On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions
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Publication:2504735
DOI10.1007/s10231-003-0088-yzbMath1150.35035OpenAlexW1580929281MaRDI QIDQ2504735
Publication date: 1 February 2007
Published in: Annali di Matematica Pura ed Applicata. Serie Quarta (Search for Journal in Brave)
Full work available at URL: https://doi.org/10.1007/s10231-003-0088-y
Variational methods involving nonlinear operators (47J30) Singular perturbations in context of PDEs (35B25) Nonlinear elliptic equations (35J60)
Related Items (27)
On upper semicontinuity of the Allen–Cahn twisted eigenvalues ⋮ Stable phase interfaces in the van der Waals–Cahn–Hilliard theory ⋮ Critical points via \(\Gamma \)-convergence: general theory and applications ⋮ Phase transition layers for Fife-Greenlee problem on smooth bounded domain ⋮ Clustering of boundary interfaces for an inhomogeneous Allen-Cahn equation on a smooth bounded domain ⋮ Connectivity of boundaries by clustering phase transition layers of Fife-Greenlee problem on smooth bounded domain ⋮ A finite difference method for a conservative Allen-Cahn equation on non-flat surfaces ⋮ Interfaces with boundary intersection for an inhomogeneous Allen-Cahn equation in three-dimensional case ⋮ Interior interfaces with (or without) boundary intersection for an anisotropic Allen-Cahn equation ⋮ Stable transition layer for the Allen–Cahn equation when the spatial inhomogeneity vanishes on a nonsmooth hypersurface in ℝn ⋮ Interface foliation for an inhomogeneous Allen-Cahn equation in Riemannian manifolds ⋮ Concentration on surfaces for a singularly perturbed Neumann problem in three-dimensional domains ⋮ Transition layer for the heterogeneous Allen-Cahn equation ⋮ Solutions with multiple catenoidal ends to the Allen-Cahn equation in \(\mathbb{R}^3\) ⋮ On De Giorgi's conjecture in dimension \(N\geq 9\) ⋮ Phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation ⋮ LAYERED SOLUTIONS WITH CONCENTRATION ON LINES IN THREE-DIMENSIONAL DOMAINS ⋮ Catenoidal layers for the Allen-Cahn equation in bounded domains ⋮ Numerical simulation and error estimation of the time-dependent Allen-Cahn equation on surfaces with radial basis functions ⋮ On spikes concentrating on line-segments to a semilinear Neumann problem ⋮ Interior layers for an inhomogeneous Allen-Cahn equation ⋮ The Toda system and clustering interfaces in the Allen-Cahn equation ⋮ Solutions with transition layer and spike in an inhomogeneous phase transition model ⋮ Clustering phase transition layers with boundary intersection for an inhomogeneous Allen-Cahn equation ⋮ Asymptotic behavior of Allen-Cahn-type energies and Neumann eigenvalues via inner variations ⋮ Multiple-end solutions to the Allen-Cahn equation in \(\mathbb R^2\) ⋮ Curve-Like Concentration Layers for a Singularly Perturbed Nonlinear Problem with Critical Exponents
Cites Work
- Generation and propagation of interfaces for reaction-diffusion equations
- Ginzburg-Landau equation and motion by mean curvature. II: Development of the initial interface
- Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature
- From constant mean curvature hypersurfaces to the gradient theory of phase transitions.
- Clustering layers and boundary layers in spatially inhomogeneous phase transition problems.
- Fast Reaction, Slow Diffusion, and Curve Shortening
- Local minimisers and singular perturbations
- Phase transitions and generalized motion by mean curvature
- On the convergence of stable phase transitions
- Geometrical Evolution of Developed Interfaces
- Higher energy solutions in the theory of phase transitions: A variational approach
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