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EuroSummer School

Instructional Conference on
Nonlinear Partial Differential Equations

Edinburgh, 8-18 January 2001

Scientific Programme | Conference Arrangements | Application Forms | Conference Home Page

Scientific programme

On this page you will find a list of speakers and the timetable for the course. Lecture titles, some abstracts and background reading are listed by author below the timetable. Clicking on a speaker's name in the timetable will take you to the information relating to that speaker's lecture or series
Updated 18 December 2000

SPEAKERS

A Aftalion (ENS Paris) J Kristensen (Heriot Watt)
L Ambrosio (Pisa) C Le Bris (Ecole Nationale des Ponts et Chaussees)
J M Ball (Oxford) A Quarteroni (EPFL Lausanne)
Y Brenier (Paris VI) C A Stuart (EPFL Lausanne)
G R Burton (Bath) J F Toland (Bath)
V Caselles (Barcelona) N Touzi (Paris)
N Dancer (Sydney) N S Trudinger (Canberra)
M J Esteban (Paris Dauphine)

TIMETABLE

Instructional Lectures will be delivered at three levels: introductory (I); intermediate (M) and advanced (A). Single invited lectures dealing with the application of PDEs are delivered at 4.15.

The lectures will be held in the Cedar Suite.

N.B. NEW TIMETABLE
18th December 2000

  9.15-10.45 11.15-12.45 2.00-3.30 4.15-5.15
MONDAY 8 Registration Burton 1
(I)
Stuart 1
(I)
Burton 2
(I)
TUESDAY 9 Burton 3
(I)
Stuart 2
(I)
Burton 4
(I)
Stuart 3
(I)
WEDNESDAY 10 Kristensen 1
(M)
Esteban 1
(M)
Free afternoon
THURSDAY 11 Esteban 2
(M)
Aftalion 1
(I)
Le Bris 1
(A)
Touzi
FRIDAY 12 Le Bris 2
(A)
Kristensen 2
(M)
Aftalion 2
(I)
Quarteroni
SATURDAY 13 Ambrosio 1
(M)
Ambrosio 2
(M)
Free afternoon
MONDAY 15 Ball 1
(A)
Kristensen 3
(M)
Ball 2
(A)
Ambrosio 3
(A)
TUESDAY 16 Ambrosio 4
(A)
Ball 3
(A)
Free afternoon
WEDNESDAY 17 Brenier 1
(A)
Caselles 1
(A)
Dancer Toland
THURSDAY 18 Brenier2
(A)
Brenier 3
(A)
Caselles 2
(A)

A Aftalion (ENS Paris)
Maximum principles for elliptic and parabolic equations (2 lectures)

ABSTRACT
If u is a convex function in an interval I, then it is well known that u attains its maximum on the boundary of I. This property is a form of the Maximum Principle and can be generalized in dimension N for functions u with D u ³ 0 or more generally with Mu + c (x) u ³ 0 where M is an elliptic operator and c (x) £ 0. A form of the Maximum Principle is that u reaches its maximum on the boundary if it is not constant. Another form is that if Mu + c (x) u ³ 0 in the domain and u £ 0 on the boundary then u £ 0 in the whole domain. In these lectures, we will give a review of the various forms of the Maximum Principle (MP): weak MP, strong MP, Hopf MP. We will give sufficient conditions for the MP to hold for a general elliptic operator: a usual condition is c (x) £ 0 but we will see for instance that it also holds with any c (x) for narrow domains or domains with small volume. Then we will give applications of the Maximum Principle to derive uniqueness and symmetry properties for solutions of elliptic PDEs.

For parabolic PDE's, the Maximum Principle states that the maximum is reached either on the boundary of the domain or at time t = 0. We will give various versions of the Maximum Principle for parabolic equations and give applications to uniqueness results.
REFERENCES (not essential preparatory reading)
Books
A. Friedman (1964) Partial differential equations of parabolic type . Prentice-Hall, Inc
D.Gilbarg & N.Trudinger (1983) Elliptic partial differential equations of second order. (2nd edn) Grundlehren der Mathematischen Wissenschaften,244.Springer-Verlag
M.H.Protter & H.F.Weinberger (1967)Maximum principles in differential equations. Prentice-Hall, Inc
J.Smoller (1994)Shock waves and reaction-diffusion equations. Grundlehren der Mathematischen Wissenschaften,258 . Springer-Verlag.
Research Papers
H.Berestycki & L.Nirenberg (1991) On the method of moving planes and the sliding method. Bol. Soc. Bras. Mat. 22 , 1-37.
H.Berestycki, L.Nirenberg & S.R.S. Varadhan (1994) The principal eigenvalues and maximum principle for second order elliptic operators in general domains. Comm. Pure Appl. Math. 47 , 47-92.
B.Gidas, Wei-Ming Ni & L.Nirenberg (1979) Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209--243.
J.Serrin (1971) A symmetry theorem in potential theory. Arch. Rat. Mech. Anal. 43 , 304-318.
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L Ambrosio (Pisa)
Introduction to geometric measure theory (lectures 1 & 2)
Background reading:
K. J. Falconer (1995) The geometry of fractal sets . Cambridge University Press.
W. Rudin (1987) Real and complex analysis. McGraw-Hill
The Mumford-Shah functional (lectures 3 & 4)
Background reading:
L. Ambrosio, N. Fusco & D. Pallara (2000) Functions of bounded variation and free discontinuity problems. Oxford University Press.
J. M. Morel & S. Solimini (1994) Variational models in image segmentation. Birkhauser.
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J M Ball (Oxford)
Microstructure and energy minimization (3 lectures)

