Abstract The Allen--Cahn equation is a semilinear PDE which is deeply linked to the theory
of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet …

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation
Δu+(1–u²) u= 0 in ℝ N with∂ yN u> 0 must be such that its level sets u–λ are all …

The Ohta–Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal
free boundary problem. In a proper range of the block composition parameter and the …

We consider the problem ε^ 2 Δ uV (x) u+ u^ p= 0,\;\;\;\;\; u> 0,\;\;\; u ∈ H^ 1 (\cal R^ 2), where
p> 1, ε> 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a …

We study global variational properties of the space of solutions to-ε^ 2 Δ u+ W'(u)= 0-ε 2 Δ u+
W′(u)= 0 on any closed Riemannian manifold M. Our techniques are inspired by recent …

With the tunability of the nematic liquid crystal (NLC), a broadband frequency reconfigurable
and versatile metastructure (MS) is proposed and theoretically investigated in this paper …

We construct a new class of entire solutions for the Allen–Cahn equation Δu+ (1− u2) u= 0,
in R2 (∼ C). Given k⩾ 1, we find a family of solutions whose zero level sets are, away from a …

We prove that every non‐degenerate minimal submanifold of codimension two can be
obtained as the energy concentration set of a family of critical maps for the (rescaled) …

We continue the study of 34, proving concentration phenomena for the equation− ε 2 Δ u+
u= up in a smooth bounded domain Ω⊆ R^n and with Neumann boundary conditions. The …

We consider the following semilinear elliptic equation with critical exponent: Δ u= K (x) u (n+
2)/(n-2), u> 0 in ℝ n, where n≥ 3, K> 0 is periodic in (x 1,…, xk) with 1≤ k<(n-2)/2. Under …