Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

Jimmy Lamboley; Pieralberto Sicbaldi

arxiv: 1907.08159 · v1 · pith:WDSKVJ6Nnew · submitted 2019-07-18 · 🧮 math.AP · math.OC

Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

Jimmy Lamboley , Pieralberto Sicbaldi This is my paper

Pith reviewed 2026-05-24 19:32 UTC · model grok-4.3

classification 🧮 math.AP math.OC

keywords Faber-Krahn inequalityfirst Dirichlet eigenvalueshape optimizationRiemannian manifoldfree boundary problemregularityexistence

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The pith

Minimizers of the first Dirichlet eigenvalue for fixed volume exist in compact Riemannian manifolds and are smooth except possibly on a codimension-5-or-higher set.

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

The paper studies the minimization of the first Dirichlet eigenvalue of the Laplace-Beltrami operator among domains of fixed volume inside a general Riemannian manifold. It proves that such minimizers exist whenever the manifold is compact and supplies a counter-example showing that existence can fail when the manifold is non-compact. The authors then import techniques from free-boundary regularity theory to conclude that any minimizer is smooth away from a possible residual singular set whose codimension is at least five. This supplies the first general existence-plus-regularity statement for the spectral shape-optimization problem outside the Euclidean setting.

Core claim

In any compact Riemannian manifold, a domain of prescribed volume that minimizes the first Dirichlet eigenvalue exists; moreover, every such minimizer is smooth up to a possible residual set of codimension five or higher, obtained by viewing the problem as a free-boundary problem for the eigenfunction.

What carries the argument

The first Dirichlet eigenvalue functional λ₁(Ω) together with the free-boundary regularity theory that yields smoothness outside a codimension-5 set.

If this is right

Existence holds on every compact Riemannian manifold.
The same codimension bound on the singular set that is known in Euclidean space continues to hold on curved manifolds.
Non-existence is possible once the manifold is allowed to be non-compact.
The overdetermined boundary condition satisfied by the eigenfunction on the free boundary remains the same as in the Euclidean case.

Where Pith is reading between the lines

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

Numerical shape-optimization algorithms on compact manifolds can safely restrict attention to smooth domains.
The same existence-plus-regularity statement is likely to hold for other spectral functionals whose Euler-Lagrange conditions produce free-boundary problems.
If the singular set is shown to be empty in a given manifold, then every minimizer is globally smooth.

Load-bearing premise

The tools already developed for free-boundary problems apply directly to the eigenvalue minimization problem without any extra curvature or injectivity-radius assumptions on the manifold.

What would settle it

A concrete example, in some compact Riemannian manifold, of a volume-constrained λ₁-minimizer whose singular set has codimension four or less.

read the original abstract

In this paper, we study the minimization of $\lambda_{1}(\Omega)$, the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets $\Omega$ of fixed volume in a Riemmanian manifold $(M,g)$. In the Euclidian setting (when $(M,g)=(\mathbb{R}^n,e)$), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds $(M,g)$. In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

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.

Desk Editor's Note private letter to a colleague

The paper gives existence of Faber-Krahn minimizers on compact manifolds plus a codim-5 regularity result via free-boundary methods.

read the letter

The two things to know are that the authors prove existence of volume-constrained minimizers for the first Dirichlet eigenvalue on any compact Riemannian manifold, and they obtain a regularity theorem stating that the minimizer is smooth outside a residual set of codimension at least 5. They also supply a counter-example showing existence can fail when the manifold is non-compact. These are the concrete new statements in the abstract. The existence part follows the usual direct-method route once one has suitable compactness for sets of fixed volume, which is standard once the manifold is compact. The regularity step recasts the problem as a free-boundary obstacle problem for the eigenfunction and invokes existing free-boundary theory. That move is reasonable and explains the codimension-5 threshold, which matches what is known in the Euclidean free-boundary literature. The work is therefore a direct extension of the classical Faber-Krahn theory to the manifold setting where explicit solutions are unavailable. The main soft spot is the transfer of free-boundary regularity. The Laplace-Beltrami operator in normal coordinates carries curvature-dependent lower-order terms that can affect monotonicity formulas and blow-up analysis. The abstract states the result for arbitrary compact manifolds without listing curvature or injectivity-radius hypotheses, so the reduction step needs explicit verification in the proofs. If those terms are controlled or absorbed, the claim holds; if not, the codimension-5 conclusion may require extra assumptions. The paper is aimed at researchers in spectral geometry and shape optimization who already know the Euclidean theory and free-boundary techniques. A reader looking for the first general existence and regularity statements on manifolds will find the results useful. The claims are specific enough and the approach is grounded enough that the paper deserves a serious referee rather than a desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper proves existence of minimizers for the first Dirichlet eigenvalue of the Laplace-Beltrami operator subject to a fixed-volume constraint on compact Riemannian manifolds, supplies a counter-example showing that existence can fail on certain non-compact manifolds, and establishes partial regularity: any minimizer is smooth outside a closed residual set of Hausdorff dimension at most n-5. The regularity argument proceeds by reducing the shape-optimization problem to a free-boundary obstacle problem for the first eigenfunction and invoking existing free-boundary regularity theorems.

