For a bounded open set D⊂RdD\subset\mathbb{R}^d with Lipschitz boundary, k≥2k\geq 2 and Robin parameter β>0\beta>0, consider the shape optimization problem of minimizing ∑i=1kλ1(Ωi,β)\sum_{i=1}^{k}\lambda_1(\Omega_i,\beta) over kk-tuples of mutually disjoint open sets contained in DD, where λ1(Ω,β)\lambda_1(\Omega,\beta) is the first eigenvalue of the Laplacian with Robin boundary condition ∂νu+βu=0\partial_\nu u+\beta u=0. Bucur, Fragalà and Giacomini (2018) proved that an optimal partition exists and, as a first regularity step, that the jump sets of the interfaces are Ahlfors regular and essentially closed. They posed, as the natural open question stemming from their existence theorem, establishing the regularity of the free boundaries in an optimal kk-tuple, and suggested investigating whether the one-phase Robin free-boundary regularity results (Caffarelli–Kriventsov; Kriventsov) remain valid in this multiphase context.

Background

The Robin Laplacian on an open set Ω⊂Rd\Omega\subset\mathbb{R}^d is the Laplace operator supplemented with the boundary condition ∂νu+βu=0\partial_\nu u+\beta u=0 on ∂Ω\partial\Omega, where ν\nu is the outward unit normal and β>0\beta>0 is a fixed parameter. Unlike the Dirichlet condition (u=0u=0), the Robin condition couples the Dirichlet and Neumann data through the boundary, and the first eigenvalue admits the variational characterization λ1(Ω,β)=inf⁡u∈H1(Ω)∖{0}∫Ω∣∇u∣2 dx+β∫∂Ωu2 dHd−1∫Ωu2 dx,\lambda_1(\Omega,\beta)=\inf_{u\in H^1(\Omega)\setminus\{0\}}\frac{\int_\Omega|\nabla u|^2\,dx+\beta\int_{\partial\Omega}u^2\,d\mathcal{H}^{d-1}}{\int_\Omega u^2\,dx}, so that boundary area enters the energy directly; here Hd−1\mathcal{H}^{d-1} denotes (d−1)(d-1)-dimensional Hausdorff measure.

The optimal partition problem studied by Bucur, Fragalà and Giacomini is inf⁡{∑i=1kλ1(Ωi,β) : (Ω1,…,Ωk)∈A(D)},\inf\left\{\sum_{i=1}^{k}\lambda_1(\Omega_i,\beta)\ :\ (\Omega_1,\dots,\Omega_k)\in\mathcal{A}(D)\right\}, where A(D)\mathcal{A}(D) is the family of kk-tuples of pairwise disjoint open subsets of a given bounded open set DD with Lipschitz boundary, each with Hd−1\mathcal{H}^{d-1}-countably rectifiable boundary of finite measure. The problem is a multiphase free discontinuity problem: the unknown includes both the cells Ωi\Omega_i and their shared interfaces. Its analogue for the first Dirichlet eigenvalue — the theory of spectral minimal partitions initiated by Conti–Terracini–Verzini, Caffarelli–Lin and Helffer–Hoffmann-Ostenhof–Terracini — is well developed, and for the Dirichlet case a nearly complete regularity theory of optimal partitions is now available: Spolaor, Trey and Velichkov proved full regularity of the interfaces in dimension two, and Ognibene and Velichkov subsequently settled boundary points and the local structure near triple junctions (three C1,αC^{1,\alpha}-arcs meeting at 120∘120^\circ), as summarized in their 2026 survey.

For the Robin problem, Bucur–Fragalà–Giacomini (Calc. Var. PDE 2018) proved existence of an optimal kk-tuple through a relaxed formulation posed on vector-valued functions of special bounded variation (SBV — the class of functions whose discontinuities concentrate on a countably rectifiable jump set) whose components, the first eigenfunctions of the cells, have mutually disjoint supports. Viewing the eigenfunctions as a vector field, the interfaces of the partition appear as its jump set. They showed that each cell is open, that the union of the interfaces is essentially closed, and that the jump sets are Ahlfors regular in DD — meaning that the (d−1)(d-1)-measure of the interface inside any small ball centered on it is comparable above and below by ρd−1\rho^{d-1}, a quantitative flatness-free size estimate obtained via Mumford–Shah-type quasi-minimality (the jump set behaves like that of a minimizer of the Mumford–Shah image-segmentation energy). This is the current state of knowledge: nothing is known about the higher regularity of the interfaces, that is, whether they are smooth hypersurfaces outside a thin singular set.

