Topological invariants for discontinuous mappings Differential topology in Sobolev spaces Paweł Goldstein [email protected] Kolokwium MIM, 7 listopada 2019 TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 1/21 Two reasons: • natural: folding, breaking, changes of state; • technical: necessary because of the available mathematical tools. Introduction Why do we need • non-differentiable • discontinuous mappings in real-life applications? TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 2/21 Introduction Why do we need • non-differentiable • discontinuous mappings in real-life applications? Two reasons: • natural: folding, breaking, changes of state; • technical: necessary because of the available mathematical tools. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 2/21 Minima of accumulative, i.e, integral quantities ) we measure distance to minimizer with integral expressions. Introduction Variational principles • Fermat’s principle (Hero of Alexandria, Pierre de Fermat): Light travels through media along paths of shortest time. • Extremal action principle (Euler, Maupertuis): Bodies travel along paths locally minimizing the reduced action (integral of the momentum). Other examples: Dirichlet’s principle for harmonic maps, Einstein-Hilbert action in general relativity etc. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 3/21 Introduction Variational principles • Fermat’s principle (Hero of Alexandria, Pierre de Fermat): Light travels through media along paths of shortest time. • Extremal action principle (Euler, Maupertuis): Bodies travel along paths locally minimizing the reduced action (integral of the momentum). Other examples: Dirichlet’s principle for harmonic maps, Einstein-Hilbert action in general relativity etc. Minima of accumulative, i.e, integral quantities ) we measure distance to minimizer with integral expressions. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 3/21 What all maps? C1(Bn), with distance d(f , g) = jE(f ) − E(g)j? Problems: • d is not a metric (remedy: add kf − gkL2 to d(f , g)) • space is not complete ) existence of minima is not guaranteed. Introduction Harmonic maps and W1,2 Canonical example: harmonic maps A map f : Bn ! Rm is minimizing harmonic if it is a minimum of the Dirichlet energy Z E(f ) = jDf j2 Bn among all maps with the same boundary values as f . TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 4/21 C1(Bn), with distance d(f , g) = jE(f ) − E(g)j? Problems: • d is not a metric (remedy: add kf − gkL2 to d(f , g)) • space is not complete ) existence of minima is not guaranteed. Introduction Harmonic maps and W1,2 Canonical example: harmonic maps A map f : Bn ! Rm is minimizing harmonic if it is a minimum of the Dirichlet energy Z E(f ) = jDf j2 Bn among all maps with the same boundary values as f . What all maps? TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 4/21 What all maps? Problems: • d is not a metric (remedy: add kf − gkL2 to d(f , g)) • space is not complete ) existence of minima is not guaranteed. Introduction Harmonic maps and W1,2 Canonical example: harmonic maps A map f : Bn ! Rm is minimizing harmonic if it is a minimum of the Dirichlet energy Z E(f ) = jDf j2 Bn among all maps with the same boundary values as f . C1(Bn), with distance d(f , g) = jE(f ) − E(g)j? TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 4/21 What all maps? Introduction Harmonic maps and W1,2 Canonical example: harmonic maps A map f : Bn ! Rm is minimizing harmonic if it is a minimum of the Dirichlet energy Z E(f ) = jDf j2 Bn among all maps with the same boundary values as f . C1(Bn), with distance d(f , g) = jE(f ) − E(g)j? Problems: • d is not a metric (remedy: add kf − gkL2 to d(f , g)) • space is not complete ) existence of minima is not guaranteed. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 4/21 But if n ≥ 2, then this space contains discontinuous functions: 1 log(log(e + )) 2 W1,2(Bn). jxj Introduction Harmonic functions and W1,2 Right choice: Sobolev space W1,2(Bn) – the completion of C1 (or C¥) maps, in the norm q kf kW1,2 = kf kL2 + E(f ) TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 5/21 Introduction Harmonic functions and W1,2 Right choice: Sobolev space W1,2(Bn) – the completion of C1 (or C¥) maps, in the norm q kf kW1,2 = kf kL2 + E(f ) But if n ≥ 2, then this space contains discontinuous functions: 1 log(log(e + )) 2 W1,2(Bn). jxj TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 5/21 Definition (Sobolev space) W1,p(U) = ff 2 Lp(U) : Dif 2 Lp(U) for i = 1, . , ng If p ≤ n = dim U and n ≥ 2, then W1,p(U) contains discontinuous functions. Introduction Weak derivative and Sobolev spaces Definition 1 A function g 2 Lloc is the weak derivative in the i-th direction of 1 f 2 Lloc(U), if Z ¶ Z = − 2 ¥( ) f i f g f for any f Co U . U ¶x U = i 2 1( ) i = ¶ We write g D f . If f C U , then D f ¶xi f . TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 6/21 Introduction Weak derivative and Sobolev spaces Definition 1 A function g 2 Lloc is the weak derivative in the i-th direction of 1 f 2 Lloc(U), if Z ¶ Z = − 2 ¥( ) f i f g f for any f Co U . U ¶x U = i 2 1( ) i = ¶ We write g D f . If f C U , then D f ¶xi f . Definition (Sobolev space) W1,p(U) = ff 2 Lp(U) : Dif 2 Lp(U) for i = 1, . , ng If p ≤ n = dim U and n ≥ 2, then W1,p(U) contains discontinuous functions. