Abundance of cusps in semi-linear operators

Transcription

1 Abundance of cusps in semi-linear operators Carlos Tomei, PUC-Rio with Marta Calanchi (U. Milano) and André Zaccur (PUC-Rio) RISM, Varese, September 2015

2 A simple example This will be essentially proved: other situations will be just quoted. 1/5

3 A simple example Ω R n is open, bounded, smooth, connected. This will be essentially proved: other situations will be just quoted. 1/5

4 A simple example Ω R n is open, bounded, smooth, connected. Take standard boundary conditions (SBC) for b : H 2 b H0. (Dirichlet, Neumann, periodic, Ω compact manifold, are SBC) This will be essentially proved: other situations will be just quoted. 1/5

5 A simple example Ω R n is open, bounded, smooth, connected. Take standard boundary conditions (SBC) for b : H 2 b H0. (Dirichlet, Neumann, periodic, Ω compact manifold, are SBC) We require simplicity of the k-th eigenvalue µ k,b of b. This will be essentially proved: other situations will be just quoted. 1/5

6 A simple example Ω R n is open, bounded, smooth, connected. Take standard boundary conditions (SBC) for b : H 2 b H0. (Dirichlet, Neumann, periodic, Ω compact manifold, are SBC) We require simplicity of the k-th eigenvalue µ k,b of b. The smooth nonlinearities f : R R satisfy f is Morse; f (c) = 0, c Hb 2 = c / σ( b). This will be essentially proved: other situations will be just quoted. 1/5

7 A simple example Ω R n is open, bounded, smooth, connected. Take standard boundary conditions (SBC) for b : H 2 b H0. (Dirichlet, Neumann, periodic, Ω compact manifold, are SBC) We require simplicity of the k-th eigenvalue µ k,b of b. The smooth nonlinearities f : R R satisfy f is Morse; f (c) = 0, c Hb 2 = c / σ( b). x l, x r with f (x l ) = f (x r ) = µ k,b, f (x l )f (x r ) < 0. This will be essentially proved: other situations will be just quoted. 1/5

8 A simple example Ω R n is open, bounded, smooth, connected. Take standard boundary conditions (SBC) for b : H 2 b H0. (Dirichlet, Neumann, periodic, Ω compact manifold, are SBC) We require simplicity of the k-th eigenvalue µ k,b of b. The smooth nonlinearities f : R R satisfy f is Morse; f (c) = 0, c Hb 2 = c / σ( b). x l, x r with f (x l ) = f (x r ) = µ k,b, f (x l )f (x r ) < 0. Theorem (Calanchi,T., Zaccur) There is g C 0,α for which the equation F(u) = b u f (u) = g has (at least) three solutions. (more later) This will be essentially proved: other situations will be just quoted. 1/5

9 The context We think of differential operators in geometric terms. Good for the eyes, for numerics. 2/5

10 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. Good for the eyes, for numerics. 2/5

11 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. (Ambrosetti-Prodi) f : R R convex, ran f σ( b ) = {µ 1,b }. Then F : H 2 b H0 is a global fold. Good for the eyes, for numerics. 2/5

12 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. (Ambrosetti-Prodi) f : R R convex, ran f σ( b ) = {µ 1,b }. Then F : Hb 2 H0 is a global fold. Same for F : C 2,α b C 0,α, where C 2,α b = Hb 2 C2,α. Good for the eyes, for numerics. 2/5

13 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. (Ambrosetti-Prodi) f : R R convex, ran f σ( b ) = {µ 1,b }. Then F : Hb 2 H0 is a global fold. Same for F : C 2,α b C 0,α, where C 2,α b = Hb 2 C2,α. (Cafagna-Donati, Malta-Saldanha-T.) For f (x) = x 2k+1 x, F (u) = D p u f (u) is a global cusp. Good for the eyes, for numerics. 2/5

14 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. (Ambrosetti-Prodi) f : R R convex, ran f σ( b ) = {µ 1,b }. Then F : Hb 2 H0 is a global fold. Same for F : C 2,α b C 0,α, where C 2,α b = Hb 2 C2,α. (Cafagna-Donati, Malta-Saldanha-T.) For f (x) = x 2k+1 x, F (u) = D p u f (u) is a global cusp. (Ruf) For 0 < λ < µ 2,N, F (u) = N u + u 3 λu is tantalizingly close to a cusp. Good for the eyes, for numerics. 2/5

