MS-E1991 - Calculus of Variations, 07.01.2019-04.04.2019
This course space end date is set to 04.04.2019 Search Courses: MS-E1991
Topic outline
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The Lebesgue differentiation theorem in metric measure spaces.
Material: Section 3.4 in [HKST] and [H] pages 4 and 12-13.
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Paths in a metric space.
Material: [HKST] Section 5.1 and P. Hajlasz, Sobolev spaces on metric-measure spaces. (Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002)), 173--218, Contemp. Math. , 338, Amer. Math. Soc., Providence, RI, 2003.
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The goal is to prove the following theorem: A complete doubling space carries a doubling measure.
Material: Chapter 13 in [H].
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David, Guy; Snipes, Marie A non-probabilistic proof of the Assouad embeddingtheorem with bounds on the dimension. Anal. Geom. Metr. Spaces 1 (2013), 36–41.
An embedding theorem of Assouad asserts that, upon replacing the metric of a doubling metric space by a root of d, the resulting ‘snowflaked’ space is bi-Lipschitz embeddable into a finite-dimensional Euclidean space.
Material:
[H] p. 98, 112 and 121.
Assouad, Patrice Plongements lipschitziens dans
Rn . (French) [Lipschitz embeddings inRn ] Bull. Soc. Math. France 111 (1983), no. 4, 429–448. -
The goal is to construct a doubling measure that is not absolutely continuous with respect the Lebesgue measure.
Material:
Garnett, John; Killip, Rowan; Schul, Raanan A doubling measure on
ℝd can charge a rectifiable curve. Proc. Amer. Math. Soc. 138 (2010), no. 5, 1673–1679. -
Modulus of a curve family.
Material: Sections 5.2 and 5.3 in [HKST].
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Completeness of the Sobolev spaces on metric measure spaces.
Material: [HKST] and [BB].
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BV functions on metric measure spaces.
Material:
Miranda, Michele, Jr. Functions of bounded variation on "good'' metric spaces. J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004.
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The goal is to prove the following theorem: Assume that X is a doubling metric measure space that supports a p-Poincare inequality. Then Lipschitz functions are dense in the Sobolve space.
Material: Section 8.2 in [HKST] and Section 5.3 in [BB].
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The goal of this project is to show that every complete and doubling metric measure space that supports a Poincare inequality is quasiconvex.
Material: Section 8.3 in [HKST] and Section 4.5 in [BB].
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Material:
The goal is study the existence and regularity of minimizers of functionals of linear growth in metric measure spaces. There are counterexamples to show that in general, minimizers are not continuous and can have jump discontinuities inside the domain.
Hakkarainen, Heikki; Kinnunen, Juha; Lahti, Panu Regularity of minimizers of the area functional in metric spaces. Adv. Calc. Var. 8 (2015), no. 1, 55–68.
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Clarendon Press, Oxford, 2000.
G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-Dimensional Variational Problems. An Introduction, Oxford Lecture Ser. Math. Appl. 15, Oxford University Press, Oxford, 1998. -
The goal is to study a comparison principle for super- and subminimizers on parabolic space-time cylinders and a uniqueness result for minimizers of a boundary value problem.
Material:
Kinnunen, Juha; Masson, Mathias Parabolic comparison principle and quasiminimizers in metric measure spaces. Proc. Amer. Math. Soc. 143 (2015), no. 2, 621–632.
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The goal is to study the existence of a solution to variational problems by the direct methods in the calculus of variations. This project is in the classical Euclidean case.
Material:
Evans, Lawrence C. Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
Giusti, Enrico Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
Jost, Jürgen; Li-Jost, Xianqing Calculus of variations. Cambridge Studies in Advanced Mathematics, 64. Cambridge University Press, Cambridge, 1998.
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The goal is to study the Euler-Lagrange equation of a variational problem. This project is in the classical Euclidean case.
Material:
Evans, Lawrence C. Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
Giusti, Enrico Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
Jost, Jürgen; Li-Jost, Xianqing Calculus of variations. Cambridge Studies in Advanced Mathematics, 64. Cambridge University Press, Cambridge, 1998.
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Relaxation method in the calculus of variations. This project is in the classical Euclidean case.
Material:
Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
Attouch, Hedy; Buttazzo, Giuseppe; Michaille, Gérard Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization. MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. xii+634 pp. ISBN: 0-89871-600-4
Jost, Jürgen; Li-Jost, Xianqing Calculus of variations. Cambridge Studies in Advanced Mathematics, 64. Cambridge University Press, Cambridge, 1998.
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Γ -convergence in the calculus of variations.Material:
Attouch, Hedy; Buttazzo, Giuseppe; Michaille, Gérard Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization. MPS/SIAM Series on Optimization, 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA, 2006. xii+634 pp. ISBN: 0-89871-600-4
Dal Maso, Gianni An introduction to
Γ -convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993.Jost, Jürgen; Li-Jost, Xianqing Calculus of variations. Cambridge Studies in Advanced Mathematics, 64. Cambridge University Press, Cambridge, 1998.
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Regularity of the gradient of a miniminizer
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