Nonconvex Functionals Related to Multiphase Systems
Augusto Visintin · SIAM Journal on Mathematical Analysis · 1990
Let $\Omega $ be a bounded domain of $\mathbb{R}^N (N \geqq 1)$, with $\phi $ a (nonconvex) lower semicontinuous function $\mathbb{R} \to \mathbb{R} \cup \{ + \infty \} $, such that for any $u \in L^1 (\Omega )$, $\Phi (u): = \int_\Omega \phi (u(x))dx > - \infty $. Let $\Lambda :L^1 (\Omega ) \to [0, + \infty ]$ fulfill the generalized co-area formula$\Lambda (u) = \int_\mathbb{R} \Lambda (H(u - s))ds( \leqq + \infty )$ for all $u \in L^1 (\Omega )$, where $H(\xi ) = 0$ if $\xi < 0$, $H(\xi ) = 1$ if $\xi \geqq 0$. For instance, \[ \begin{gathered} V(u): = \int_\Omega {| { abla u} |} = \sup \left\{ {\int_\Omega u \,{\operatorname{div}}\,\eta\, dx:\eta \in C_c^1 (\Omega )^N ,| \eta | \leqq 1} \right\}, \hfill \\ \Lambda _r (u): = \int {\int_{\Omega ^2 } {| {u(x) - u(y)}|| {x - y} |^{ - (N + r)} } } dx\,dy\quad (0 < r < 1), \hfill \\ \tilde \Lambda _r (u): = \int_{\mathbb{R}^ + } {h^{ - (1 + r)} } dh\int_\Omega {\left( {\mathop {{\operatorname{ess}}\sup }\limits_{B_h (x) \cap \Omega } u - \mathop {{\operatorname{ess}}\inf }\limits_{B_h (x) \cap \Omega } u} \right)} dx\quad (0 < r < 1), \hfill \\ \end{gathered}\] where $B_h (x): = \{ y:| {y - x} | \leqq h\} $. Here it is proven that if $\Lambda = \Lambda ^{ * * } $, then for any $u \in L^1 (\Omega )$, $\partial (\Phi + \Lambda )(u) = \partial \Phi (u) + \partial \Lambda (u)$ in $L^\infty (\Omega )$, and $(\Phi + \Lambda )^{ * * } (u) = \Phi ^{ * * } (u) + \Lambda (u)$. This and another result entail that for any $\xi \in L^\infty (\Omega )$, if u is an absolute (relative, respectively) minimum of $\Psi _\xi :v \mapsto \int_\Omega {[\phi (v(x)) - \xi (x)v(x)]} dx + \Lambda (v)$ in $L^1 (\Omega )$, then there exists $\tilde \xi \in L^\infty (\Omega )$ such that almost everywhere in $\Omega $, $u(x)$ is an absolute (relative, respectively) minimum of $y \mapsto \phi (y) - \tilde \xi (x)y$ in $\mathbb{R}$. Hence, for both sorts of minima, certain values are a priori excluded from $u(\Omega )$, which can be nonconvex. This can represent the occurrence of a phase structure, i.e., pattern formation. If $\phi $ is the free-energy density function of some substance, $\Lambda $ can model the phase interaction contribution to the global free energy. The absolute and relative minima of $\Psi _\xi $ are related to the stable and metastable equilibrium states, respectively. Solid-liquid systems are discussed in particular. The proposed model accounts for supercooling andsuperheating effects. If $\Lambda = V$, the mean curvature of the solid-liquid interface $\mathcal{S}$ is as prescribed by the Gibbs–Thomson law. If $\Lambda = \Lambda _r (0 < r < 1)$, $\mathcal{S}$ can be more irregular, as in dendritic formations and snowflakes. This model can be extended to include mushy regions.