Two neighboring theories indicate both the target and the obstacles. On the one hand, for one-phase free boundary problems with Robin-type boundary conditions — notably the thermal insulation problem, in which one minimizes a Dirichlet energy plus a boundary penalization over one unknown phase — Caffarelli and Kriventsov proved that the free boundary is locally a smooth hypersurface near flat'' points, and Kriventsov later proved that flatness propagates (flat implies smooth''), yielding C1,αC^{1,\alpha} regularity of a large portion of the free boundary. Bucur–Fragalà–Giacomini suggested investigating whether these one-phase regularity results remain valid in their multiphase partition setting. On the other hand, for two-phase free boundary problems with a Robin transmission condition across the interface (a condition relating the jump of the normal derivatives of the two phases to the jump of the solution, arising in energy problems but not in the spectral partition problem), Guarino Lo Bianco, La Manna and Velichkov recently obtained existence and regularity results in dimension two. The multiphase spectral Robin case combines the difficulties of both neighbors: several phases compete, the governing state functions are eigenfunctions whose boundary condition lives on the unknown interface itself, and monotonicity-formula and epiperimetric-inequality techniques (inequalities comparing a blow-up cone with a competitor, which drive modern free-boundary regularity proofs) have not yet been adapted to Robin jump conditions at multi-junction points.

Problem Statement

Let D⊂RdD\subset\mathbb{R}^d be a bounded open set with Lipschitz boundary, k≥2k\geq 2, and β>0\beta>0. Let (Ω1,…,Ωk)(\Omega_1,\dots,\Omega_k) be an optimal kk-tuple for the partition problem min⁡{∑i=1kλ1(Ωi,β) : (Ω1,…,Ωk)∈A(D)},\min\left\{\sum_{i=1}^{k}\lambda_1(\Omega_i,\beta)\ :\ (\Omega_1,\dots,\Omega_k)\in\mathcal{A}(D)\right\}, as constructed by Bucur–Fragalà–Giacomini, where A(D)\mathcal{A}(D) is the class of kk-tuples of pairwise disjoint open subsets of DD with Hd−1\mathcal{H}^{d-1}-countably rectifiable boundaries, and λ1(Ω,β)\lambda_1(\Omega,\beta) is the first eigenvalue of the Laplacian with Robin boundary condition ∂νu+βu=0\partial_\nu u+\beta u=0 on ∂Ω\partial\Omega. Establish the regularity of the free boundaries of an optimal kk-tuple — the interfaces ∂Ωi∩D\partial\Omega_i\cap D along which neighboring cells meet, equivalently the jump set of the vector of first eigenfunctions of the cells — beyond the known first step of Ahlfors regularity and essential closedness, and investigate whether the one-phase Robin free-boundary regularity results of Caffarelli–Kriventsov and Kriventsov remain valid in this multiphase setting. The regularity question is whether the interfaces are of class C1,αC^{1,\alpha} outside an explicitly identified singular set (specifying the points covered and the dimension), or whether the one-phase Robin free-boundary regularity results of Caffarelli–Kriventsov and Kriventsov fail to extend to this multiphase setting. In the Dirichlet benchmark, a complete regularity theory comprised smoothness of the interfaces outside a small singular set (in dimension two, a finite set), classification of the blow-up limits of the interfaces at interface points, the local structure at junctions where three or more cells meet, and regularity up to the fixed boundary; this benchmark is listed for orientation only, not as a set of separately mandatory deliverables.

References

  1. Dorin Bucur, Ilaria Fragalà, Alessandro Giacomini, "Optimal partitions for Robin Laplacian eigenvalues", Calculus of Variations and Partial Differential Equations 57 (2018), Art. 122. DOI 10.1007/s00526-018-1393-z, https://doi.org/10.1007/s00526-018-1393-z (preprint arXiv:1803.03813, https://arxiv.org/abs/1803.03813)
  2. Dorin Bucur, Ilaria Fragalà, "On the honeycomb conjecture for Robin Laplacian eigenvalues", Communications in Contemporary Mathematics 21 (2019), 1850007. DOI 10.1142/S0219199718500074, https://doi.org/10.1142/S0219199718500074
  3. Alessandro Giacomini, Dorin Bucur, Ilaria Fragalà, "Multiphase free discontinuity problems: monotonicity formula and regularity results", Annales de l'Institut Henri Poincaré C, Analyse non linéaire 38 (2021), 1553–1582. DOI 10.1016/j.anihpc.2020.12.003, https://doi.org/10.1016/j.anihpc.2020.12.003
  4. Dorin Bucur, Alessandro Giacomini, "Minimization of the k-th eigenvalue of the Robin-Laplacian", Journal of Functional Analysis 277 (2019), 643–687. DOI 10.1016/j.jfa.2019.04.009, https://doi.org/10.1016/j.jfa.2019.04.009
  5. Luis A. Caffarelli, Dennis Kriventsov, "A free boundary problem related to thermal insulation", Communications in Partial Differential Equations 41 (2016), 1149–1182. DOI 10.1080/03605302.2016.1199038, https://doi.org/10.1080/03605302.2016.1199038
  6. Dennis Kriventsov, "A free boundary problem related to thermal insulation: flat implies smooth", Calculus of Variations and Partial Differential Equations 58 (2019), Art. 78. DOI 10.1007/s00526-019-1509-0, https://doi.org/10.1007/s00526-019-1509-0
  7. Serena Guarino Lo Bianco, Domenico Angelo La Manna, Bozhidar Velichkov, "A two-phase problem with Robin conditions on the free boundary", Journal de l'École polytechnique — Mathématiques 8 (2021), 1–25. DOI 10.5802/jep.139, https://doi.org/10.5802/jep.139
  8. Serena Guarino Lo Bianco, Domenico Angelo La Manna, Bozhidar Velichkov, "Free boundary cluster with Robin condition on the transmission interface", Calculus of Variations and Partial Differential Equations 64 (2024), Art. 4. DOI 10.1007/s00526-024-02864-9, https://doi.org/10.1007/s00526-024-02864-9 (preprint arXiv:2204.00381, https://arxiv.org/abs/2204.00381)
  9. Luca Spolaor, Baptiste Trey, Bozhidar Velichkov, "Free boundary regularity for a multiphase shape optimization problem", Communications in Partial Differential Equations 45 (2020), 77–108. DOI 10.1080/03605302.2019.1658773, https://doi.org/10.1080/03605302.2019.1658773 (preprint arXiv:1810.06963, https://arxiv.org/abs/1810.06963)
  10. Roberto Ognibene, Bozhidar Velichkov, "Boundary regularity of the free interface in spectral optimal partition problems", preprint (2024), arXiv:2404.05698, https://arxiv.org/abs/2404.05698
  11. Roberto Ognibene, Bozhidar Velichkov, "Structure of the free interfaces near triple junction singularities in harmonic maps and optimal partition problems", Archive for Rational Mechanics and Analysis 250 (2026), Art. 77. DOI 10.1007/s00205-026-02221-4, https://doi.org/10.1007/s00205-026-02221-4 (preprint arXiv:2412.00781, https://arxiv.org/abs/2412.00781)
  12. Roberto Ognibene, Bozhidar Velichkov, "A survey on the optimal partition problem", La Matematica 5 (2026), Art. 6. DOI 10.1007/s44007-025-00176-8, https://doi.org/10.1007/s44007-025-00176-8 (preprint arXiv:2510.08241, https://arxiv.org/abs/2510.08241)
  13. James B. Kennedy, João P. Ribeiro, "Cheeger cuts and Robin spectral minimal partitions of metric graphs", Journal d'Analyse Mathématique 158 (2025), 111–145. DOI 10.1007/s11854-025-0419-7, https://doi.org/10.1007/s11854-025-0419-7 (preprint arXiv:2310.02701, https://arxiv.org/abs/2310.02701)
  14. Mickaël Nahon, "Existence and regularity of optimal shapes for spectral functionals with Robin boundary conditions", Journal of Differential Equations 335 (2022), 69–102. DOI 10.1016/j.jde.2022.07.001, https://doi.org/10.1016/j.jde.2022.07.001
  15. Simone Cito, Alessandro Giacomini, "Minimization of the k-th eigenvalue of the Robin-Laplacian with perimeter constraint", Calculus of Variations and Partial Differential Equations 63 (2024), Art. 244. DOI 10.1007/s00526-024-02872-9, https://doi.org/10.1007/s00526-024-02872-9
  16. Hanna N. Kim, Richard S. Laugesen, "Two disks maximize the third Robin eigenvalue: positive parameters", Annales mathématiques du Québec 49 (2025), 403–419. DOI 10.1007/s40316-025-00254-x, https://doi.org/10.1007/s40316-025-00254-x (preprint arXiv:2501.15029, https://arxiv.org/abs/2501.15029)
  17. M. Conti, S. Terracini, G. Verzini, "An optimal partition problem related to nonlinear eigenvalues", Journal of Functional Analysis 198 (2003), 160–196. DOI 10.1016/S0022-1236(02)00105-2, https://doi.org/10.1016/S0022-1236(02)00105-2
  18. Luis A. Caffarelli, Fang Hua Lin, "An optimal partition problem for eigenvalues", Journal of Scientific Computing 31 (2007), 5–18. DOI 10.1007/s10915-006-9114-8, https://doi.org/10.1007/s10915-006-9114-8
  19. B. Helffer, T. Hoffmann-Ostenhof, S. Terracini, "Nodal domains and spectral minimal partitions", Annales de l'Institut Henri Poincaré, Analyse non linéaire 26 (2009), 101–138. DOI 10.1016/j.anihpc.2007.07.004, https://doi.org/10.1016/j.anihpc.2007.07.004
  20. Luca Esposito, Lorenzo Lamberti, Giovanni Pisante, "Ahlfors-regularity for minimizers of a multiphase optimal design problem", preprint (2025), arXiv:2506.15861, https://arxiv.org/abs/2506.15861