Significance. If the reduction and the transfer of free-boundary techniques are justified, the work supplies the first general existence-plus-partial-regularity theory for the Faber-Krahn problem on arbitrary compact Riemannian manifolds, paralleling the classical regularity theory for the isoperimetric problem. The explicit counter-example for non-compact manifolds is a useful clarification of the compactness requirement.

major comments (2)

[Regularity theory section] Regularity theory section (the paragraph beginning 'we then focus on the regularity theory...'): The claim that solutions are C^∞ outside a residual set of codimension ≥5 rests on reducing the eigenvalue minimization to a free-boundary obstacle problem and directly quoting Euclidean free-boundary results. The manuscript does not verify that the lower-order curvature terms arising from the Laplace-Beltrami operator in geodesic normal coordinates preserve the Weiss-type monotonicity formula or the blow-up classification needed for the dimension-reduction argument. This verification is load-bearing for the codimension-5 statement, since the cited free-boundary theorems are typically proved under vanishing curvature or under uniform bounds on sectional curvature and positive injectivity radius.
[Existence theorem for compact manifolds] Existence theorem for compact manifolds (the statement and proof sketch of the existence result): The direct-method argument relies on lower semi-continuity of λ₁ with respect to L¹-convergence of characteristic functions. The manuscript should explicitly confirm that the lower-order terms coming from the metric do not destroy the necessary compactness or semi-continuity when the manifold is only assumed compact and without boundary, rather than assuming the result follows verbatim from the Euclidean case.

minor comments (2)

[Abstract] Abstract: 'Riemmanian' is misspelled.
Notation: the symbol for the first eigenvalue is written both as λ₁(Ω) and λ_{1}(Ω); a single consistent notation should be adopted throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments. We address each major comment below and will revise the manuscript accordingly to strengthen the justifications.

read point-by-point responses

Referee: The claim that solutions are C^∞ outside a residual set of codimension ≥5 rests on reducing the eigenvalue minimization to a free-boundary obstacle problem and directly quoting Euclidean free-boundary results. The manuscript does not verify that the lower-order curvature terms arising from the Laplace-Beltrami operator in geodesic normal coordinates preserve the Weiss-type monotonicity formula or the blow-up classification needed for the dimension-reduction argument.

Authors: We agree that a more explicit verification is desirable. In the revised manuscript, we will include a detailed computation showing that in geodesic normal coordinates, the perturbation terms from the metric are of lower order and do not interfere with the monotonicity of the Weiss functional or the classification of homogeneous blow-ups, as these terms vanish upon rescaling. This follows from the smoothness of the metric and standard estimates in Riemannian geometry. We will add this as a subsection in the regularity part. revision: yes
Referee: The direct-method argument relies on lower semi-continuity of λ₁ with respect to L¹-convergence of characteristic functions. The manuscript should explicitly confirm that the lower-order terms coming from the metric do not destroy the necessary compactness or semi-continuity when the manifold is only assumed compact and without boundary, rather than assuming the result follows verbatim from the Euclidean case.

Authors: On a compact Riemannian manifold, the volume measure is comparable to the Euclidean one locally with uniform constants due to compactness, and the gradient term in the Rayleigh quotient is controlled similarly. The lower semi-continuity of the eigenvalue under L1 convergence of sets can be proved by approximating with the Euclidean case locally and using the compactness of the manifold to cover with finitely many charts. We will add a short paragraph in the existence section to detail this argument rather than referring to the Euclidean case. revision: yes

Circularity Check

0 steps flagged

No circularity; regularity follows from external free-boundary theory

full rationale

The paper establishes existence on compact manifolds and regularity (smoothness outside a residual set of codimension at least 5) by reducing the Faber-Krahn minimization problem to a free-boundary obstacle problem for the first eigenfunction and then invoking standard tools from free-boundary regularity theory. No derivation step in the abstract or described chain reduces the claimed results to a fitted parameter, a self-citation load-bearing premise, or a definition that presupposes the output. The cited free-boundary results are treated as independent external input rather than self-referential or author-overlapping theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard facts from Riemannian geometry and elliptic PDE theory; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math The first Dirichlet eigenvalue of the Laplace-Beltrami operator is well-defined and continuous with respect to domain variation in the Riemannian setting.
Invoked implicitly for the minimization problem to make sense.
domain assumption Compactness of the manifold implies compactness in the space of domains of fixed volume for the direct method in the calculus of variations.
Used to obtain the existence result on compact manifolds.

pith-pipeline@v0.9.0 · 5734 in / 1333 out tokens · 30472 ms · 2026-05-24T19:32:33.026211+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

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[1] [1]

Aguilera, H

N. Aguilera, H. W . Alt, and L. A. Caffarelli. An optimizat ion problem with volume constraint. SIAM J. Control Optim., 24(2):191–198, 1986

work page 1986

[2] [2]

