Singular perturbations and first order PDE on manifolds

Singular perturbations and first order PDE on manifolds

In this Note we present some results concerning the concentration of sequences of first eigenfunctions on the limit sets of a Morse–Smale dynamical system on a compact Riemannian manifold. More precisely a renormalized sequence of eigenfunctions converges to a measure μ concentrated on the hyperbolic sets of the field. The coefficients which appear in the limit measure can be characterized usin...

First and Second Order Perturbations of Hypersurfaces

In this paper we find the first and second order perturbations of the induced metric and the extrinsic curvature of a non-degenerate hypersurface Σ in a spacetime (M, g), when the metric g is perturbed arbitrarily to second order and the hypersurface itself is allowed to change perturbatively (i.e. to move within spacetime) also to second order. The results are fully general and hold in arbitra...

On microlocal analyticity of solutions of first-order nonlinear PDE

— We study the microlocal analyticity of solutions u of the nonlinear equation ut = f(x, t, u, ux) where f(x, t, ζ0, ζ) is complex-valued, real analytic in all its arguments and holomorphic in (ζ0, ζ). We show that if the function u is a C2 solution, σ ∈ CharLu and 1 i σ([Lu, Lu]) < 0 or if u is a C3 solution, σ ∈ CharLu, σ([Lu, Lu]) = 0, and σ([Lu, [Lu, Lu]]) 6= 0, then σ / ∈WFau. Here WFau de...

First-Order Singular and Discontinuous Differential Equations

Discrete System Order-reduction via Balancing Transformation Using Singular Perturbations

Order-reduction techniques based on system balancing are developed for linear discrete systems using the method of singular perturbations. The generalized residualization technique is introduced with three variants as improved tools for orderreduction at low and medium frequencies. The newly introduced techniques have the same theoretical upper error bound with respect to the H∞ norm of the red...