Andrea Braides

The Γ-limit of higher-order singular perturbations of the Perona–Malik functional is analyzed. The energies considered combine the critically scaled logarithmic term with a k -th order regularization designed to balance bulk and interfacial effects. A compactness result is obtained, and the Γ-limit is identified as a free-discontinuity functional on SBV, given by the sum of the Dirichlet energy and a surface term proportional to the jump amplitude to the power 1 k {\frac{1}{k}} . The surface density is characterized through a one-dimensional optimal-profile problem with homogeneous boundary conditions on derivatives up to order k - 1 {k-1} . As a consequence, the limit of the same energies at a different scaling is determined. That scaling had been previously studied in the second-order case to address the so-called staircasing phenomenon.

We investigate the stability with respect to homogenization of classes of integrals arising in the control-theoretic interpretation of some Hamilton–Jacobi equations. The prototypical case is the homogenization of energies with a Lagrangian consisting of the sum of a kinetic term and a highly oscillatory potential [Formula: see text], where [Formula: see text] is periodic and [Formula: see text] is a nonnegative perturbation thereof. We assume that [Formula: see text] has zero average in tubular domains oriented along a dense set of directions. Stability then holds true; that is, the resulting homogenized functional is identical to that for [Formula: see text]. We consider various extensions of this case. As a consequence of our results, we obtain stability for the homogenization of some steady-state and time-dependent, first-order Hamilton–Jacobi equations with convex Hamiltonians and perturbed periodic potentials. Finally, we show with an example that, for negative [Formula: see text], stability may not hold. Our study revisits and, depending on the different assumptions, complements results obtained by P.-L. Lions, P. Souganidis, and their collaborators using PDE techniques.

Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized some earlier Gamma-convergence results by himself and S. Mortola contained in a seminal paper, validating the so-called minimal-interface criterion. I will give an overview of some recent results on perturbations with fractional and higher-order seminorms both in the framework of phase transitions and of free-discontinuity problems, relating these results with the Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limit analysis for fractional Sobolev seminorms, and with the theory of Gamma-expansions.

We consider the problem of the homogenization of non-local quadratic energies defined on \delta δ -periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale \varepsilon ε . For kernels with unbounded support we show that we may have three regimes: (i) \varepsilon<\!<\delta ε <  ⁣ < δ , for which the \Gamma Γ -limit even in the strong topology of L^2 L 2 is 0 0 ; (ii) \frac\varepsilon\delta\to\kappa ε δ → κ , in which the energies are coercive with respect to a convergence of interpolated functions, and the limit is governed by a non-local homogenization formula parameterized by \kappa κ ; (iii) \delta<\!<\varepsilon δ <  ⁣ < ε , for which the \Gamma Γ -limit is computed with respect to a coarse-grained convergence and exhibits a separation-of-scales effect; namely, it is the same as the one obtained by formally first letting \delta\to 0 δ → 0 (which turns out to be a pointwise weak limit, thanks to an iterated use of Jensen's inequality), and then, noting that the outcome is a nonlocal energy studied by Bourgain, Brezis and Mironescu, letting \varepsilon\to0 ε → 0 . A slightly more complex description is necessary for case (ii) if the kernel is compactly supported.