AI 中文总结
研究具有空间依赖势阱的固-固相变的二阶莫迪卡-莫托拉泛函，忽略弹性能密度框架无关性假设，在适当条件下证明该泛函\(\Gamma\)-收敛到局部界面能。
AI 中文摘要
我们研究形如\[ E_\varepsilon[u]:= \int_\Omega \frac{1}{\varepsilon} W(x, \nabla u) + \varepsilon|\nabla^2 u|^2 \, dx \]的二阶莫迪卡-莫托拉泛函，它模拟非均匀介质中的固-固相变。我们忽略弹性能密度\(W\)中的框架无关性假设，同时允许空间依赖的逐点相容势阱。在对\(W\)的适当假设和相关秩一连接的正则性条件下，我们证明\(E_\varepsilon\) \(\Gamma\)-收敛到为合适的层压型构型定义的局部界面能。
英文摘要
We investigate a second-order Modica-Mortola functional of the form \[ E_\varepsilon[u] := \int_Ω\frac{1}{\varepsilon} W(x, \nabla u) + \varepsilon|\nabla^2 u|^2 \, dx, \] which models solid--solid phase transitions in heterogeneous media. We neglect the assumption of frame indifference in the elastic energy density $W$ while allowing for space-dependent, pointwise-compatible wells. Under suitable assumptions on $W$ and regularity conditions on the associated rank-one connection, we prove that $E_\varepsilon$ $Γ$-converges to a local interfacial energy defined for suitable laminate-type configurations.
