Abstract
Continuing the program initiated in Golovaty et al. (SIAM J Math Anal 51(1):276–320, 2018), we analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain \(\Omega \) we investigate the \(\varepsilon \rightarrow 0\) asymptotics of the variational problem
within various parameter regimes for \(L_\varepsilon > 0.\) Here \(u:\Omega \rightarrow \mathbb {R}^2\) and W is a potential vanishing on the unit circle and at the origin. When \(\varepsilon \ll L_\varepsilon \rightarrow 0\), we show that these functionals \(\Gamma \)-converge to a constant multiple of the perimeter of the phase boundary and the divergence penalty is not felt. However, when \(L_\varepsilon \equiv L > 0\), we find that a tangency requirement along the phase boundary for competitors in the conjectured \(\Gamma \)-limit becomes a mechanism for development of singularities. We establish criticality conditions for this limit and under a non-degeneracy assumption on the potential we prove the compactness of energy bounded sequences in \(L^2\). The role played by this tangency condition on the formation of interfacial singularities is investigated through several examples: each of these examples involves analytically rigorous reasoning motivated by numerical experiments. We argue that generically, “wall” singularities between \(\mathbb {S}^1\)-valued states of the kind analyzed in Golovaty et al. (SIAM J Math Anal 51(1):276–320, 2018) are expected near the defects along the phase boundary.
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03 May 2023
A Correction to this paper has been published: https://doi.org/10.1007/s00205-023-01879-4
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Acknowledgements
PS, MN and RV acknowledge the support from NSF DMS-1362879 and a Simons Collaboration grant 585520. RV also acknowledges the support from an Indiana University College of Arts and Sciences Dissertation Year Fellowship. The research of RV was also partially supported by the Center for Nonlinear Analysis at Carnegie Mellon University and by NSF DMS-1411646. DG acknowledges the support from NSF DMS-1729538.
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Appendix
Appendix
We present here the Proof of Theorem 3.15. See Fig. 4 for a guide to the notation.
Proof
The derivation of (5.18) follows the same general lines as those appearing in the proof of Theorem 3.14. However, a major complicating consideration is that it is no longer possible to assume that the deforming vector field X is normal to all four curves \(\Gamma _{ij}\) since they all meet at p. Instead we will have to incorporate tangential components of X along these four curves as well.
To this end, we assume simply that \(X\in C_0^1(B(p,R);\mathbb {R}^2)\) and again introduce the map \(\Psi \) via (3.56). We assume that each \(\Gamma _{ij}\) is smoothly parametrized by arclength through a map \(r_{ij}:[0,s_0]\rightarrow \Gamma _{ij}\) for some \(s_0>0\) with \(r_{ij}(0)=p\). Then we replace (3.55) by
where
As a consequence of the compact support of X, we have that
but we stress that none of these functions is assumed to vanish at \(s=0\), namely at the location of the junction P.
We now deform each region \(\Omega _j\), for \(j=0,1,2,3\) by the map \(\Psi \) to form four contiguous regions \(\Omega _j^t:=\Psi (\Omega _j,t)\) and we deform the four boundary curves \(\Gamma _{ij}\) to form four new boundary curves \(\Gamma _{ij}^t:=\Psi (\Gamma _{ij},t).\) Of course the junction point P is also carried along by this flow.
The four curves \(\Gamma _{ij}^t\) are parametrized by \(s\mapsto \Psi (r_{ij}(s),t)\) which we denote by \(r_{ij}^t(s)\) though s no longer represents arclength. Indeed one calculates that
from which it follows that
where \(\kappa _{ij}(s)\) denotes the curvature of \(\Gamma _{ij}\) at \(r_{ij}(s)\) (compare with (3.61)) and we have invoked the Frenet relations \(\tau _{ij}'=\kappa _{ij}\nu _{ij}\) and \(\nu _{ij}'=-\kappa _{ij}\tau _{ij}\). A related calculation goes to show that the unit normal \(\nu _{ij}^t\) to \(\Gamma _{ij}^t\) is given by
Now in the ball B(p, R) the unperturbed critical point is given by
and we wish to perturb it into a new function \(u^t\) given by
To carry this out, as in the previous proof, we extend the domain of definition of \(u_{j}\) to a neighborhood of \(\Omega _{j}\) in such a way that the extension is constant along the normals to the boundary of its original domain of definition. Then we introduce three functions \(\phi _1,\phi _2\) and \(\phi _3\) such that
so as to preserve the required \(\mathbb {S}^1\)-valued nature of \(u_j^t.\)
We must also take care to preserve the property \(u^t\in H_\mathrm {div}\,\) in the sense of (3.38) and this requires that the following four conditions hold to O(t) along \(\Gamma _{01},\Gamma _{12},\Gamma _{23}\) and \(\Gamma _{03}\) respectively:
We note that the first and last of these conditions implies at \(t=0\) that either \(u_1\equiv \tau _{01}\) or \(\equiv -\tau _{01}\) along \(\Gamma _{01}\) and likewise either \(u_3\equiv \tau _{03}\) or \(\equiv -\tau _{03}\) along \(\Gamma _{03}\).
