Bifurcations of extremals of Fredholm functionals.

*(Russian, English)*Zbl 1108.47055
Sovrem. Mat., Fundam. Napravl. 12, 3-140 (2004); translation in J. Math. Sci., New York 145, No. 6, 5311-5454 (2007).

The present monograph is a contemporary survey of works on variational methods in bifurcation theory. The subject-matter treated is very useful and with rich applications. In a wider sense, the research direction which is traditional for the Voronezh mathematical school is the analysis of smooth functionals on smooth Banach manifolds with applications to boundary value problems of mathematical physics.

After auxiliary material on Morse theory and singularity theory of smooth mappings in Ch. 1, the authors present the reduction scheme of A. Poincaré created for finite-dimensional variational problems with the aim of reduction of their dimensions. Its generalization, based on the change of linear fibering \(p:\mathbb{R}^n \to M\) by smooth submersion, allows to formulate a general reduction principle similar to the Morse–Bott reduction theorem. Then, in 1.6, some finite-dimensional examples of reduction are given, such as the function of height on the vertically disposed two-dimensional torus, the Legendre transformation realizing the relation between Lagrangian and Hamiltonian formalisms, postcritical equilibria of \((n+1)\)-linked elastic chain of rigid rods of unit lengths joined by elastic hinges, small reduction in the bifurcations problem of multidimensional gyroscopes, stationary relations. Further bifurcations of extremals are studied from minimum points with singularities of multidimensional folds.

In Ch. 2, after general information about regular extremals of Fredholm functionals, Morse lemma and regular extremals stability, the notions of key functions for Ljapunov–Schmidt and Morse–Bott nonlocal schemes are introduced, which are included then in the general reducing scheme with relevant key function. A finite-dimensional reduction of a smooth functional \(V\) on an open subset \(O\) of a smooth Banach manifold \(M\) is a triple \(\{p,\varphi, N \}\), where \(N\) is a smooth finite-dimensional manifold, \(p\) is a smooth submersion from \(O\) on \(N\), \(\varphi\) is a smooth mapping from \(N\) in \(O\) with the property \(p\cdot \varphi=I\), under the following conditions: \(\varphi(\xi)\) is unique critical point of the restriction \(V|_{p^{-1}(\xi)}\) for \(\forall \xi \in N\) and the second differential \(\frac{\partial^2}{\partial x^2}\left(V|_{p^{-1}(\xi)}\right)(\varphi(\xi))(h,h)\), \(h\in T_{\varphi(\xi)}(p^{-1}(\xi))\) is the nongenerate square form.

The function \(W(\xi):=V(\varphi(\xi))\) is the key function. Then the marginal mapping \(\varphi\) determines the one-to-one correspondence between the sets of critical points of the key function \(W(\xi)\) and the functional \(V\) which are simultaneously generated or not. When \(\varphi(\xi)\) is the point of global minimum of \(V|_{p^{-1}(\xi)}\) for all \(\xi \in N\), i.e., the reduction is elliptic, the corresponding critical points have the same Morse index. Extensions of reduction schemes are discussed. Further, their application to the natural mechanical system-differential equation \(\ddot{x}+Ax-\nabla \omega(x)=0\), \(x\in \mathbb{R}^n\) with boundary conditions \(x(0)=x(1)=0\) is given. Integrable reducing schemes are determined.

Ch. 3 is devoted to topological comparison of reducing schemes where the central problem is the comparison of smooth equivalence types for the relevant key functions. The results presented here are established on the base of the test for smooth equivalence functions, using R. Thom’s homotopic method. The main condition of smooth equivalence here is the possibility of smooth deformation of reducing schemes in one to another. At the global comparison, the condition of coercivity conservation at the deformation is used. A counterexample is given that shows the essence of homotopy condition of reducing schemes for the global comparison of their key functions. A modification of the theorem about key functions global (for the brevity of presentation) equivalence under continuous group symmetry conditions is obtained with the aid of the standard averaging approach for compact Lie groups. As an example, the generalized Duffing equation for the system of linked oscillators \(\ddot{x}+\lambda x-\nabla P(x)=0\), \(x\in \mathbb{R}^n\), \(x(0)=a\), \(x(1)=b\), is considered, where \(P(x)=\sum_{i,j=1}^na_{i,j}x^2_ix^2_j\), \(a_{i,j}=a_{j,i}\), is a quartic form with parallelepipedal symmetry.

