Nonexistence of Solutions for Dirichlet Problems with Supercritical Growth in Tubular Domains

We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ ( u ) = 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ⁢ ( Γ k ) {T_{\varepsilon}(\Gamma_{k})} with thickness ε > 0 {{\varepsilon}>0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of ℝ n {\mathbb{R}^{n}} . Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε > 0 {{\varepsilon}>0} small enough. When k ≥ 2 {k\geq 2} or Γ k {\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f.

Thin domain limit and counterexamples to strong diamagnetism

We study the magnetic Laplacian and the Ginzburg–Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some nonlinear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little–Parks experiment, for a thin cylindrical sample.