction operations for multilinear forms and multivectors


6. Oriented and non-oriented Grassmannian




2. Area and coarea [1, §3.2.1-12] (2 lectures)




1. Approximate Jacobian


2. Area formula, i.e., a generalisation of the change of variable formula


3. The coarea formula




3. Rectifiable sets and measures [1, §3.2.14-22] (2 lectures)




1. Formulation of the Whitney extension theorem and the Rademacher theorem on differentiability of Lipschitz functions


2. Definition and examples of rectifiable and purely unrectifiable sets


3. Briefly on area and coarea formulas for maps between rectifiable sets


4. Hausdorff measure truncated to a manifold as the surface area measure


5. Several characterisations of rectifiable sets and measures without proofs, e.g. Preiss (1987) and Azzam and Tolsa (2015)


6. Rectifiable measures as weak limits of sequences of smooth manifolds




4. Varifolds [10] (4 lectures)




1. General, rectifiable, and integral varifolds


2. Flat norm, convergence, and compactness of families of varifolds (by Tikhonov's theorem)


3. Tangent measures and varifolds


4. Push-forward of a varifold by a Lipschitz map


5. Second fundamental form and mean curvature of embedded smooth manifolds


6. First variation of the varifold with respect to an anisotropic integrand and the generalised mean curvature


7. Proof of monotonicity of density quotients and some illustrative remarks on the consequences


8. Weak maximum principle based on [8].




5. Ellipticity (4 lectures)




1. Definitions of Almgren ellipticity (AE) [11] and the Atomic Condition (AC) [4].


2. Dependence of the notion of ellipticity on the choice of competitors.


3. Overview of the consequences of ellipticity:
   
   
   
   • existence of minima in the class of rectifiable varifolds [12];
   
   
   • rectifiability of critical points [4];
   
   
   • partial regularity of minima [11].
   
   
   
   


4. condition BC and relations between AE and AC [6].


5. Geometrical characterisation of AC.


6. Scalar atomic condition (SAC) and regularity of critical graphs [7].


7. Constructions of k-dimensional translation invariant measures in Rⁿ and the ellipticity problem [5].

Exercises

1. Operations on multi-vectors and multi-linear forms.


2. Various characterizations of embed manifolds beyond the material of Analysis II (applications of the constant rank theorem).


3. The Grassmannian as an embedded manifold.


4. The Hausdorff measure in a normed space expressed by the integral of some integrand with respect to the Euclidean Hausdorff measure.


5. An example showing that a Hausdorff measure is not a product of lower dimensional Hausdorff measures.


6. Derivation of the formula for the anisotropic perimeter of a set.


7. Holmes-Thompson measure.


8. Relationship between the weak topology and the product topology.


9. Relationship between the flat norm and week convergence of measures.


10. Calculation of derivatives of the Jacobian and other functions whose domain is a set of linear maps.