Free-boundary regularity, general partitions, and grouping for the optimal partition problem of the principal eigenvalue of linearly coupled elliptic systems

Luo and Zhang (2019) proved that, for a four-equation Schrödinger system with linear coupling functions and nonlinear competitive couplings, the $\beta\to\infty$ limit configurations segregate into two groups, are Lipschitz continuous (via a variant of Almgren's monotonicity formula), satisfy local Pohozaev extremality conditions, and generate optimal partitions for the principal eigenvalue of linearly coupled elliptic systems. The fine regularity and geometry of the free boundary between cells, the general $k$-partition and $n$-component theory, and the complete description of grouping phenomena in the singular limit go beyond what that work establishes and remain open. This record poses that open problem: develop the interface regularity theory (regular hypersurfaces outside a small stratified singular set) for optimal partitions of principal eigenvalues of linearly coupled elliptic systems, carry the limit-configuration construction and analysis to general $k$ and general $n$, and classify the possible grouping and multiplicity patterns in the singular limit.

Background

An optimal partition problem asks for the best way to split a bounded domain $\Omega\subset\mathbb{R}^N$ into $k$ disjoint cells $\omega_1,\dots,\omega_k$, where 'best' means minimizing a functional of the cells — classically the sum of their first Dirichlet eigenvalues (the smallest values for which $-\Delta u=\lambda u$ with zero boundary data admits nontrivial solutions). Conti–Terracini–Verzini (2003) and Caffarelli–Lin (2006/2007) introduced this problem and conjectured that optimal interfaces are smooth hypersurfaces except on a small singular set. The union of the interfaces between cells is the free boundary, and its regularity is governed by two tools. Almgren's frequency function of a solution $u$ at a point is the ratio $N(r)=\frac{r\int_{B_r(x)}|\nabla u|^2}{\int_{B_r(x)}|u|^2}$, which measures the growth of $u$ on small balls; when $N(r)$ is monotone nondecreasing in $r$, blow-up limits of $u$ (rescalings zooming into $x$) are homogeneous functions, and this homogeneity is the engine of interface regularity. The Pohozaev identity is a Rellich-type integral identity on balls that, in partition problems, yields extremality conditions — geometric inequalities constraining admissible interfaces, analogous to first-variation conditions in shape optimization. For scalar (single-equation, Laplacian) spectral partitions the regularity program is complete in its main lines: Tavares–Terracini (2012) connected the problem to sign-changing solutions of competing elliptic systems; Ramos–Tavares–Terracini (2016) proved Lipschitz regularity of the (rescaled) eigenfunctions and regularity of the interfaces together with the extremality conditions; Alper (2020) proved that the singular set of the free interface has finite $(N-2)$-dimensional Hausdorff measure, is rectifiable, and admits a stratification analogous to that of minimal surfaces.

Luo–Zhang (2019) vectorized the construction side of this program. A Schrödinger system here means a system of coupled semilinear elliptic equations for components $u_1,\dots,u_n$ (stationary states of interacting populations or coupled Bose–Einstein condensates); the coupling is linear when the equations contain terms proportional to the other components with constant coefficients, and nonlinear competition of strength $\beta>0$ penalizes overlap of the supports. Luo–Zhang studied a four-equation system with linear coupling functions and nonlinear couplings, allowing the associated elliptic operators to be indefinite (not sign-definite). As $\beta\to\infty$ they proved that the limit configurations segregate — the components have asymptotically disjoint supports, clustered into two groups; a variant of Almgren's monotonicity formula yields Lipschitz continuity of the limit profiles; a local Pohozaev identity yields extremality conditions; and they used the limit profiles to construct an optimal partition for the principal eigenvalue of elliptic systems. For a linearly coupled elliptic system, the principal eigenvalue is the spectral value playing the role of the first eigenvalue, characterized through the existence of a positive eigenfunction vector for the coupled system. What the paper does not do: it develops no fine free-boundary regularity theory (interface regularity, size and stratification of the singular set), it treats only the case leading to segregation into two groups in detail (general $k$-partitions and general $n$-component systems are not carried out there), and it gives no complete description of grouping — the ways components may cluster into multiple groups in the limit. These are the open directions the problem below formalizes.

Problem Statement

Develop the regularity and structure theory for the optimal partition problem for the principal eigenvalue of linearly coupled elliptic systems introduced by Luo–Zhang (2019), whose formulation fixes the admissible class of partitions, the cost functional, and the notion of principal eigenvalue for possibly indefinite linearly coupled systems; the regularity and structure theory requested below is not developed in that work. Concretely:

1. Free-boundary regularity. Prove that the interfaces separating the cells of an optimal partition for the principal eigenvalue of the linearly coupled elliptic system are regular — locally smooth hypersurfaces outside a closed singular set — and establish the size and stratification of that singular set, in analogy with the scalar theory, where Alper proved the singular set has finite $(N-2)$-dimensional Hausdorff measure and admits a minimal-surface-type stratification. This includes the optimal regularity of the vector-valued eigenfunctions up to the interface.

2. General partitions. Carry out the limit-configuration construction, the monotonicity and Lipschitz analysis, and the extremality conditions for general $k$-partitions and general $n$-component systems; the published work treats in detail only the case in which the limit configurations segregate into two groups and does not carry out the general case.

3. Grouping classification. Give a complete description of the grouping phenomena in the singular limit $\beta\to\infty$: which clusterings of the components into groups can occur for optimal partitions of the system, and which multiplicity phenomena arise in the limit problem. Here a grouping is a partition of the $n$ components into groups of components whose supports coincide in the limit, and the multiplicity phenomena are the possible sizes of these groups (how many components may collapse into a common group) together with the number of distinct limit groupings that can occur.

These three deliverables form one coherent scientific objective — the detailed regularity and structure theory of the vectorial spectral partition problem. Extensions to broader system classes — different nonlinear interaction exponents, more general matrix-valued linear couplings with weaker regularity, and space dimensions beyond the treated regimes — are recorded as a natural continuation of this program, not as part of the completion boundary of the problem stated here.

References

1. Haijun Luo and Zhitao Zhang, 'Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems', Advanced Nonlinear Studies 19 (2019), 693–715. DOI 10.1515/ans-2019-2057, https://doi.org/10.1515/ans-2019-2057
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