Variational aspects of phase transitions with prescribed mean curvature
We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curv...
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| description | We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asymptotics that are sharp to order \(o(\varepsilon^2)\) and \(C^{2,\alpha}\) estimates on multiplicity-one phase transition layers. |
| doi_str_mv | 10.48550/arxiv.2011.00358 |
| format | Article |
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These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. 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| subjects | Curvature Eigenvalues Hyperspaces Lower bounds Mathematical analysis Mathematics - Analysis of PDEs Mathematics - Differential Geometry Phase transitions Riemann manifold Transition layers Upper bounds |
| title | Variational aspects of phase transitions with prescribed mean curvature |
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