Substituting (5.3) and (5.5) into the four conditions of (5.7), and expanding to O(t), a tedious but straight-forward calculation leads to the following requirements relating the traces of the \(\phi _j\) to \(h^{\nu }_{ij,s}\):
for \(s\in [0,s_0]\), where s in the subscript denotes the derivative with respect to s
With the perturbations of the four curves \(\Gamma _{ij}\) and three functions \(u_j^t\) defined, we are ready to compute the variation of \(E_0\) in a neighborhood of the junction point P. Carrying out the calculation (3.70) in \(\Omega _j\) for \(j=1,2,3\) and then applying the divergence theorem we find with the aid of (3.39) that
where we have used the fact that \(u_1^\perp \cdot \nu _{01}=u_1\cdot \tau _{01}\), \(u_1^\perp \cdot \nu _{12}=u_1\cdot \tau _{12}\), etc.
Now we appeal to the relations (5.8)–(5.11), along with the conditions \(u_2\cdot \tau _{12}=-u_1\cdot \tau _{12}\) and \(u_3\cdot \tau _{23}=-u_2\cdot \tau _{23}\) and perform an integration by parts to find
We turn now to calculating the variations of the four jump energies. We begin by invoking (5.4) to compute
Thus,
To compute the variation in the jump energies over \(\Gamma _{12}^t\) and \(\Gamma _{23}^t\) requires an expansion to O(t) of the quantities \(u^t\cdot \nu _{12}^t\) and \(u^t\cdot \nu _{23}^t.\) Substituting the expression for \(r_{12}^t\) from (5.3) into the formula for \(u_1^t\) from (5.6) and Taylor expanding in t we find with the use of (5.5) that along \(\Gamma _{12}^t\) we have
where \(u_1\) and \(\phi _1\) in the expression above are evaluated at \(x=r_{12}(s)\) and \(u_1'=\frac{{\hbox {d}}}{\hbox {d}s}u_1(r_{12}(s)).\) In the last line we have also used that our extension of \(u_1\) was constant along \(\nu _{12}\) to eliminate the term \(\nabla u_1\cdot \nu _{12}\) that would other have been present upon Taylor expanding.
Similarly, we calculate that along \(\Gamma _{23}\) we have
From (5.14) and (5.15), along with (5.4) we can compute that
Now, since
and
we have that
and
Using these last two identities in (5.16) and integrating by parts implies that
Then invoking the criticality condition (3.40) from Theorem 3.11 and integrating by parts we can rewrite this identity as
Now we can combine (5.12), (5.13) and (5.17), and through a use of the criticality conditions (3.53) and (3.54) of Theorem 3.14 we find that all integrals over the four curves \(\Gamma _{ij}\) drop, leaving only
Recall now that \(h^{\tau }_{01}(0)=X(p)\cdot \tau _{01}(0)\), \(h^{\nu }_{01}(0)=X(p)\cdot \nu _{01}(0)\), etc. Thus, the arbitrary value of the vector X(p), implies that a vanishing first variation \(\frac{{\hbox {d}}}{\hbox {d}t}_{|_t=0} E_0(u^t)=0\) leads to the necessary condition at a junction P of the form
where all quantities above are evaluated at the junction P. \(\quad \square \)
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Golovaty, D., Novack, M., Sternberg, P. et al. A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants. Arch Rational Mech Anal 236, 1739–1805 (2020). https://doi.org/10.1007/s00205-020-01501-x
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DOI: https://doi.org/10.1007/s00205-020-01501-x