pith. sign in

arxiv: 2605.00694 · v1 · submitted 2026-05-01 · 🧮 math.AP · math.OC

Unstable free boundary problems in optimal control theory: existence and regularity

Lorenzo Ferreri , Idriss Mazari-Fouquer , Rapha\"el Prunier This is my paper

Pith reviewed 2026-05-09 18:51 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords optimal controlfree boundary regularitybang-bang propertysemilinear elliptic equationsunstable free boundary problemsexistence and regularity
0
0 comments X

The pith

Optimal controls for a broad class of semilinear elliptic problems are bang-bang with free boundaries that are smooth outside sets of dimension at most d-2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that maximizers of integrals of the form ∫ ψ(Θ_m) - c ∫ m, where Θ_m solves the semilinear equation -ΔΘ_m = m Θ_m + B(x, Θ_m) under the pointwise constraint 0 ≤ m ≤ 1, must be characteristic functions χ_{E*}. It further shows that the free boundary ∂E* is smooth except possibly on a closed set of Hausdorff dimension at most d-2. A parallel result holds for the volume-constrained version of the same problem, where in two dimensions the free boundary reduces to a finite union of smooth curves. The argument proceeds by recasting the optimality condition as an unstable free boundary problem and then establishing the required non-degeneracy and regularity properties for blow-ups. If these statements hold, many control problems in mathematical physics and biology reduce to the geometric task of selecting an optimal domain rather than an arbitrary density function.

Core claim

We prove that for a large class of problems of the form maximize ∫ ψ(Θ_m) - c ∫ m subject to -ΔΘ_m = m Θ_m + B(x, Θ_m) and 0 ≤ m ≤ 1 a.e., the solution m* equals χ_{E*} and ∂E* is smooth up to a (d-2)-dimensional subset. For the volume-constrained analogue maximize ∫ ψ(Θ_m) subject to the same PDE, the same pointwise bound on m, and fixed ∫ m = m_0, the optimal m* is again bang-bang and, when d = 2, ∂E* is a finite union of smooth curves. The proof is obtained by reducing the problem to an unstable free boundary problem whose regularity is analyzed via a new combination of optimal-control, free-boundary, and measure-theoretic tools that secure non-degeneracy of blow-ups even though the free

What carries the argument

Reduction of the optimal-control problem to an unstable free-boundary problem, followed by blow-up analysis that yields non-degeneracy despite a sign-changing Laplacian and the absence of energy minimization.

If this is right

  • The optimal control is necessarily 0 or 1 almost everywhere.
  • The free boundary is C^∞ outside a closed singular set of Hausdorff dimension at most d-2.
  • In the two-dimensional volume-constrained case the free boundary consists of finitely many C^∞ curves.
  • The same regularity conclusions apply to a large family of nonlinearities B and payoff functions ψ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction technique may apply to other control problems whose state equations are semilinear but whose optimality conditions produce non-variational free boundaries.
  • Numerical schemes could be designed to optimize directly over characteristic functions of domains whose boundaries satisfy the derived regularity.
  • The non-degeneracy arguments developed for sign-changing Laplacians could be tested on related free-boundary problems arising in phase-transition models.
  • If the singular set is empty for generic data, many optimal designs in applications would possess globally smooth interfaces.

Load-bearing premise

The reduction to the unstable free-boundary problem succeeds in proving non-degeneracy of blow-ups even though the free boundary is not minimizing and the Laplacian of the state function changes sign.

What would settle it

An explicit example in which an optimal m* takes a value strictly between 0 and 1 on a positive-measure set, or in which the free boundary of E* contains a singularity of positive (d-1)-dimensional Hausdorff measure in dimension d > 2, would disprove the main claims.

Figures

Figures reproduced from arXiv: 2605.00694 by Idriss Mazari-Fouquer, Lorenzo Ferreri, Rapha\"el Prunier.

