Learning objectives
The course gives an overview of Sobolev spaces with applications to the study of elliptic, second order, linear partial differential equations both in L^p space (p in (1,+oo)) and in spaces of bounded and Hoelder continuous functions.
Prerequisites
Calculus and advanced calculus. Linear algebra. Topology. Measure theory and integration.
Linear functional analysis.
Course unit content
An overview on the L^p spaces.
weak derivatives and the Sobolev spaces W^{k,p}.
Some characterization of the Sobolev spaces W^{k,p}. The trace of the a function in a Sobolev space. Variational formulation of boundary value elliptic problems. Lax and Milgram lemma. Regularizing weak solutions. Compact operators. The Riesz-Thorin's and Marcinkiewicz interpolation theorems. Riesz potentials.. Continuity method. Boundary value problems for elliptic equations in L^p-spaces and in spaces of bounded and hoelder continuous functions.
Full programme
An overview on the L^p spaces.
weak derivatives and the Sobolev spaces W^{k,p}.
Some characterization of the Sobolev spaces W^{k,p}. The trace of the a function in a Sobolev space. Variational formulation of boundary value elliptic problems. Lax and Milgram lemma. Regularizing weak solutions. Compact operators. The Riesz-Thorin's and Marcinkiewicz interpolation theorems. Riesz potentials.. Continuity method. Boundary value problems for elliptic equations in L^p-spaces and in spaces of bounded and hoelder continuous functions.
Bibliography
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer Verlag 2011.
L.C. Evans, Partial differential equations, 2nd Edition, American Mathematical Society 2010.
D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, 2nd Edition, Springer Verlag 1983.
Notes by the teacher.
Teaching methods
Lectures
Assessment methods and criteria
The exam consists of an oral part which is aimed at evaluating the knowledge of the abstract results seen during the course, their proofs and the skills in using such results.
Other information
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