Asymptotic Analysis

We consider systems of coupled Schrödinger equations which appear in nonlinear optics and binary Bose–Einstein condensation. Namely, we prove that for most μ,ν∈R, a,b>0, the system −Δu=μu+au³−βuv², −Δv=νv+bv³−βvu², u,v∈H¹₀(B₁(0)), where B₁(0) is the unit ball of R³, admits a family of radially symmetric positive solutions (u_β,v_β) provided the interaction parameter β>0 is sufficiently large. By using a Morse index technique we deduce that these solutions are bounded uniformly in β, hence their limit functions as β→∞ undergo the phenomenon of phase segregation.

In this paper, we study the Dirichlet problem for an elliptic equation, in which the 1-Laplacian operator and lower-order terms appear. We introduce a suitable definition of solution and prove the existence of, at least, one bounded solution in BV(Ω) having a negligible jump part. Moreover, a uniqueness result for small positive data is proved, and explicit examples of solutions are shown.

The asymptotic behaviour of a class of elliptic equations with highly oscillating coefficients, in a perforated periodic domain, is analyzed. We consider, in each period, two types of holes and we impose, on their boundaries, a Signorini and, respectively, a Neumann condition. Using the periodic unfolding method, we prove that the limit problem contains two additional terms, a right-hand side term and a “strange” one.

In this paper, we study the scattering theory for a 2×2 matrix Schrödinger operator

P=−h²d²/dx²I₂+V(x)+hR(x,hD_x)

on L²(R)⌖L²(R), where V(x) is a real diagonal matrix, the eigenvalues of which are never equal. Under some assumptions of analyticity and decay at infinity of V, we describe the asymptotic behavior of the scattering matrix S=(s_ij)_1≤i,j≤4 associated with P when the semi-classical parameter h goes to zero. Moreover, we obtain the estimate

‖S₁₂‖+‖S₂₁‖=O(e^−δ/h),

where S₁₂ and S₂₁ are the two off-diagonal elements of S and δ>0 is a constant which is explicitly related to the behavior of V(x) in the complex domain.

In this paper, we prove the existence of a uniform attractor for non-autonomous extensible beam equation with localized damping.

We consider linear control systems of the form y′(t)=Ay(t)+Bu(t) where A generates a strongly continuous semigroup of contractions (e^tA)_t≥0 on an infinite-dimensional Hilbert space Y. We suppose that the control is unbounded in the sense that the linear control operator B is bounded from the (Hilbert) control space U to some larger space W such that Y⊂W, but not from U into Y. Taking into account eventual control saturation, we study the problem of stabilization by (possibly nonlinear) monotone feedback of the form u(t)=−f(B^*y(t)). We extend to the unbounded monotone feedback context weak and strong stability results which have been established for linear feedback systems. These results are based on weak observability involving the unstable subspace. This fact is illustrated by a heat equation with singular control. In the particular case where the system can be reduced to a second-order evolution equation of the form z″(t)+𝒜z(t)=ℬu(t), we establish decay estimates with (eventually) saturating feedback and under a strong observability property. We show that when the initial data are sufficiently regular, we obtain exponential and polynomial decay. We establish also that when the control operator ℬ is bounded, the regularity assumption can be dispensed with. Applications to the wave equation with distributed control and pointwise control are considered. We establish for these systems non-uniform exponential decay with general saturating feedback. For a string, we present an explicit saturating feedback leading to exponential decay of the energy. Moreover, the degree of regularity of the initial data are related to the Diophantine properties of the actuator position.

In this paper, we consider the problem of pricing perpetual American put options with volatility driven by two other processes. By using a perturbation approach, we obtain approximate but explicit closed-form pricing formulae for the option and optimal exercise prices, respectively, under a general multi-scale SV (stochastic volatility) model. A key feature of the expansion methodology employed here is to balance the two SV processes, while dealing with the free boundary conditions properly. It turns out that in the current formulae, the fast volatility factor does not play an explicit role, while the slow factor is quite crucial, a phenomenon that is shown to be quite reasonable through our discussions.

We consider a parabolic equation with fast oscillating periodic coefficients, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new in a bounded domain, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform null controllability result is the first in the multidimensional setting for parabolic equations with oscillating coefficients. Finally, we prove that the sequence of null controls converges to the optimal null control of the homogenized equation when the period tends to zero.

In a previous work by the first author with J. Turi [Appl. Math. Optim. 58(1) (2008), 1–27], a stochastic variational inequality has been introduced to model an elasto-plastic oscillator with noise. A major advantage of the stochastic variational inequality is to overcome the need to describe the trajectory by phases (elastic or plastic). This is useful, since the sequence of phases cannot be characterized easily. In particular, when a change of regime occurs, there are numerous small elastic phases which may appear as an artefact of the Wiener process. However, it remains important to have informations on both the elastic and plastic phases. In order to reconcile these contradictory issues, we introduce an approximation of stochastic variational inequalities by imposing artificial small jumps between phases allowing a clear separation of the elastic and plastic regimes. In this work, we prove that the approximate solution converges on any finite time interval, when the size of jumps tends to 0.