Ground states of a two phase model with cross and self attractive interactions (Q2823009)

The authors analyze the existence of minimizers to an energy functional involved in the case where two phases, represented by two subsets \(E_{1}\) and \(E_{2}\) of \(\mathbb{R}^{N}\), interact with themselves and with the other phase and written as NEWLINE\[NEWLINE\mathcal{F} (E_{1},E_{2})=c_{11}J_{K}(E_{1},E_{1})+c_{22}J_{K}(E_{2},E_{2})+(c_{12}+c_{21})J_{K}(E_{1},E_{2}),NEWLINE\]NEWLINE with NEWLINE\[NEWLINEJ_{K}(E_{i},E_{j})=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\chi _{E_{i}}(x)\chi _{E_{j}}(y)K(x-y)dxdyNEWLINE\]NEWLINE for some \(K\in L_{loc}^{1}( \mathbb{R}^{N};\mathbb{R})\). The minimizers of the energy \(\mathcal{F}\) have to satisfy \(E_{1}\cap E_{2}=\varnothing \) and \(|E_{i}|=m_{i}\) for given masses \(m_{i}\). Assuming that \(c_{12}+c_{21}=-2\), the authors introduce the relaxed problem obtained when replacing the characteristic functions \(\chi _{E_{i}}\) by functions \(f_{i}\in L^{1}(\mathbb{R}^{N};\mathbb{R}^{+})\) in the above energy and considering the class \(\mathcal{A}_{m_{1},m_{2}}= \{(f_{1},f_{2})\in L^{1}(\mathbb{R}^{N};\mathbb{R}^{+})\times L^{1}(\mathbb{R }^{N};\mathbb{R}^{+}):\int_{\mathbb{R}^{N}}f_{i}(x)dx=m_{i}\), \( f_{1}+f_{2}\leq 1\) a.\,e. in \(\mathbb{R}^{N}\}\) of admissible functions.NEWLINENEWLINE They establish properties of a minimizer \((f_{1},f_{2})\) of this relaxed problem. They prove an existence result for every \(c_{11},c_{22}\leq 0\) and they prove a compactness property of a minimizer in \(\mathcal{A} _{m_{1},m_{2}}\). In the case where \(c_{12}+c_{21}\leq -2\), called strongly attractive case, the authors establish a necessary and sufficient condition for the existence of a minimizer to the relaxed problem if \(c_{11}=c_{22}=-1\) , or if \(c_{11}=-1\) and \(c_{22}<-1\), or if \(c_{22}=-1\) and \(c_{11}<-1\), or if \(c_{22}<-1\) and \(-1<c_{11}\leq 0\), or if \(c_{11}<-1\) and \(-1<c_{22}\leq 0\) .NEWLINENEWLINE In the weakly attractive case corresponding to \( c_{12}+c_{21}>-2\), the authors prove an existence result for a minimizer which is unique up to translations and they provide the structure of this minimizer. Taking \(K\) as the Coulomb kernel, the authors prove existence results in the strongly and in the weakly attractive cases. In the latter case, they first prove an existence result if \(-1<c_{11},c_{22}\leq 0\), then distinguishing between the case \(N=1\) and \(N\geq 2\), for which they establish further properties of these minimizers. They also provide the form of these minimizers for almost all the values of the coefficients.