Many materials (e.g. alloys) undergo solid phase transformations involving a change of shape at some critical temperature. Such phase transformations result in patterns of fine microstructure, whose morphology is important for determining the macroscopic response of the material. A central model for such materials is nonlinear elasticity, with a stored-energy function that is 'non-elliptic'. The analysis of minimizers and minimizing sequences for the total elastic energy then provides key information about microstructure morphology. Crucial to this analysis are tools for passing from microscales to macroscales such a weak convergence, Young measures and quasiconvexity.
Useful background is provided by:
S Muller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems, Lecture notes in Maths 1713, (Springer-Verlag, Berlin), 85-210 1999.
P. Pedregal, Parametrized measures and variational principles, Progress in nonlinear differential equations and their applications 30 (Birkhauser, Basel), 1991.
P. Pedregal, Variational Methods in Nonlinear elasticity (SIAM, Philadelphia), 2000.
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Y Brenier (Paris VI)
Lect 1. Volume preserving maps and incompressible fluids I
Lect 2. Volume preserving maps and incompressible fluids II
Lect 3. Hydrodynamic limits of Plasma equations
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G R Burton (Bath)
Lecture 1: Nonlinear Functional Analysis
Weak convergence, weak compactness and convexity are important tools for proving existence in the Calculus of Variations. With this application in mind we will study weak continuity and semicontinuity in a nonlinear context.

Preparation should include familiarity with the basics of linear functional analysis, including Banach spaces, the Hahn-Banach theorem, reflexivity, weak and weak* topologies, and the (Banach-Alaoglu) theorem on weak* compactness of a ball, as covered in:
G.F. Simmons (1963) Introduction to Topology and Modern Analysis, ch. 9, pp, 211-34. McGraw-Hill, New York.
or
H. Brezis (1983) Analyse fonctionelle - Théorie et applications, ch. I, pp. 1-7 and ch III, pp. 33-43. Masson, Paris.
Lectures 2-4: Function Spaces
The lectures will revolve around the notion of a rearrangement of a function. In the context of variational problems posed on Sobolev spaces, various symmetrisation procedures allow the functions in a given minimising sequence to be rearranged into more symmetrical versions to form a minimising sequence with better compactness properties.

The set of all (generally unsymmetric) rearrangements of a given function is of interest in its own right, being a physically natural constraint set for some variational problems in fluid mechanics. Preparation should include familiarity with Lebesgue integration, the definitions of Sobolev spaces, and the statements of the Sobolev embedding theorem and the Rellich-Kondrachov compact embedding theorem, as covered in:
R.A. Adams (1970) Sobolev Spaces, pp. 12-27, 44-46, 65-67, 95-98, and 143-144 Academic Press, San Diego.
or
H. Brezis (1983) Analyse fonctionelle -Théorie et applications, pp. 54-62, 149-157, and 162-171. Masson, Paris.
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V Caselles (Barcelona)
PDEs in image processing - statics & dynamics (2 lectures)
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N Dancer (Sydney)
Peak solutions of nonlinear elliptic equations and their stability.
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M J Esteban (Paris Dauphine)
Variational methods in mathematical physics (2 lectures)

This course, which follows that of Professor C.A. Stuart, will be devoted to the presentation of the modern tools of non-compact variational problems. These problems appear quite often in the modelling of physical phenomena, and more particularly in quantum mechanics and atomic physics. Non-compact phenomena are also related to the consideration of the so-called critical exponents in equations coming from differential geometry.

The main method to be described in this course is the so-called concentration-compactness method, introduced by P.-L. Lions to deal with general non-compact situation in variational situations. Previous ways to treat this difficulty include the work of some geometers like Uhlenbeck, Schoen, etc. and also in PDEs including the critical exponents, the work of Brezis and Nirenberg.
Good preparation for this course are, by order of difficulty, the books of :
O. Kavian (1993) Introduction la theorie des points critiques et applications aux problèmes elliptiques . (French) [Introduction to critical point theory and applications to elliptic problems]. Mathématiques & Applications [Mathematics & Applications], 13 . Springer-Verlag, Paris.
M. Willem (1996) Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhauser.
M. Struwe (1996) Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. (2nd edn).Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 34. Springer-Verlag.
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J Kristensen (Heriot Watt)
Linear and multilinear algebra for nonlinear systems (3 lectures)
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C Le Bris (Ecole Nationale des Ponts et Chaussees)
PDEs in physics and chemistry - variational methods in atomic physics (2 lectures)
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A Quarteroni (EPFL Lausanne)
Mathematical models for blood dynamics
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C A Stuart (EPFL Lausanne)
The calculus of variations (3 lectures)
A .dvi file of the abstract is available to download here.
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J F Toland (Bath)
PDEs in wave theories
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N Touzi (Paris)
Mathematical modelling of the hedging problem in finance
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