Figure 1
Figure 1. Figure 1: Illustration of the construction of a “long” part of the nodal set. on the sequence {η˜rk ◦ Rk}k∈IN for a suitable sequence of rotations {Rk}k∈IN, we assume without loss of generality that (3.3.2) ˜ηrk → k→∞ η˜0 in C 1,γ . Remark 3.3.1. Of course, we say we can make this assumption without loss of generality, but this is because we only consider here the length of the nodal set {η = 0}. In general, the que… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the repeating patterns. In particular, B0 = 0. As I0 is the only sign component of length π, we can use (4.1.1) and apply the analysis of Case 1 to S(0; 1) \ I0 to obtain a finite sequence of sign components {Ik}1≤k≤N such that Ik = (θ0,k; θ1,k) with π = θ0,1 < θ1,1 < · · · < θ1,N = 2π, and the analysis of Case 1 entails φ ′ ̸= 0 on ∪ N k=1 ∂Ik. In particular, φ ′ (0) ̸= 0, in contradiction… view at source ↗
Figure 3
Figure 3. Figure 3: In two dimensions, such an accumulation of smaller and smaller compo￾nents can not happen. In two dimensions, this would allow to conclude that there exists only a finite number of connected components. In higher dimensions, this does not allow to conclude. 4.4. Regularity in higher dimensions for penalised problems This section is devoted to the proof of the only part of Theorem 2 that remains to be dealt… view at source ↗
read the original abstract

We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form ``maximise $\int \psi(\Theta_m)-c\int m$ where $-\Delta \Theta_m=m\Theta_m+B(x,\Theta_m)$, under the constraint $0\leq m\leq 1$ a.e.", the solution $m^*$ is bang-bang, in the sense that $m^*=\chi_{E^*}$, and that $\partial E^*$ is smooth up to a $(d-2)$-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem ``maximise $\int \psi(\Theta_m)$ where $-\Delta \Theta_m=m\Theta_m+B(x,\Theta_m)$, under the constraint $0\leq m\leq 1$ a.e and $\int m=m_0$" are bang-bang in the sense that $m^*=\chi_{E^*}$ and that, in the two-dimensional case, $\partial E^*$ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau \& Weiss and Chanillo, Kenig \& To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes the first general regularity results for a class of constrained optimal control problems of the form maximize ∫ ψ(Θ_m) − c ∫ m subject to −ΔΘ_m = m Θ_m + B(x, Θ_m) with 0 ≤ m ≤ 1 a.e. It proves that the optimal control m* is bang-bang (m* = χ_{E*}) and that the free boundary ∂E* is smooth outside a (d−2)-dimensional singular set. For the volume-constrained variant (fixed ∫ m = m_0), the same bang-bang property holds, and in two dimensions ∂E* consists of finitely many smooth curves. The argument proceeds by reduction to an unstable free-boundary problem, with a new non-degeneracy analysis for blow-ups that blends optimal-control, free-boundary, and measure-theoretic tools to handle the non-minimizing character of the functional and the sign-changing Laplacian.

Significance. If the central claims hold, the paper supplies the first general existence-plus-regularity theory for these unstable free-boundary problems arising in mathematical physics and biology. The explicit handling of non-degeneracy in a non-minimizing, sign-changing setting extends the classical results of Monneau–Weiss and Chanillo–Kenig–To and demonstrates a workable synthesis of optimal-control and free-boundary techniques. This opens the door to further applications and to analogous regularity statements for related control problems.

minor comments (3)
  1. The abstract states that the non-degeneracy step is obtained by 'blending tools from optimal control theory, free boundary and measure theory,' but the main text should contain an explicit roadmap (e.g., a diagram or numbered list of lemmas) showing precisely which tool supplies which estimate; this would make the logical flow easier to follow.
  2. In the volume-constrained case the two-dimensional regularity statement is stated only for d=2; a brief remark on the obstruction to extending the finite-union-of-curves conclusion to higher dimensions would be helpful.
  3. The bibliography should include full citations for the cited works of Monneau–Weiss and Chanillo–Kenig–To, together with any recent extensions that are used in the blow-up analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive report, including the accurate summary of our results and the recognition of their significance in extending regularity theory to unstable free-boundary problems in optimal control. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent external regularity theorems plus novel non-degeneracy argument

full rationale

The paper reduces the optimal-control problem to an unstable free-boundary problem and obtains bang-bang optimality plus almost-everywhere smoothness of the free boundary. The regularity step invokes external results of Monneau & Weiss and Chanillo-Kenig-To, which are independent of the present authors. The only novel technical content is the non-degeneracy of blow-ups in the sign-changing, non-minimizing setting; this is achieved by a blend of optimal-control, free-boundary and measure-theoretic tools that is not shown to reduce to any fitted parameter, self-definition, or self-citation chain. No equation or lemma in the supplied text equates a claimed output to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard elliptic regularity for the state PDE and existence of optimal controls for the given class; the novelty lies in adapting these to the unstable sign-changing setting rather than introducing new axioms or entities.

