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Nonconforming FEMs avoid the strong restrictions of conforming FEMs. So discontinuous ansatz and test functions, approximate test integrals, and strong forms are admitted. This allows the proof of convergence for the full spectrum of linear to fully nonlinear equations and systems of orders 2 and 2m. General fully nonlinear problems only allow strong forms and enforce new techniques and C¹ FEs. Variational crimes for FEs violating regularity and boundary conditions are studied in ℝ² for linear and quasilinear problems. Essential tools are the anticrime transformations. The relations between the strong and weak form of the equations allow the usually technical proofs for consistency. Due to the dominant role of FEMs, numerical solutions for five classes of problems are only presented for FEMs. Most remain valid for the other methods as well: vari-ational methods for eigenvalue problems, convergence theory for monotone operators (quasilinear problems), FEMs for fully nonlinear elliptic problems and for nonlinear boundary conditions, and quadrature approximate FEMs. We thus close several gaps in the literature.

This chapter presents the basic functional analysis and abstract error estimates used in the book. A summary of the relevant theory of linear variational problems, compactness, and the Fredholm alternative is presented. This is followed by a more detailed discussion, with proofs, of the corresponding error estimates including Cea’s lemma, Babuska-Brezzi theory for mixed problems, and convergence theory for collectively compact operators. The Hilbert-Schmidt theory of eigenvalues and error estimates for eigenvalues are also briefly mentioned.

This chapter formulates and proves the jump mechanism. It constructs a variational problem which proves forcing equivalence for the original Hamiltonian using Definition 6.18. It first constructs a special variational problem for the slow mechanical system. A solution of this variational problem is an orbit “jumping” from one homology class to the other. The chapter then modifies this variational problem for the fast time-periodic perturbation of the slow mechanical system. This is achieved by applying the perturbative results established in Chapter 7. Recall the original Hamiltonian system near a double resonance can be brought to a normal form and this normal form, in turn, is related to the perturbed slow system through coordinate change and energy reduction. The variational problem for the perturbed slow system can then be converted to a variational problem for the original.

This chapter presents the basic functional analysis and abstract error estimates used in the book. A summary of the relevant theory of linear variational problems,compactness, and the Fredholm alternative is presented. This is followed by a more detailed discussion, with proofs, of the corresponding error estimates including Cea’s lemma, Babuska-Brezzi theory for mixed problems, and convergence theory for collectively compact operators. The Hilbert-Schmidt theory of eigenvalues and error estimates for eigenvalues are also briefly mentioned.

The variational point of view on exceptional structures in dimensions 6, 7 and 8 is one of Nigel Hitchin’s seminal contributions. One feature of this point of view is that it motivates the study of boundary value problems, for structures with prescribed data on a boundary. This chapter considers the case of 7 dimensions and G₂ structures. It briefly reviews a general framework and then goes on to examine in more detail symmetry reductions to dimensions 4 and 3. In the latter case, the chapter presents an interesting variational problem related to the real Monge–Ampère equation and describes a generalization of this.

This chapter discusses a variational formulation of boundary value problems in small deformation solid mechanics. It begins by introducing the important principle of virtual power, and shows that it encapsulates Cauchy’s traction law, and the local form of the basic balance of forces (equation of equilibrium), and the local from of the balance of moments (symmetry of the stress). Since the principle of virtual power encapsulates both the equation of equilibrium and the Cauchy relation for tractions, it can be used to formulate and solve boundary-value problems in solid mechanics in a variational or weak sense. Specifically, it is shown how the displacement problem of linear elastostatics may be formulated variationally using the principle of virtual power.

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. It depends on the parameter b > 0. The behavior of the minimum values as well as of the minimizing trajectories is studied when b → 0 and when b → +∞. Finally, certain heat kernels are computed in terms of the minimum value of the action. The aforementioned variational problem has already been considered previously as a warm up to the more general problem on Riemannian manifolds, in order to study in detail the small time asymptotics of the hypoelliptic heat kernel.