Phase transitions in multiphase media taking into account the interface surface energy by an integral of higher derivatives of the displacement field (Q2773602)

The author studies the minimization of energy functional NEWLINE\[NEWLINE I(u,\chi,\tau,\sigma)= \sigma\|u\|_H^p + \int_{\Omega}\chi^{\lambda}(x) F^{\lambda}\bigl(\dot{u}(x),u(x),x,\tau\bigr) dx NEWLINE\]NEWLINE which acts on the set of functions NEWLINE\[NEWLINE X = \bigl\{\{u,\chi\}: u\in H = W_2^2(\Omega,\mathbb R^m)\cap \overset\circ{W}^1_2(\Omega,\mathbb R^m),\;\chi = (\chi^+,\chi^-,\chi^0)\bigr\}, NEWLINE\]NEWLINE where \(0 < p <1\), \(\sigma > 0\), \(\Omega\subset\mathbb R^m\) is a bounded domain with Lipschitz boundary, \(\chi^{\lambda}\) are measurable characteristic functions, \(\lambda = +, -, 0\), and \(\chi^+(x) + \chi^-(x) + \chi^0(x) = 1\) for a.e. \(x\in\Omega\). The functional is minimized under the additional constraint NEWLINE\[NEWLINE \int_{\Omega}\bigl(\rho^+\chi^+(x) + \rho^-\chi^-(x) + \rho^0\chi^0(x)\bigr) dx = M, NEWLINE\]NEWLINE where \(|\Omega|\min\{\rho^+, \rho^-, \rho^0\} \leq M \leq |\Omega|\max\{\rho^+, \rho^-, \rho^0\}\).NEWLINENEWLINENEWLINEThe author proves existence of an equilibrium state for the considered functional and studies dependence of equilibrium states on parameters \(\sigma\) and \(\tau\). In addition, he compares two approaches for regularizing the energy functional which are based on using higher derivatives of the displacement field and the area of interface surface.