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Real-analytic properties of Sobolev functions. Change of variables in integral for Lipschitz transformations – area
and coarea formula. Differentiation of convex functions. Recommended for master students of mathematical
analysis.
Last update: T_KMA (02.05.2013)
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The exam is oral. The required knowledge corresponds to the sylabus at the presented extent Last update: Malý Jan, prof. RNDr., DrSc. (29.10.2019)
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L. C. Evans, R. F. Gariepy: Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, 1992
W. P. Ziemer: Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Springer-Verlag, New York, 1989 Last update: Malý Jan, prof. RNDr., DrSc. (29.10.2019)
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The exam is oral. The required knowledge corresponds to the sylabus at the presented extent Last update: Malý Jan, prof. RNDr., DrSc. (29.10.2019)
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1. Real-analytic properties of Sobolev functions
Riesz potential estimate
Beppo Levi's characterization
Embedding theorems
Approximate differentiability, a.e. differentiability
Examples concerning discontinuity and non-differentiability
2. Change of variables in integral for Lipschitz transforms
Area formula
Coarea formula
Sard type theorems
Luzin's (N) condition
3. Differentiation of convex functions
Lipschitz estimates
Zajíček's theorem (informatively)
Alexandrov's theorem Last update: Kaplický Petr, doc. Mgr., Ph.D. (09.06.2015)
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Measures, Radon-Nikodym theorem, Lebesgue integral, Radon measures, convolution, smoothing by convolution, strong, weak and weak* convergence in Banach spaces, elements of theory of distributions, Lipschitz functions and mappings, Hahn-Banach theorem, Hausdorff measure, L^p spaces and spaces of continuous functions Last update: Malý Jan, prof. RNDr., DrSc. (02.05.2018)
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