Ch. 4 is devoted to reducing schemes on Banach manifolds. After auxiliary material on Banach manifolds with Riemannian metric, the existence theorem of a unique critical point of absolute minimal value on the closure of a geodetically convex domain \(\mathcal{O}\in M\) for the index zero Fredholmian family \(V_t\) functionals is proved under the conditions of its geodesic convexity on \(\mathcal{O}\) for all \(t\), propriety of the first codifferential family of mappings \(\nabla V_t(\cdot)\) on \(\overline{\mathcal{O}}\times [0,1]\) and \(\nabla V_t(x)\neq 0\;\forall (x,t)\in \partial \mathcal{O}\times [0,1]\). As an example is given the functional of the action on the manifold of spherical loops. Then, in Ch. 4, the reducing submersions on Banach manifolds are studied with application to elastic equilibrium forms of a circular plate uniformly compressed on the edge (along normals) described by the von Karman system and to equilibrium configurations of the linear lengthwise compressed Kirchhoff rod with rigid fastening of edges. Further \(G\)-invariant functionals on Banach \(G\)-manifolds are investigated, where, after the passage to the restriction of the functional on local transversal to orbit equivariant, analogs of Morse properties and finite-dimensionality are obtained. Here, the behavior of critical orbits under perturbations is studied. By the Raleygh–Schrödinger perturbation theory of linear operators, pencils, caustics geometry and their parametrizations are studied.

In Ch. 5, based on the analysis of smooth functions behaviour near critical points lying on the manifold boundary, the investigation of boundary and corner extremals is made. Classification of types of arising singularities and a description of relevant distributions of bifurcating Morse extremals are realized based on the reduction to key functions of several variables. The coincidence of symmetry and boundary singularities leads to new bifurcation effects which are important in applications to elasticity theory and crystals theory.

The final Ch. 6 contains many applications. These are Morse–Bott reduction for the Kirchhoff rod, bifurcations of equilibrium forms of an elastic plate lengthwise compressed and hingely fastened on its edge \(\Omega_a=[0,1]\times [0,1]\), to phase transitions in segnetoelectric crystals, to variational boundary value problems for ODE of fourth order and the two-modal bifurcation of elastic beam wave motions with 1:2 resonance.

The reader may take note of the interesting article on the same subject as the reviewed monograph, given by A. V. Trenogin and N. A. Sidorov [“Potentiality conditions of branching equation and bifurcation points of nonlinear operators”, Uzbek Math. J. No. 2, 40–49 (1992; MR 95c:47074)].

After auxiliary material on Morse theory and singularity theory of smooth mappings in Ch. 1, the authors present the reduction scheme of A. Poincaré created for finite-dimensional variational problems with the aim of reduction of their dimensions. Its generalization, based on the change of linear fibering \(p:\mathbb{R}^n \to M\) by smooth submersion, allows to formulate a general reduction principle similar to the Morse–Bott reduction theorem. Then, in 1.6, some finite-dimensional examples of reduction are given, such as the function of height on the vertically disposed two-dimensional torus, the Legendre transformation realizing the relation between Lagrangian and Hamiltonian formalisms, postcritical equilibria of \((n+1)\)-linked elastic chain of rigid rods of unit lengths joined by elastic hinges, small reduction in the bifurcations problem of multidimensional gyroscopes, stationary relations. Further bifurcations of extremals are studied from minimum points with singularities of multidimensional folds.

In Ch. 2, after general information about regular extremals of Fredholm functionals, Morse lemma and regular extremals stability, the notions of key functions for Ljapunov–Schmidt and Morse–Bott nonlocal schemes are introduced, which are included then in the general reducing scheme with relevant key function. A finite-dimensional reduction of a smooth functional \(V\) on an open subset \(O\) of a smooth Banach manifold \(M\) is a triple \(\{p,\varphi, N \}\), where \(N\) is a smooth finite-dimensional manifold, \(p\) is a smooth submersion from \(O\) on \(N\), \(\varphi\) is a smooth mapping from \(N\) in \(O\) with the property \(p\cdot \varphi=I\), under the following conditions: \(\varphi(\xi)\) is unique critical point of the restriction \(V|_{p^{-1}(\xi)}\) for \(\forall \xi \in N\) and the second differential \(\frac{\partial^2}{\partial x^2}\left(V|_{p^{-1}(\xi)}\right)(\varphi(\xi))(h,h)\), \(h\in T_{\varphi(\xi)}(p^{-1}(\xi))\) is the nongenerate square form.

