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This is the basic course about the theory of partial differential equations. The notion of a weak (distributional) solution and the corresponding function spaces will be introduced and we establish the theory for (linear) elliptic equations.
Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (11.09.2013)
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At the end of the semester there will be an exam, that will have written and oral part. Students are supposed to provide the knowledge of the theory taught during the semester.
Students are obliged to solve homeworks correctly for passing the tutorials. It is also a necessary condition in order to pass the exam. Last update: Kaplický Petr, doc. Mgr., Ph.D. (11.10.2024)
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L. C. Evans: Partial Differential Equations, AMS, 2010 D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, 2001 Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (10.09.2013)
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According to sylabus Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (27.09.2020)
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General notation of weak solutions
Sobolev spaces: definition and basic overview of its properties, embedding and trace theorems
Weak solutions to linear elliptic equations on bounded domains, various boundary conditions, solution by the use of the Riesz representation theorem and the use of the Lax-Milgram theorem, compactness of the solution operator, eigen-values and eigen-vectors of the solution operator, Fredholm-like theorems and their applications, maximum principle for weak solution, $W^{2,2}$ and higher regularity, symmetric operators and the equivalence with minimizing of a quadratic functional
Bochner spaces: defintion and basic overview of its properties, Gelfand triple, integration by parts formula, embeddding.
Weak solutions to linear parabolic equations, various boundary conditions, construction of a solution via Galerkin method, uniqueness and regularity of solution.
Weak solution to linear hyperbolic equation, various boudary codition, construction of a solution via Galerkin method, uniqueness of solution, finite speed of propagation. Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (04.10.2018)
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Basic knowledge of the mathematical analysis, measure theory (including the Lebesgue spaces) and the classical theory of PDEs is needed. Furthermore, starting from the middle of the semester also some basic facts from the functional analysis will be required (Riesz representation theorem for Hilbert spaces, spectrum of selfadjoint compact operators, weak convergence). Last update: Bulíček Miroslav, doc. RNDr., Ph.D. (27.09.2020)
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