* This work was supported by National Science Foundation grant DMS 9404431.In this paper we prove that the Newton method applied to the generalized equation y ∈ f(x) + F(x) with a C^1 function f and a set-valued map F acting in Banach spaces, is locally convergent uniformly in the parameter y if and only if the map (f +F)^(−1) is Aubin continuous at the reference point. We also show that the Aubin continuity actually implies uniform Q-quadratic convergence provided that the derivative of f is Lipschitz...

We prove the existence and uniform decay rates of global solutions for a hyperbolic system with a discontinuous and nonlinear multi-valued term and a nonlinear memory source term on the boundary.

We consider complex-valued solutions $u_{\epsilon }$ of the Ginzburg–Landau equation on a smooth bounded simply connected domain $\Omega$ of ${ℝ}^N$, $N\ge 2$, where $\epsilon \>0$ is a small parameter. We assume that the Ginzburg–Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where $M_0$ is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of $u_{\epsilon }$, as $\epsilon \to 0$, is to establish uniform $L^p$ bounds for the gradient, for some $p\>1$. We review some recent techniques developed in...

We consider complex-valued solutions uE of the Ginzburg–Landau equation on a smooth bounded simply connected domain Ω of ${ℝ}^N$, N ≥ 2, where ε > 0 is a small parameter. We assume that the Ginzburg–Landau energy $E_{\epsilon }\left(u_{\epsilon }\right)$ verifies the bound (natural in the context) $E_{\epsilon }\left(u_{\epsilon }\right)\le M_0|log\epsilon |$, where M0 is some given constant. We also make several assumptions on the boundary data. An important step in the asymptotic analysis of uE, as ε → 0, is to establish uniform Lp bounds for the gradient, for some p>1. We review some...

The mathematical model of a beam on a unilateral elastic subsoil of Winkler's type and with free ends is considered. Such a problem is non-linear and semi-coercive. The additional assumptions on the beam load ensuring the problem solvability are formulated and the existence, the uniqueness of the solution and the continuous dependence on the data are proved. The cases for which the solutions need not be stable with respect to the small changes of the load are described. The problem is approximated...

We consider the problem of localizing an inaccessible piece $I$ of the boundary of a conducting medium $\Omega$, and a cavity $D$ contained in $\Omega$, from boundary measurements on the accessible part $A$ of $\partial \Omega$. Assuming that $g(t,\sigma )$ is the given thermal flux for $\left(t,\sigma \right)\in (0,T)\times A$, and that the corresponding output datum is the temperature $u(T_0,\sigma )$ measured at a given time $T_0$ for $\sigma \in A_{\mathrm{out}}\subset A$, we prove that $I$ and $D$ are uniquely localized from knowledge of all possible pairs of input-output data $(g,u{\left(T_0\right)}_{\mid A_{\mathrm{out}}})$. The same result holds when a mean value of the temperature...

We consider the problem of localizing an inaccessible piece I of the boundary of a conducting medium Ω, and a cavity D contained in Ω, from boundary measurements on the accessible part A of ∂Ω. Assuming that g(t,σ) is the given thermal flux for (t,σ) ∈ (0,T) x A, and that the corresponding output datum is the temperature u(T0,σ) measured at a given time T0 for σ ∈ Aout ⊂ A, we prove that I and D are uniquely localized from knowledge of all possible pairs of input-output data $(g,u{\left(T_0\right)}_{\mid A_{\mathrm{out}}})$. The same result...

For α∈(0, 1) an α-trimming, P∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f≤1/(1−α), in the way P∗(B)=∫Bf(x)P(dx). If P, Q are probability measures on euclidean space, we consider the problem of obtaining the best L2-Wasserstein approximation between: (a) a fixed probability and trimmed versions of the other; (b) trimmed versions of both probabilities. These best trimmed approximations naturally...

In the setting of the optimal transportation problem we provide some conditions which ensure the existence and the uniqueness of the optimal map in the case of cost functions satisfying mild regularity hypothesis and no convexity or concavity assumptions.

For external magnetic field hex ≤ Cε–α, we prove that a Meissner state solution for the Chern-Simons-Higgs functional exists. Furthermore, if the solution is stable among all vortexless solutions, then it is unique.