Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. ·...
Transcript of Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. ·...
![Page 1: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/1.jpg)
Reaction-diffusion equations and minimal surfaces
Xavier Cabre
ICREA and UPC, Barcelona
Frontiers of Mathematics and Applications III - 2012
Summer Course UIMP 2012, Santander (Spain), August 13-17, 2012
Xavier Cabre Reaction-diffusion equations and minimal surfaces
![Page 2: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/2.jpg)
CONTENTS:
1 The Allen-Cahn equation and minimal surfaces
• Foliations and minimality
• The Simons cone
• Saddle-shaped solutions
=⇒ Weights from axial symmetry: sm−1tk−1ds dt
2 Antisymmetry for problems with weights
3 The explosion problem in axially symmetric domains
4 Isoperimetric inequalities with homogeneous weights
Xavier Cabre Reaction-diffusion equations and minimal surfaces
![Page 3: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/3.jpg)
![Page 4: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/4.jpg)
![Page 5: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/5.jpg)
![Page 6: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/6.jpg)
![Page 7: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/7.jpg)
![Page 8: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/8.jpg)
![Page 9: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/9.jpg)
![Page 10: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/10.jpg)
![Page 11: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/11.jpg)
![Page 12: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/12.jpg)
![Page 13: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/13.jpg)
![Page 14: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/14.jpg)
![Page 15: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/15.jpg)
![Page 16: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/16.jpg)
![Page 17: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/17.jpg)
![Page 18: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/18.jpg)
![Page 19: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/19.jpg)
![Page 20: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/20.jpg)
![Page 21: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/21.jpg)
![Page 22: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/22.jpg)
![Page 23: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/23.jpg)
![Page 24: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/24.jpg)
![Page 25: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/25.jpg)
![Page 26: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/26.jpg)
![Page 27: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/27.jpg)
![Page 28: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/28.jpg)
![Page 29: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/29.jpg)
![Page 30: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/30.jpg)
![Page 31: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/31.jpg)
![Page 32: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/32.jpg)
![Page 33: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/33.jpg)
2 Antisymmetry —or oddness— for problems with weights
minu(±L)=±m
∫ L
−L
{1
2u2 +
1
4(1− u2)2
}a(x)dx
Xavier Cabre Reaction-diffusion equations and minimal surfaces
![Page 34: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/34.jpg)
![Page 35: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/35.jpg)
![Page 36: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/36.jpg)
![Page 37: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/37.jpg)
![Page 38: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/38.jpg)
![Page 39: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/39.jpg)
![Page 40: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/40.jpg)
![Page 41: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/41.jpg)
![Page 42: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/42.jpg)
3 The explosion problem in axially symmetric domains
• Mean curvature of the level sets and
a geometric Sobolev inequality
• Regularity up to n ≤ 7 in axially symmetric domains
Xavier Cabre Reaction-diffusion equations and minimal surfaces
![Page 43: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/43.jpg)
![Page 44: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/44.jpg)
![Page 45: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/45.jpg)
![Page 46: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/46.jpg)
![Page 47: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/47.jpg)
![Page 48: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/48.jpg)
![Page 49: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/49.jpg)
4 Isoperimetric inequalities with homogeneous weights
• Monomial weights
• Homogeneous weights in cones
• Homogeneous weights in exterior domains
BASIC TOOL:
Cabre’s proof of the classical isoperimetric inequality; see:
Butl. Soc. Catalana Mat. 15 (2000) in CATALAN
Discrete Contin. Dyn. Syst. 20 (2008).
Xavier Cabre Reaction-diffusion equations and minimal surfaces
![Page 50: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/50.jpg)
![Page 51: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/51.jpg)
![Page 52: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/52.jpg)
![Page 53: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/53.jpg)
![Page 54: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/54.jpg)
![Page 55: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/55.jpg)
![Page 56: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/56.jpg)
![Page 57: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/57.jpg)
![Page 58: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/58.jpg)
![Page 59: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/59.jpg)
![Page 60: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/60.jpg)
![Page 61: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/61.jpg)
![Page 62: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/62.jpg)
![Page 63: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/63.jpg)
![Page 64: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/64.jpg)
![Page 65: Xavier Cabr e - UAMverso.mat.uam.es/.../uimp2012/courses/UIMP2012-cabre.pdf · 2017. 12. 1. · Xavier Cabr e ICREA and UPC, Barcelona Frontiers of Mathematics and Applications III](https://reader033.fdocuments.in/reader033/viewer/2022060823/609cb2b6de07325da66d4d95/html5/thumbnails/65.jpg)