axioms (2)
  • standard math Elliptic regularity theory applies to solutions of the state equation −ΔΘ_m = m Θ_m + B(x, Θ_m)
    Invoked to analyze the PDE satisfied by the state variable Θ_m
  • domain assumption Optimal controls exist for the large class of problems considered
    The regularity statements are made for solutions whose existence is presupposed

pith-pipeline@v0.9.0 · 5602 in / 1446 out tokens · 49193 ms · 2026-05-09T18:51:21.024197+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    Andersson, H

    J. Andersson, H. Shahgholian, and G. S. Weiss. Uniform regularity close to cross singularities in an unstable free boundary problem.Comm. Math. Phys., 296(1):251–270, 2010

    work page 2010

  2. [2]

    Andersson, H

    J. Andersson, H. Shahgholian, and G. S. Weiss. On the singularities of a free boundary through Fourier expansion. Invent. Math., 187(3):535–587, 2012

    work page 2012

  3. [3]

    Andersson and G

    J. Andersson and G. S. Weiss. Cross-shaped and degenerate singularities in an unstable elliptic free boundary problem.Journal of Differential Equations, 228:633–640, 2006

    work page 2006

  4. [4]

    J. E. Andersson, H. Shahgholian, and G. S. Weiss. The singular set of higher dimensional unstable obstacle type problems.Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., 24(1):123–146, 2013

    work page 2013

  5. [5]

    Barles and C

    G. Barles and C. Georgelin. A simple proof of convergence for an approximation scheme for computing motions by mean curvature.SIAM Journal on Numerical Analysis, 32(2):484–500, apr 1995

    work page 1995

  6. [6]

    Berestycki, F

    H. Berestycki, F. Hamel, and L. Roques. Analysis of the periodically fragmented environment model. I: Species persistence.J. Math. Biol., 51(1):75–113, 2005

    work page 2005

  7. [7]

    Berestycki, L

    H. Berestycki, L. Nirenberg, and S. R. S. Varadhan. The principal eigenvalue and maximum principle for second- order elliptic operators in general domains. InCollected papers. Volume II: PDE, SDE, diffusions, random media. Edited by Rajendra Bhatia, Abhay Bhatt and K. R. Parthasarathy, pages 388–433. Berlin: Springer; New Dehli: Hindustan Book Agency, 2012

    work page 2012

  8. [8]

    Bintz and S

    J. Bintz and S. Lenhart. Optimal resource allocation for a diffusive population model.Journal of Biological Systems, 28(04):945–976, dec 2020

    work page 2020

  9. [9]

    Braida, J

    B. Braida, J. Dalphin, C. Dapogny, P. Frey, and Y. Privat. Shape and topology optimization for maximum proba- bility domains in quantum chemistry.Numerische Math., To appear

    work page

  10. [10]

    Buttazzo, J

    G. Buttazzo, J. Casado-D´ ıaz, and F. Maestre. On the regularity of optimal potentials in control problems governed by elliptic equations.Adv. Calc. Var., 17(4):1341–1364, 2024

    work page 2024

  11. [11]

    Buttazzo, J

    G. Buttazzo, J. Casado-D´ ıaz, and F. Maestre. Optimization problems for elliptic PDEs. Preprint, arXiv:2601.01591 [math.OC] (2026), 2026

    work page arXiv 2026

  12. [12]

    L. A. Caffarelli. The regularity of free boundaries in higher dimensions.Acta Math., 139:155–184, 1978

    work page 1978

  13. [13]