15 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. (Ambrosetti-Prodi) f : R R convex, ran f σ( b ) = {µ 1,b }. Then F : Hb 2 H0 is a global fold. Same for F : C 2,α b C 0,α, where C 2,α b = Hb 2 C2,α. (Cafagna-Donati, Malta-Saldanha-T.) For f (x) = x 2k+1 x, F (u) = D p u f (u) is a global cusp. (Ruf) For 0 < λ < µ 2,N, F (u) = N u + u 3 λu is tantalizingly close to a cusp. (Burghelea-Saldanha-T.) The critical set of F(u) = D 2 D u f (u) is a union of hyperplanes, one for each eigenvalue in ran f. Good for the eyes, for numerics. 2/5

16 The context We think of differential operators in geometric terms. (Dolph-Hammerstein) Let L : X Y Y self-adjoint, c = min σ(l) > 0 and f : Y Y (c ɛ)-lipschitz. Then L + f : Y Y is a Lipschitz homeomorhism. (Ambrosetti-Prodi) f : R R convex, ran f σ( b ) = {µ 1,b }. Then F : Hb 2 H0 is a global fold. Same for F : C 2,α b C 0,α, where C 2,α b = Hb 2 C2,α. (Cafagna-Donati, Malta-Saldanha-T.) For f (x) = x 2k+1 x, F (u) = D p u f (u) is a global cusp. (Ruf) For 0 < λ < µ 2,N, F (u) = N u + u 3 λu is tantalizingly close to a cusp. (Burghelea-Saldanha-T.) The critical set of F(u) = D 2 D u f (u) is a union of hyperplanes, one for each eigenvalue in ran f. (Calanchi-Saldanha-T.) {q C 0,α λ 1 ( b q) = 0} is a hyperplane. Good for the eyes, for numerics. 2/5

17 The basic tools, some questions Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

18 The basic tools, some questions Singularity theory obtains local normal forms of functions. Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

19 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

20 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Said differently, we only handle Morin singularities: folds, cusp, swallowtails... Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

21 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Said differently, we only handle Morin singularities: folds, cusp, swallowtails... Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

22 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Said differently, we only handle Morin singularities: folds, cusp, swallowtails... One may classify a singular point of F, Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

23 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Said differently, we only handle Morin singularities: folds, cusp, swallowtails... One may classify a singular point of F, or find possible folds, cusps... Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

24 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Said differently, we only handle Morin singularities: folds, cusp, swallowtails... One may classify a singular point of F, or find possible folds, cusps... or, harder, show that F is a global fold, cusp... Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

25 The basic tools, some questions Singularity theory obtains local normal forms of functions. A lucky break: we only consider Jacobians with kernel of dimension at most 1, of Fredholm index 0. Said differently, we only handle Morin singularities: folds, cusp, swallowtails... One may classify a singular point of F, or find possible folds, cusps... or, harder, show that F is a global fold, cusp... Proposition (Morin, Church, Micheletti, Tarantello, Ruf...) A zero u c C 2,α b of Λ : C 2,α b C 0,α, u (λ k (u), δ k (u) = λ k (u), φ k (u)) for which λ k (u c ) 0 and ρ(u c ) = δ k (u c ), φ k (u c ) 0 is a cusp of F. Cusps and economics: Yves Balasko, The equilibrium manifold, MIT Press. 3/5

26 A cusp for F(u) = b u f (u) Liberté, egalité, Gateaux differentiabilité. 4/5

27 A cusp for F(u) = b u f (u) We search for a zero of Λ as a two valued u c first, u c = f (x l ) χ θ + f (x r ) χ c θ Liberté, egalité, Gateaux differentiabilité. 4/5

28 A cusp for F(u) = b u f (u) We search for a zero of Λ as a two valued u c first, u c = f (x l ) χ θ + f (x r ) χ c θ for which λ k (u c ) and δ k (u c ) are L.I., Liberté, egalité, Gateaux differentiabilité. 4/5

29 A cusp for F(u) = b u f (u) We search for a zero of Λ as a two valued u c first, u c = f (x l ) χ θ + f (x r ) χ c θ for which λ k (u c ) and δ k (u c ) are L.I., and then mollify. Liberté, egalité, Gateaux differentiabilité. 4/5

