Error estimates related to singular perturbations of first-order equations and systems

First-Order Singular and Discontinuous Differential Equations

Improved Error Estimates for First Order Sigma-delta Systems

A sigma-delta modulator converts a sequence of real numbers bounded by 1 into a digital sequence with entries +1 and 1 only. When the input sequence consists of samples of a bandlimited function, oversampled by a factor with respect to its Nyquist frequency, we prove that filtering the digital output with an appropriate kernel reproduces the original bandlimited function up to an error of at mo...

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...

Singular perturbations to semilinear stochastic heat equations

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε = 0. A similar effect is also observed for some (formally) small stocha...

Singular perturbations of weakly coupled systems of Hamilton-Jacobi equations

In this paper we study singular perturbations of weakly coupled systems of elliptic equations. A model problem is given by small random perturbations of random evolution processes. In this setting we give a PDE proof of large deviation results, analogous to those studied in [5], [10, 11]. An essential tool in our approach is the weak KAM theory introduced in [15].