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...

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...

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.

Uniqueness of the optimal control is obtained by assuming certain conditions on the crowding effect of the species. Moreover, an approximation procedure for the unique optimal control is developed.

We prove an upper bound for the Aviles–Giga problem, which involves the minimization of the energy $E_{\epsilon }\left(v\right)=\epsilon {\int }_{\Omega }|{\nabla }^2{v|}^{2}dx+{\epsilon }^{-1}{\int }_{\Omega }{\left(1-\right|\nabla v|}^2{)}^{2}dx$ over $v\in H^2\left(\Omega \right)$, where $\epsilon >0$ is a small parameter. Given $v\in W^{1,\infty }\left(\Omega \right)$ such that $\nabla v\in BV$ and $\left|\nabla v\right|=1$ a.e., we construct a family $\left\{v_{\epsilon }\right\}$ satisfying: $v_{\epsilon }\to v$ in $W^{1,p}\left(\Omega \right)$ and $E_{\epsilon }\left(v_{\epsilon }\right)\to \frac{1}{3}{\int }_{J_{\nabla v}}{\left|{\nabla }^{+}v-{\nabla }^{-}v\right|}^{3}d{\mathscr{H}}^{N-1}$ as $\epsilon$ goes to 0.

Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.