Existence of minimizers for the SDRI model in $\mathbb{R}^n$: Wetting and dewetting regimes with mismatch strain

TL;DR

This work extends the SDRI variational model to all dimensions $n\ge 2$, establishing existence and partial regularity of energy-minimizing configurations $(A,u)$ under a volume constraint and with a mismatch strain encoding wetting and dewetting. A novel $\tau_{\mathcal{C}}$-topology couples $L^1$ convergence of the crystal region with a.e. convergence of displacements in the GSBD framework, enabling lower semicontinuity and compactness of the coupled elastic-surface energy $\mathcal{F}=\mathcal{W}+\mathcal{S}$. The paper also shows that the approach remains valid under Dirichlet boundary conditions and extends to $p$-growth elastic densities, connecting to existing 2D results and literature models on voids and Griffith-type energies. Decay estimates yield density bounds and essential closedness of interfaces, leading to partial regularity of minimizers and a deeper understanding of wetting/dewetting morphologies in higher dimensions. The results thus provide a robust variational理论 for SDRI phenomena with mismatch strain in realistic 3D settings, broadening the mathematical toolkit for crystalline morphology under elastic and capillarity-driven instabilities.

Abstract

The existence and the regularity results obtained in [37] for the variational model introduced in [36] to study the optimal shape of crystalline materials in the setting of stress-driven rearrangement instabilities (SDRI) are extended from two dimensions to any dimensions $n\geq2$. The energy is the sum of the elastic and the surface energy contributions, which cannot be decoupled, and depend on configurational pairs consisting of a set and a function that model the region occupied by the crystal and the bulk displacement field, respectively. By following the physical literature, the ``driving stress'' due to the mismatch between the ideal free-standing equilibrium lattice of the crystal with respect to adjacent materials is included in the model by considering a discontinuous mismatch strain in the elastic energy. Since two-dimensional methods and the methods used in the previous literature where Dirichlet boundary conditions instead of the mismatch strain and only the wetting regime were considered, cannot be employed in this setting, we proceed differently, by including in the analysis the dewetting regime and carefully analyzing the fine properties of energy-equibounded sequences. This analysis allows to establish both a compactness property in the family of admissible configurations and the lower-semicontinuity of the energy with respect to the topology induced by the $L^1$-convergence of sets and a.e.\ convergence of displacement fields, so that the direct method can be applied. We also prove that our arguments work as well in the setting with Dirichlet boundary conditions.

Figures (4)

• Figure 1: The partitions of the substrate and the free crystal respectively into the families of Caccioppoli sets $\{S^i\}_{i\geq1}$ and $\{F^i\}_{i\geq0},$ used in the proof of the $\tau_\mathcal{C}$-compactness, are illustrated by assigning different line patterns to the crystal phases that interact with the substrate, and a dotted pattern to the remaining "hanging phase" $F^0.$
• Figure 3: The partition of $A = \bigcup_{i\ge0} F^i$ and the construction of $B_k^\delta:=A_k\setminus G_k^\delta$ in Proposition \ref{['prop:pass_to_good_seq_compacte']}. The set $G_{\delta,k}$ is a finite union of holes along the boundaries $F^i\cup \bigcup_{j\ne i} S^j$ in which $u_k - a_k^i$ converges. Note that the sets $\{F^i\setminus G_k^\delta\}_{i=0}^m$ partition $B_k^\delta.$ Since $F^0$ is a "hanging" component of $A,$ i.e., not linked to the substrate, it is reasonable to assume that the elastic energy in $F^0$ is $0.$ Then we define the displacement fields $v_k^\delta$ as follows: in $S^i\cup (F^i\setminus G_k^\delta)$ for $i=1,\ldots,m$ we set $v_k^\delta:=u_k - a_k^i$ and in $F^0\setminus G_k^\delta$ we write $v_k^\delta:=u_0.$ Finally, since $A_k\setminus A$ may present large trace portions along $\partial S$ on which $v_k^\delta$ forms a jump, we need to change the values of $v_k^\delta$ in $R_\delta^i\setminus A$ near $S^i.$
• Figure 4: The sets $E_k$ and $E$ in Proposition \ref{['prop:estimate_red_boundary']}.
• Figure 5: Construction of holes $C_1^j,$$C_2^j$ and $D_k^j.$

Theorems & Definitions (50)

• Theorem 1.1: Existence of minimizing configurations
• Theorem 1.2: Regularity results for minimizing configurations
• Remark 2.1: Values of displacements outside a set
• Definition 2.2
• Remark 2.3: A priori bounds
• Theorem 2.4: Existence of minimizing configurations
• Theorem 2.5: Lower semicontinuity
• Theorem 2.6: Compactness
• Theorem 2.7: Properties of minimizing configurations)
• Theorem 2.8: Elastic density with $p$-growth