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 6/21 • nice embedding properties, ∗ Banach spaces with uniformly convex norm ) reflexive ) weak compactness of bounded subsets. ∗ means that sufficiently nice functionals attain their minima (minimizing sequences have weakly convergent subsequences). Of interest are also Orlicz-Sobolev functions W1,P(U) = ff 2 L1(U) : P(jDif j) 2 L1(U) for i = 1, . , ng, e.g., for P(t) = tn/ log(e + t) and p < n, W1,p ⊃ W1,P ⊃ W1,n. Introduction Sobolev and Orlicz-Sobolev spaces Advantages of Sobolev spaces: • smooth functions are dense, TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 7/21 ∗ Banach spaces with uniformly convex norm ) reflexive ) weak compactness of bounded subsets. ∗ means that sufficiently nice functionals attain their minima (minimizing sequences have weakly convergent subsequences). Of interest are also Orlicz-Sobolev functions W1,P(U) = ff 2 L1(U) : P(jDif j) 2 L1(U) for i = 1, . , ng, e.g., for P(t) = tn/ log(e + t) and p < n, W1,p ⊃ W1,P ⊃ W1,n. Introduction Sobolev and Orlicz-Sobolev spaces Advantages of Sobolev spaces: • smooth functions are dense, • nice embedding properties, TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 7/21 Of interest are also Orlicz-Sobolev functions W1,P(U) = ff 2 L1(U) : P(jDif j) 2 L1(U) for i = 1, . , ng, e.g., for P(t) = tn/ log(e + t) and p < n, W1,p ⊃ W1,P ⊃ W1,n. Introduction Sobolev and Orlicz-Sobolev spaces Advantages of Sobolev spaces: • smooth functions are dense, • nice embedding properties, ∗ Banach spaces with uniformly convex norm ) reflexive ) weak compactness of bounded subsets. ∗ means that sufficiently nice functionals attain their minima (minimizing sequences have weakly convergent subsequences). TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 7/21 Introduction Sobolev and Orlicz-Sobolev spaces Advantages of Sobolev spaces: • smooth functions are dense, • nice embedding properties, ∗ Banach spaces with uniformly convex norm ) reflexive ) weak compactness of bounded subsets. ∗ means that sufficiently nice functionals attain their minima (minimizing sequences have weakly convergent subsequences). Of interest are also Orlicz-Sobolev functions W1,P(U) = ff 2 L1(U) : P(jDif j) 2 L1(U) for i = 1, . , ng, e.g., for P(t) = tn/ log(e + t) and p < n, W1,p ⊃ W1,P ⊃ W1,n. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 7/21 local orientation: If F: W ⊂ Rn ! F(W) ⊂ Rn is C1 and JF(x) = det DF(x) > 0, then F preserves local orientation at x. Introduction Topology encoded in the derivative invertibility: Assume X, Y are Banach, F: X ! Y is C1. If DF(x) : X ! Y is a linear isomorphism (1) then F is a local homeomorphism at x. If (1) holds for all x 2 X, then F is an open mapping. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 8/21 Introduction Topology encoded in the derivative invertibility: Assume X, Y are Banach, F: X ! Y is C1. If DF(x) : X ! Y is a linear isomorphism (1) then F is a local homeomorphism at x. If (1) holds for all x 2 X, then F is an open mapping. local orientation: If F: W ⊂ Rn ! F(W) ⊂ Rn is C1 and JF(x) = det DF(x) > 0, then F preserves local orientation at x. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 8/21 If F: M ! N is C1, then 1 Z deg F = JF(x) dx vol(N) M Other topological invariants (Hopf index, characteristic classes) of a mapping can be determined if the derivative is known. Introduction Topology encoded in the derivative Brouwer degree: Let M, N be closed manifolds of the same dimension. TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 9/21 Other topological invariants (Hopf index, characteristic classes) of a mapping can be determined if the derivative is known. Introduction Topology encoded in the derivative Brouwer degree: Let M, N be closed manifolds of the same dimension. If F: M ! N is C1, then 1 Z deg F = JF(x) dx vol(N) M TOPOLOGICAL INVARIANTS FOR DISCONTINUOUS MAPPINGS 9/21 Introduction Topology encoded in the derivative Brouwer degree: Let M, N be closed manifolds of the same dimension. If F: M ! N is C1, then 1 Z deg F = JF(x) dx vol(N) M Other topological invariants (Hopf index, characteristic classes) of a mapping can be determined if the derivative is known.