    R. S. Cantrell and C. Cosner. Diffusive logistic equations with indefinite weights: Population models in disrupted environments II.SIAM Journal on Mathematical Analysis, 22(4):1043–1064, jul 1991

    work page 1991

  14. [14]

    R. S. Cantrell and C. Cosner. The effects of spatial heterogeneity in population dynamics.J. Math. Biol., 29(4):315– 338, 1991

    work page 1991

  15. [15]

    C´ ea, A

    J. C´ ea, A. Gioan, and J. Michel. Quelques resultats sur l'identification de domaines.Calcolo, 10(3-4):207–232, sep 1973

    work page 1973

  16. [16]

    Chambolle, I

    A. Chambolle, I. Mazari-Fouquer, and Y. Privat. Stability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems.Math. Ann., 392(3):4181–4219, 2025

    work page 2025

  17. [17]

    Chanillo, D

    S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes.Commun. Math. Phys., 214(2):315–337, 2000. FREE BOUNDARY PROBLEMS IN OPTIMAL CONTROL 45

    work page 2000

  18. [18]

    Chanillo, D

    S. Chanillo, D. Grieser, and K. Kurata. The free boundary problem in the optimization of composite membranes. InDifferential geometric methods in the control of partial differential equations. Proceedings of the 1999 AMS- IMS-SIAM joint summer research conference, University of Colorado, Boulder, CO, USA, June 27–July 01, 1999, pages 61–81. Providence, RI...

    work page 1999

  19. [19]

    Chanillo and C

    S. Chanillo and C. E. Kenig. Weak uniqueness and partial regularity for the composite membrane problem.J. Eur. Math. Soc. (JEMS), 10(3):705–737, 2008

    work page 2008

  20. [20]

    Chanillo, C

    S. Chanillo, C. E. Kenig, and T. To. Regularity of the minimizers in the composite membrane problem inR 2.J. Funct. Anal., 255(9):2299–2320, 2008

    work page 2008

  21. [21]

    Chemin.Perfect incompressible fluids

    J.-Y. Chemin.Perfect incompressible fluids. Transl. from the French by Isabelle Gallagher and Dragos Iftimie, volume 14 ofOxf. Lect. Ser. Math. Appl.Oxford: Clarendon Press, 1998

    work page 1998

  22. [22]

    S. J. Cox and J. R. McLaughlin. Extremal eigenvalue problems for composite membranes, i.Applied Mathematics & Optimization, 22(1):153–167, July 1990

    work page 1990

  23. [23]

    J. M. Diamond. The island dilemma: Lessons of modern biogeographic studies for the design of natural reserves. Biological Conservation, 7(2):129–146, Feb. 1975

    work page 1975

  24. [24]

    L. C. Evans. Convergence of an algorithm for mean curvature motion.Indiana University mathematics journal, pages 533–557, 1993

    work page 1993

  25. [25]

    H. Federer. The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension.Bull. Am. Math. Soc., 76:767–771, 1970

    work page 1970

  26. [26]

    Ferreri, D

    L. Ferreri, D. Mazzoleni, B. Pellacci, and G. Verzini. Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem.J. Math. Pures Appl. (9), 205:33, 2026. Id/No 103815

    work page 2026

  27. [27]

    Ferreri and G

    L. Ferreri and G. Verzini. Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditions.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 224:25, 2022. Id/No 113103

    work page 2022

  28. [28]

    Ferreri and G

    L. Ferreri and G. Verzini. Asymptotic properties of an optimal principal Dirichlet eigenvalue arising in population dynamics.J. Funct. Anal., 287(7):51, 2024. Id/No 110543

    work page 2024

  29. [29]

    Henrot and M

    A. Henrot and M. Pierre.Shape variation and optimization. A geometrical analysis, volume 28 ofEMS Tracts Math.Z¨ urich: European Mathematical Society (EMS), 2018

    work page 2018

  30. [30]

    Heo and Y

    J. Heo and Y. Kim. On the fragmentation phenomenon in the population optimization problem.Proc. Am. Math. Soc., 149(12):5211–5221, 2021

    work page 2021

  31. [31]

