(in Polish) Geometryczny rachunek wariacyjny 1000-1M26GRW
Submanifolds embedded in R^n (descriptive tools); [Federer1969, §§3.1.19–20]
Area and coarea formulae for Lipschitz functions; [Federer1969, §§3.2.2–12]
Countably rectifiable sets; [Federer1969, §§3.2.14–15]
Approximate differentiation and tangent vectors; [Federer1969, §§3.2.16–19]
Area and coarea formulae on rectifiable sets (overview); [Federer1969, §§3.2.20–22]
Rectifiable and integral varifolds; [Allard1972, §§3.1–2]
First variation with respect to an anisotropic functional; [Allard1972, §§4.1–4], [DPDRG2018, Appendix A]
Mean curvature and the second fundamental form in the anisotropic setting; [Allard1986, §2.1], [DPDRH2019, §2]
Weak maximum principle; [White2010]
Monotonicity formula and its consequences; [Allard1972, §5.1], [Menne2016, §4]
Compactness theorem for integral and rectifiable varifolds; [Allard1972, §5.6 and §6.4]
Ellipticity of functionals according to Almgren; [Almgren1968], [FK2018]
The atomic condition and its relation to ellipticity; [DRK2020]
Existence and regularity of minimisers (overview/informational); [Almgren1968]
Regularity of critical points (informational); [Allard1986]
In the tutorials:
Tensor product and exterior power of vector spaces
Orientation of a linear subspace
The Jacobian and its differential
Models of the Grassmannian: orthogonal projections or simple unit k-vectors
Examples of naturally occurring functionals whose ellipticity is unknown (Holmes–Thompson, Busemann–Hausdorff, Minkowski content with respect to a non-Euclidean norm)
Convexity of the Lagrangian, and ellipticity in the orientable case
Discussion of an open problem: existence/construction of elliptic functionals
Discussion of an open problem: does Almgren's ellipticity imply the atomic condition?
Discussion of an open problem: does the weak maximum principle hold in higher codimensions?
Course coordinators
Assessment criteria
Credit for the tutorials is given on the basis of attendance.
Final exam - oral
Bibliography
[Simon2014] Leon Simon, Introduction to geometric measure theory, 2014, Tsinghua Lectures, Vol. 2, No. 2
[Federer1969] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.
[Allard1972] William K. Allard. On the first variation of a varifold. Ann. of Math. (2), 95:417–491, 1972.
[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.
[DPDRG2018] Guido De Philippis, Antonio De Rosa, and Francesco Ghiraldin. Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies. Comm. Pure Appl. Math., 71(6):1123–1148, 2018.
[DPDRH2019] De Philippis, G., De Rosa, A., Hirsch, J.: The area blow-up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete Contin. Dyn. Syst. – A. 39(12), 7031–7056, 2019.
[White2010] Brian White. The maximum principle for minimal varieties of arbitrary codimension. Comm. Anal. Geom., 18(3):421–432, 2010.
[Menne2016] Ulrich Menne. Weakly differentiable functions on varifolds. Indiana Univ. Math. J., 65(3):977–1088, 2016.
[Almgren1968] F. J. Almgren, Jr. Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. of Math. (2), 87:321–391, 1968.
[FK2018] Yangqin Fang and Sławomir Kolasiński. Existence of solutions to a general geometric elliptic variational problem. Calc. Var. Partial Differential Equations, 57(3):Art. 91, 71, 2018.
[DRK2020] Antonio De Rosa and Sławomir Kolasiński. Equivalence of the ellipticity conditions for geometric variational problems. Comm. Pure Appl. Math., 73(11):2473–2515, 2020.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes: