We consider the Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions...

We consider the Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions...

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turing instability and the interpretation refers to adaptive evolution. By analogy with other formalisms used in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sum of Dirac masses) will happen in the limit of small mutations. In the present work we study this asymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation. We prove the convergence analytically...

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space  [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.

We consider the second order initial value problem in a Hilbert space, which is a singular perturbation of a first order initial value problem. The difference of the solution and its singular limit is estimated in terms of the small parameter $\epsilon .$ The coefficients are commuting self-adjoint operators and the estimates hold also for the semilinear problem.

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system $-\Delta u+u^3+\beta u{v}^2=\lambda u,-\Delta v+v^3+\beta u^{2}v=\mu v,u,v\in H_0^1\left(\Omega \right),u,v>0$, as the interspecies scattering length $\beta$ goes to $+\infty$. For this system we consider the associated energy functionals $J_{\beta },\beta \in (0,+\infty )$, with $L^2$-mass constraints, which limit $J_{\infty }$ (as $\beta \to +\infty$) is strongly irregular. For such functionals, we construct multiple critical points via a common...

Si studiano soluzioni positive dell’equazione $-{ϵ}^2\mathrm{\Delta }u+u=u^p$ in $\mathrm{\Omega }$, dove $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ , $p>1$ ed $ϵ$ è un piccolo parametro positivo. Si impongono in genere condizioni al bordo di Neumann. Quando $ϵ$ tende a zero, dimostriamo esistenza di soluzioni che si concentrano su curve o varietà.

Consider the family uₜ = Δu + G(u), t > 0, $x\in {\Omega }_{\epsilon }$, ${\partial }_{{\nu }_{\epsilon }}u=0$, t > 0, $x\in \partial {\Omega }_{\epsilon }$, $\left(E_{\epsilon }\right)$ of semilinear Neumann boundary value problems, where, for ε > 0 small, the set ${\Omega }_{\epsilon }$ is a thin domain in ${ℝ}^l$, possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of ${ℝ}^l$. If G is dissipative, then equation $\left(E_{\epsilon }\right)$ has a global attractor ${}_{\epsilon }$. We identify a “limit” equation for the family $\left(E_{\epsilon }\right)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${}_{\epsilon }$ as ε → 0⁺.