    Hinterm¨ uller, C.-Y

    M. Hinterm¨ uller, C.-Y. Kao, and A. Laurain. Principal eigenvalue minimization for an elliptic problem with indefinite weight and robin boundary conditions.Applied Mathematics & Optimization, 65(1):111–146, dec 2011

    work page 2011

  32. [32]

    H. Ishii. A generalization of the Bence, Merriman and Osher algorithm for motion by mean curvature.Curvature flows and related topics, pages 111–127, 1995

    work page 1995

  33. [33]

    Ishii, G

    H. Ishii, G. E. Pires, and P. E. Souganidis. Threshold dynamics type approximation schemes for propagating fronts. Journal of the Mathematical Society of Japan, 51(2), apr 1999

    work page 1999

  34. [34]

    Jost.Partial differential equations, volume 214 ofGrad

    J. Jost.Partial differential equations, volume 214 ofGrad. Texts Math.New York, NY: Springer, 3rd revised and expanded ed. edition, 2013

    work page 2013

  35. [36]

    C.-Y. Kao, Y. Lou, and E. Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains.Math. Biosci. Eng., 5(2):315–335, 2008

    work page 2008

  36. [37]

    Kao and S

    C.-Y. Kao and S. A. Mohammadi. Extremal rearrangement problems involving Poisson’s equation with Robin boundary conditions.Journal of Scientific Computing, 86(3), feb 2021

    work page 2021

  37. [38]

    C.-Y. Kao, S. A. Mohammadi, and B. Osting. Linear convergence of a rearrangement method for the one-dimensional Poisson equation.J. Sci. Comput., 86(1):18, 2021. Id/No 6

    work page 2021

  38. [39]

    T. Kato. Schr¨ odinger operators with singular potentials.Isr. J. Math., 13, 1973

    work page 1973

  39. [40]

    Kinderlehrer and L

    D. Kinderlehrer and L. Nirenberg. Regularity in free boundary problems.Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 4:373–391, 1977

    work page 1977

  40. [41]

    Kriventsov and G

    D. Kriventsov and G. S. Weiss. Rectifiability, finite Hausdorff measure, and compactness for non-minimizing Bernoulli free boundaries.Commun. Pure Appl. Math., 78(3):545–591, 2025

    work page 2025

  41. [42]

    Lam and Y

    K.-Y. Lam and Y. Lou.Introduction to reaction-diffusion equations. Theory and applications to spatial ecology and evolutionary biology. Lect. Notes Math. Model. Life Sci. Cham: Springer, 2022

    work page 2022

  42. [43]

    Lamboley, A

    J. Lamboley, A. Laurain, G. Nadin, and Y. Privat. Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions.Calc. Var. Partial Differ. Equ., 55(6):37, 2016. Id/No 144

    work page 2016

  43. [44]

    Laux and F

    T. Laux and F. Otto. Convergence of the thresholding scheme for multi-phase mean-curvature flow.Calculus of Variations and Partial Differential Equations, 55(5):1–74, 2016

    work page 2016

  44. [45]

    Laux and D

    T. Laux and D. Swartz. Convergence of thresholding schemes incorporating bulk effects.Interfaces and Free Bound- aries, 19(2):273–304, 2017

    work page 2017

  45. [46]

    Y. Lou. On the effects of migration and spatial heterogeneity on single and multiple species.J. Differ. Equations, 223(2):400–426, 2006

    work page 2006

  46. [47]

    Mazari, G

    I. Mazari, G. Nadin, and Y. Privat. Optimal location of resources maximizing the total population size in logistic models.J. Math. Pures Appl. (9), 134:1–35, 2020. 46 LORENZO FERRERI, IDRISS MAZARI-FOUQUER, AND RAPHA ¨EL PRUNIER

    work page 2020

  47. [48]

    Mazari, G

    I. Mazari, G. Nadin, and Y. Privat. Optimisation of the total population size for logistic diffusive equations: bang- bang property and fragmentation rate.Commun. Partial Differ. Equations, 47(4):797–828, 2022

    work page 2022

  48. [49]

    Mazari, G

    I. Mazari, G. Nadin, and Y. Privat. Some challenging optimization problems for logistic diffusive equations and their numerical modeling. InNumerical control. Part A, pages 401–426. Amsterdam: Elsevier/North Holland, 2022

    work page 2022

  49. [50]

