FUNCTIONAL ANALYSIS 1
cod. 14864

Academic year 2007/08
3° year of course - Second semester
Professor
Academic discipline
Analisi matematica (MAT/05)
Field
Formazione analitica
Type of training activity
Characterising
56 hours
of face-to-face activities
6 credits
hub:
course unit
in - - -

Learning objectives

The course provides an introduction to the classical theory of linear elliptic partial differential equations of second order and potential theory.

Prerequisites

Basic and advanced calculus and differential geometry. <br />

Course unit content

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<br />
1. Preliminaries.<br />
Integration on manifolds. Divergence theorem and Gauss formulas. Coarea formula. <br />
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2. Harmonic functions. <br />
Definition and examples. Mean value properties. Weak and strong maximum principles. Regularity and local estimates on derivatives. Harnack's inequality. Convergence and compactness results. <br />
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3. Laplace equation. <br />
Dirichlet problem for Laplace's equation. Green's representation formula. Green function and its properties. Poisson kernel for ball and halfspace. Subharmonic and superharmonic functions. Maximum and minimum principle. Perron-Wiener-Brelot solution. Barriers. Harmonic measures and Brelot theorem. <br />
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3. Maximum principle. <br />
Hopf maximum principle and applications. <br />
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4. Poisson equation.<br />
Dirichlet problem for Poisson's equation. Newtonian potential. Holder regularity of solutions. <br />
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5. Schauder theory. <br />
Schauder estimates. Existence of $C^{2,alpha}$ by the method of continuity. <br />

Full programme

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Bibliography

Lecture notes and material from : <br />
<br />
1) D. Gilbarg - N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin 1998; <br />
<br />
2) J. Jost, Partial differential equations, Springer-Verlag, New York 2002; <br />
<br />
3) Ya-Zhe Chen - Lan-Cheng Wu, Second order elliptic equations and elliptic systems, Translations of Mathematical Monographs vol. 174, American Mathematical Society, Providence RI 1998; <br />
<br />
4) Qing Han - Fanghua Lin, Elliptic partial differential equations, Courant Lecture Notes no.1, American Mathematical Society, Providence RI 1997.

Teaching methods

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Assessment methods and criteria

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Other information

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