The function \(W(\xi):=V(\varphi(\xi))\) is the key function. Then the marginal mapping \(\varphi\) determines the one-to-one correspondence between the sets of critical points of the key function \(W(\xi)\) and the functional \(V\) which are simultaneously generated or not. When \(\varphi(\xi)\) is the point of global minimum of \(V|_{p^{-1}(\xi)}\) for all \(\xi \in N\), i.e., the reduction is elliptic, the corresponding critical points have the same Morse index. Extensions of reduction schemes are discussed. Further, their application to the natural mechanical system-differential equation \(\ddot{x}+Ax-\nabla \omega(x)=0\), \(x\in \mathbb{R}^n\) with boundary conditions \(x(0)=x(1)=0\) is given. Integrable reducing schemes are determined.

Ch. 3 is devoted to topological comparison of reducing schemes where the central problem is the comparison of smooth equivalence types for the relevant key functions. The results presented here are established on the base of the test for smooth equivalence functions, using R. Thom’s homotopic method. The main condition of smooth equivalence here is the possibility of smooth deformation of reducing schemes in one to another. At the global comparison, the condition of coercivity conservation at the deformation is used. A counterexample is given that shows the essence of homotopy condition of reducing schemes for the global comparison of their key functions. A modification of the theorem about key functions global (for the brevity of presentation) equivalence under continuous group symmetry conditions is obtained with the aid of the standard averaging approach for compact Lie groups. As an example, the generalized Duffing equation for the system of linked oscillators \(\ddot{x}+\lambda x-\nabla P(x)=0\), \(x\in \mathbb{R}^n\), \(x(0)=a\), \(x(1)=b\), is considered, where \(P(x)=\sum_{i,j=1}^na_{i,j}x^2_ix^2_j\), \(a_{i,j}=a_{j,i}\), is a quartic form with parallelepipedal symmetry.

Ch. 4 is devoted to reducing schemes on Banach manifolds. After auxiliary material on Banach manifolds with Riemannian metric, the existence theorem of a unique critical point of absolute minimal value on the closure of a geodetically convex domain \(\mathcal{O}\in M\) for the index zero Fredholmian family \(V_t\) functionals is proved under the conditions of its geodesic convexity on \(\mathcal{O}\) for all \(t\), propriety of the first codifferential family of mappings \(\nabla V_t(\cdot)\) on \(\overline{\mathcal{O}}\times [0,1]\) and \(\nabla V_t(x)\neq 0\;\forall (x,t)\in \partial \mathcal{O}\times [0,1]\). As an example is given the functional of the action on the manifold of spherical loops. Then, in Ch. 4, the reducing submersions on Banach manifolds are studied with application to elastic equilibrium forms of a circular plate uniformly compressed on the edge (along normals) described by the von Karman system and to equilibrium configurations of the linear lengthwise compressed Kirchhoff rod with rigid fastening of edges. Further \(G\)-invariant functionals on Banach \(G\)-manifolds are investigated, where, after the passage to the restriction of the functional on local transversal to orbit equivariant, analogs of Morse properties and finite-dimensionality are obtained. Here, the behavior of critical orbits under perturbations is studied. By the Raleygh–Schrödinger perturbation theory of linear operators, pencils, caustics geometry and their parametrizations are studied.

In Ch. 5, based on the analysis of smooth functions behaviour near critical points lying on the manifold boundary, the investigation of boundary and corner extremals is made. Classification of types of arising singularities and a description of relevant distributions of bifurcating Morse extremals are realized based on the reduction to key functions of several variables. The coincidence of symmetry and boundary singularities leads to new bifurcation effects which are important in applications to elasticity theory and crystals theory.

The final Ch. 6 contains many applications. These are Morse–Bott reduction for the Kirchhoff rod, bifurcations of equilibrium forms of an elastic plate lengthwise compressed and hingely fastened on its edge \(\Omega_a=[0,1]\times [0,1]\), to phase transitions in segnetoelectric crystals, to variational boundary value problems for ODE of fourth order and the two-modal bifurcation of elastic beam wave motions with 1:2 resonance.

The reader may take note of the interesting article on the same subject as the reviewed monograph, given by A. V. Trenogin and N. A. Sidorov [“Potentiality conditions of branching equation and bifurcation points of nonlinear operators”, Uzbek Math. J. No. 2, 40–49 (1992; MR 95c:47074)].

Reviewer: Irina V. Konopleva (Ul’yanovsk)

##### MSC:

47J15 | Abstract bifurcation theory involving nonlinear operators |

58E07 | Variational problems in abstract bifurcation theory in infinite-dimensional spaces |

58E09 | Group-invariant bifurcation theory in infinite-dimensional spaces |

58K35 | Catastrophe theory |

47J05 | Equations involving nonlinear operators (general) |

47N20 | Applications of operator theory to differential and integral equations |

58B05 | Homotopy and topological questions for infinite-dimensional manifolds |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

74Kxx | Thin bodies, structures |