    Mazari and D

    I. Mazari and D. Ruiz-Balet. A fragmentation phenomenon for a nonenergetic optimal control problem: optimization of the total population size in logistic diffusive models.SIAM J. Appl. Math., 81(1):153–172, 2021

    work page 2021

  50. [51]

    Mazari-Fouquer

    I. Mazari-Fouquer. Existence of optimal shapes in parabolic bilinear optimal control problems.Arch. Ration. Mech. Anal., 248(2):26, 2024. Id/No 18

    work page 2024

  51. [52]

    Mazzoleni, B

    D. Mazzoleni, B. Pellacci, and G. Verzini. Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems.SIAM J. Math. Anal., 55(4):4162–4192, 2023

    work page 2023

  52. [53]

    Merriman, J

    B. Merriman, J. Bence, and S. Osher. Diffusion generated motion by mean curvature. In L. A. Department of Mathematics, University of California, editor,CAM Report 92-33, 1992

    work page 1992

  53. [54]

    Monneau and G

    R. Monneau and G. S. Weiss. An unstable elliptic free boundary problem arising in solid combustion.Duke Math. J., 136(2):321–341, 2007

    work page 2007

  54. [55]

    G. Nadin. Optimization of the L1 norm of the solution of a Fisher-KPP equations in the small diffusivity regime. Preprint, arXiv:2112.02876 [math.AP] (2026), 2026

    work page arXiv 2026

  55. [56]

    Nagahara and E

    K. Nagahara and E. Yanagida. Maximization of the total population in a reaction–diffusion model with logistic growth.Calculus of Variations and Partial Differential Equations, 57(3):80, Apr 2018

    work page 2018

  56. [57]

    Pegon.Branched transport and fractal structures

    P. Pegon.Branched transport and fractal structures. Theses, Universit´ e Paris Saclay (COmUE), Nov. 2017

    work page 2017

  57. [58]

    Petrosyan, H

    A. Petrosyan, H. Shahgholian, and N. Uraltseva.Regularity of free boundaries in obstacle-type problems, volume 136 ofGrad. Stud. Math.Providence, RI: American Mathematical Society (AMS), 2012

    work page 2012

  58. [59]

    Pironneau.Optimal Shape Design for Elliptic Systems

    O. Pironneau.Optimal Shape Design for Elliptic Systems. Springer Berlin Heidelberg, 1984

    work page 1984

  59. [60]

    S. J. Ruuth, B. Merriman, J. Xin, and S. Osher. Diffusion-generated motion by mean curvature for filaments. Journal of Nonlinear Science, 11(6):473–493, 2001

    work page 2001

  60. [61]

    Shahgholian

    H. Shahgholian. The singular set for the composite membrane problem.Commun. Math. Phys., 271(1):93–101, 2007

    work page 2007

  61. [62]

    Soave and S

    N. Soave and S. Terracini. The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problem.Adv. Math., 334:243–299, 2018

    work page 2018

  62. [63]

    Soave and G

    N. Soave and G. Tortone. On the nodal set of solutions to some sublinear equations without homogeneity.Arch. Ration. Mech. Anal., 248(2):33, 2024. Id/No 26

    work page 2024

  63. [64]

    Swartz and N

    D. Swartz and N. K. Yip. Convergence of diffusion generated motion to motion by mean curvature.Communications in Partial Differential Equations, 42(10):1598–1643, sep 2017

    work page 2017

  64. [65]

    G. S. Weiss. Partial regularity for weak solutions of an elliptic free boundary problem.Commun. Partial Differ. Equations, 23(3-4):439–455, 1998

    work page 1998

  65. [66]

    limiting Weiss functional

    M. Yousefnezhad, C.-Y. Kao, and S. A. Mohammadi. Optimal chemotherapy for brain tumor growth in a reaction- diffusion model.SIAM Journal on Applied Mathematics, 81(3):1077–1097, jan 2021. Appendices AppendixA.Proof of some technical results A.1.Proof of Lemma 2.1.2. Proof of Lemma 2.1.2.We simply give the main elements: considerm∈ Mandh∈L ∞(Ω); define, fo...

    work page 2021