Elastic deformations on the plane and approximations (lecture II) Aldo Pratelli Department of Mathematics, University of Pavia (Italy) “Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations: Analysis and Control”, Sissa, June 20–24 2011 A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 1 / 62
Plan of the course A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
Plan of the course • Lecture I: Mappings of finite distorsion and orientation-preserving homeomorphisms . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
Plan of the course • Lecture I: Mappings of finite distorsion and orientation-preserving homeomorphisms . • Lecture II: Approximation questions: hystory, strategies and results . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
Plan of the course • Lecture I: Mappings of finite distorsion and orientation-preserving homeomorphisms . • Lecture II: Approximation questions: hystory, strategies and results . • Lecture III: Smooth approximation of (countably) piecewise affine homeomorphisms . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
Plan of the course • Lecture I: Mappings of finite distorsion and orientation-preserving homeomorphisms . • Lecture II: Approximation questions: hystory, strategies and results . • Lecture III: Smooth approximation of (countably) piecewise affine homeomorphisms . • Lecture IV: The approximation result . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
Plan of the course • Lecture I: Mappings of finite distorsion and orientation-preserving homeomorphisms . • Lecture II: Approximation questions: hystory, strategies and results . • Lecture III: Smooth approximation of (countably) piecewise affine homeomorphisms . • Lecture IV: The approximation result . • Lecture V: Bi-Lipschits extension Theorem (part 1) . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
Plan of the course • Lecture I: Mappings of finite distorsion and orientation-preserving homeomorphisms . • Lecture II: Approximation questions: hystory, strategies and results . • Lecture III: Smooth approximation of (countably) piecewise affine homeomorphisms . • Lecture IV: The approximation result . • Lecture V: Bi-Lipschits extension Theorem (part 1) . • Lecture VI: Bi-Lipschits extension Theorem (part 2) . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 2 / 62
The problem of approximating A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence u ε : Ω → R 2 made by good functions with d ( u , u ε ) ≤ ε . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence u ε : Ω → R 2 made by good functions with d ( u , u ε ) ≤ ε . • What does good mean? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence u ε : Ω → R 2 made by good functions with d ( u , u ε ) ≤ ε . • What does good mean? • What is d ( · , · )? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence u ε : Ω → R 2 made by good functions with d ( u , u ε ) ≤ ε . • What does good mean? • What is d ( · , · )? • Ah, and of course. . . u ε must be orient. pres. homeomorphisms! A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence u ε : Ω → R 2 made by good functions with d ( u , u ε ) ≤ ε . • What does good mean? (smooth / piecewise affine) • What is d ( · , · )? • Ah, and of course. . . u ε must be orient. pres. homeomorphisms! A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
The problem of approximating Let u : Ω → ∆ be an orientation-preserving homeomorphism. GOAL: Find an approximating sequence u ε : Ω → R 2 made by good functions with d ( u , u ε ) ≤ ε . • What does good mean? (smooth / piecewise affine) • What is d ( · , · )? • Ah, and of course. . . u ε must be orient. pres. homeomorphisms! BAD NEWS: Convolution does not work! (unless u , u − 1 ∈ W 2 , ∞ ) (Example by Seregin and Shilkin) A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 3 / 62
A simple idea to approximate A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. Build the affine interpolation. A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d ∗ L ∞ ? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d ∗ L ∞ ? YES (trivial). A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d ∗ L ∞ ? YES (trivial). Is it an homeomorphism? Or, at least, is it orientation preserving? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d ∗ L ∞ ? YES (trivial). Is it an homeomorphism? Or, at least, is it orientation preserving? Maybe NOT. A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
A simple idea to approximate Take a triangulation of Ω. Build the affine interpolation. Is it a good approximation for d = d ∗ L ∞ ? YES (trivial). Is it an homeomorphism? Or, at least, is it orientation preserving? Maybe NOT. BAD NEWS: Even taking “randomly” arbitrarily many points does not work! A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 4 / 62
Good results A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62
Good results The strategy of last slide (with a careful choice of points) can be adjusted. A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62
Good results The strategy of last slide (with a careful choice of points) can be adjusted. Positive results by Bing, Connell, Kirby, Moise, counterexample by Donaldson and Sullivan. A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62
Good results The strategy of last slide (with a careful choice of points) can be adjusted. Positive results by Bing, Connell, Kirby, Moise, counterexample by Donaldson and Sullivan. All this works with the distance � u − 1 − v − 1 � � � � d ( u , v ) = d ∗ L ∞ ( u , v ) = � u − v L ∞ + L ∞ . � � A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 5 / 62
What would we really like? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62
What would we really like? If u is thought as a deformation, then the energy is something of the form A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62
What would we really like? If u is thought as a deformation, then the energy is something of the form � | Du | p + h � � W ( u ) = det Du , Ω with h diverging both at 0 and + ∞ . A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62
What would we really like? If u is thought as a deformation, then the energy is something of the form � | Du | p + h � � W ( u ) = det Du , Ω with h diverging both at 0 and + ∞ . • Why exploding at 0? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62
What would we really like? If u is thought as a deformation, then the energy is something of the form � | Du | p + h � � W ( u ) = det Du , Ω with h diverging both at 0 and + ∞ . • Why exploding at 0? • Why the determinant? A. Pratelli (Pavia) Homeomorphisms and approximations SISSA, June 20–24 2011 6 / 62
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