In the Allen-Cahn theory of phase transitions, minimizers partition the domain in subregions, the sets where a minimizer is near to one or to another of the zeros of the potential. These subregions that model the phases are separated by a tiny Diffuse Interface. Understanding the shape of this diffuse interface is an important step toward the description of the structure of minimizers.We assume Dirichlet data and present general conditions on the domain and on the boundary datum ensuring the connectivity of the diffuse interface.Then we restrict to the case of two dimensions and show that the phases can be separated, in a certain optimal way, by a connected network with a well defined structure. This network is contained in the diffuse interface and is a priori unknown.Under general assumption on the potential and on the Dirichlet datum, we show that, if we assume that the phase are connected, then we can obtain precise information on the shape of the network and in turn a detailed description of the fine structure of minimizers. In particular we can characterize the shape and the size of the various phases and also how they depend on the surface tensions.
Suppose that f = (u, v) is a homeomorphism in the plane of the Sobolev class W 1,1 loc such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.
We identify the exact degree of integrability of nonnegative volume forms and the Jacobians of orientation preserving mappings from various Orlicz-Sobolev classes. An improvement takes place when the Jacobian belongs to the Orlicz space L Ψ (Ω), where Ψ grows almost linearly, that is, t 1−ε ≺ Ψ(t) ≺ t 1+ε for ε > 0. Our results amount to the principle: the further the Jacobian is from L 1 loc (Ω), the less is the improvement of integrability. In fact, as shown in [LZ], [Wu], [GIM], the largest improvement happens when the Jacobian is precisely in the space L 1 loc (Ω). Contents.
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