30 A cusp for F(u) = b u f (u) We search for a zero of Λ as a two valued u c first, u c = f (x l ) χ θ + f (x r ) χ c θ for which λ k (u c ) and δ k (u c ) are L.I., and then mollify. Any choice of θ obtains u c = µ 1,b, for which λ k (u c ) = 0. Liberté, egalité, Gateaux differentiabilité. 4/5

31 A cusp for F(u) = b u f (u) We search for a zero of Λ as a two valued u c first, u c = f (x l ) χ θ + f (x r ) χ c θ for which λ k (u c ) and δ k (u c ) are L.I., and then mollify. Any choice of θ obtains u c = µ 1,b, for which λ k (u c ) = 0. Now use the explicit formula δ k (u) = λ k (u), φ k (u)) = f (u)φ 3 k (u) to obtain some θ 0 yielding the requested u c. Ω Liberté, egalité, Gateaux differentiabilité. 4/5

32 A nonlinear trick and a normal form The trick might be more useful than the rest of the talk. 5/5

33 A nonlinear trick and a normal form Mollifying requires transversality: The trick might be more useful than the rest of the talk. 5/5

34 A nonlinear trick and a normal form Mollifying requires transversality: If ψ k,b (v) 0 at the vertex v of the sector, λ k (u c ) and δ k (u c ) are L.I. The trick might be more useful than the rest of the talk. 5/5

35 A nonlinear trick and a normal form Mollifying requires transversality: If ψ k,b (v) 0 at the vertex v of the sector, λ k (u c ) and δ k (u c ) are L.I. There is a zero of Λ in any dense subspace of H 2 b. The trick might be more useful than the rest of the talk. 5/5

36 A nonlinear trick and a normal form Mollifying requires transversality: If ψ k,b (v) 0 at the vertex v of the sector, λ k (u c ) and δ k (u c ) are L.I. There is a zero of Λ in any dense subspace of H 2 b. But... is it a cusp? A transversality condition is missing! The trick might be more useful than the rest of the talk. 5/5

37 A nonlinear trick and a normal form Mollifying requires transversality: If ψ k,b (v) 0 at the vertex v of the sector, λ k (u c ) and δ k (u c ) are L.I. There is a zero of Λ in any dense subspace of H 2 b. But... is it a cusp? A transversality condition is missing! If no zero of Λ is a cusp, terrible things happen. The trick might be more useful than the rest of the talk. 5/5

38 A nonlinear trick and a normal form Mollifying requires transversality: If ψ k,b (v) 0 at the vertex v of the sector, λ k (u c ) and δ k (u c ) are L.I. There is a zero of Λ in any dense subspace of H 2 b. But... is it a cusp? A transversality condition is missing! If no zero of Λ is a cusp, terrible things happen. What about non convex AP, as in Marta s lecture? Yes, there is a cusp there too, generically. The trick might be more useful than the rest of the talk. 5/5

39 Grazie!! Obrigado! 5

40 A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1972) D. Burghelea, N. Saldanha and C. Tomei, Results on infinite dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators, J. Differential Equations 188 (2003) V. Cafagna and G. Tarantello, Multiple solutions for some semilinear elliptic equations, Math. Ann. 276 (1987) M. Calanchi, C. Tomei and A. Zaccur, Abundance of cusps and a converse of the Ambrosetti Prodi theorem, arxiv file P.T. Church and J.G. Timourian, Global structure for nonlinear operators in differential and integral equations. II. Cusps., Topological nonlinear analysis, II PNDE, 27, Birkhäuser, Boston, MA, (1997) I. Malta, N. Saldanha and C. Tomei, Morin singularities and global geometry in a class of ordinary differential operators, TMNA 10 (1997), B. Ruf, Singularity theory and the geometry of a nonlinear elliptic equation, Ann. Scuola Norm. Sup. Pisa C1. Sc., (4), 17 (1990), 1-33, Singularity theory and bifurcation phenomena in differential equations, Topological Nonlinear Analysis, II, PNDE, 27, Birkhäuser, Boston, MA, (1997) B. Ruf, Forced secondary bifurcation in an elliptic boundary value problem, Diff. Int. Eqs. 5, 4, (1992) B. Ruf, Higher singularities and forced secondary bifurcation, SIAM J. Math.l Anal. 26, 5 (1995) Some references. 5