F. J. Almgren. Existence and regularity almost everywhe re of solutions to elliptic variational problems with con- straints. Mem. AMS, 1976

work page 1976

[3] [3]

H. W . Alt and L. A. Caffarelli. Existence and regularity f or a minimum problem with free boundary. J. Reine Angew. Math., 325:105–144, 1981

work page 1981

[4] [4]

H. W . Alt, L. A. Caffarelli, and A. Friedman. A free bounda ry problem for quasilinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 11(1):1–44, 1984

work page 1984

[5] [5]

T. Aubin. Some nonlinear problems in Riemannian geometry . Springer Monographs in Mathematics. Springer- V erlag, Berlin, 1998

work page 1998

[6] [6]

Benson, I

B. Benson, I. Blank, and J. LeCrone. Mean value theorems f or riemannian manifolds via the obstacle problem. Journal of geometric analysis , 2017

work page 2017

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Blank and Z

I. Blank and Z. Hao. The mean value theorem and basic prope rties of the obstacle problem for divergence form elliptic operators. Comm. Anal. Geom. , 23(1):129–158, 2015

work page 2015

[8] [8]

Bonﬁglioli and E

A. Bonﬁglioli and E. Laconelli. Subharmonic functions i n sub-riemannian settings. J. Eur . Math. Soc., 15:387–441, 2013

work page 2013

[9] [9]

Briançon

T. Briançon. Regularity of optimal shapes for the Dirich let’s energy with volume constraint. ESAIM Control Optim. Calc. V ar ., 10(1):99–122 (electronic), 2004

work page 2004

[10] [10]

Briançon, M

T. Briançon, M. Hayouni, and M. Pierre. Lipschitz conti nuity of state functions in some optimal shaping. Calc. V ar . Partial Differential Equations, 23(1):13–32, 2005

work page 2005

[11] [11]

Briançon and J

T. Briançon and J. Lamboley. Regularity of the optimal s hape for the ﬁrst eigenvalue of the Laplacian with volume and inclusion constraints. Ann. Inst. H. Poincaré Anal. Non Linéaire , 26(4):1149–1163, 2009

work page 2009

[12] [12]

D. Bucur. Do optimal shapes exist? Milan J. Math., 75:379–398, 2007

work page 2007

[13] [13]

Buttazzo and G

G. Buttazzo and G. Dal Maso. An existence result for a cla ss of shape optimization problems. Arch. Rational Mech. Anal., 122(2):183–195, 1993

work page 1993

[14] [14]

Cañete and M

A. Cañete and M. Ritoré. The isoperimetric problem in co mplete annuli of revolution with increasing Gauss curvature. Proc. Roy. Soc. Edinburgh Sect. A, 138(5):989–1003, 2008

work page 2008

[15] [15]

L. A. Caffarelli, D. Jerison, and C.E. Kenig. Global ene rgy minimizers for free boundary problems and full regularity in three dimensions. In Noncompact problems at the intersection of geometry, analy sis, and topology , volume 350 of Contemp. Math., pages 83–97. Amer. Math. Soc., Providence, RI, 2004

work page 2004

[16] [16]

I. Chavel. Eigenvalues in Riemannian geometry , volume 115 of Pure and Applied Mathematics . Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Rando l, With an appendix by Jozef Dodziuk

work page 1984

[17] [17]

David, M

G. David, M. Engelstein, and T. Toro. Free boundary regu larity for almost-minimizers. Preprint, 2017

work page 2017

[18] [18]

David and T

G. David and T. Toro. Regularity of almost minimizers wi th free boundary. Calc. V ar . Partial Differential Equa- tions, 54(1):455–524, 2015

work page 2015

[19] [19]

O. S. de Queiroz and L. S. Tavares. Almost minimizers for semilinear free boundary problems with variable coefﬁcients. Mathematische Nachrichten, 291(10):1486–1501, 2018

work page 2018

[20] [20]

De Silva

D. De Silva. Free boundary regularity for a problem with right hand side. Interfaces Free Bound., 13(2):223–238, 2011

work page 2011

[21] [21]

De Silva and D

D. De Silva and D. Jerison. A singular energy minimizing free boundary. to appear in J. Reine Angew. Math. , 2008

work page 2008

[22] [22]

Delay and P

E. Delay and P . Sicbaldi. Extremal domains for the ﬁrst e igenvalue in a general compact riemannian manifold. Discr . Cont. Dyn. Syst., Series A, 35(12):5799–5825, 2015. 31

work page 2015

[23] [23]

O. Druet. Asymptotic expansion of the faber-krahn proﬁ le of a compact riemannian manifold. C. R. Acad. Sci. Paris, Ser . I, 346:1163–1167, 2008

work page 2008

[24] [24]

L. C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics . American Mathemati- cal Society, Providence, RI, second edition, 2010

work page 2010

[25] [25]

L. C. Evans and R. F. Gariepy. Measure theory and ﬁne properties of functions . Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992

work page 1992

[26] [26]

Gilbarg and N

D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order . Classics in Mathematics. Springer-V erlag, Berlin, 2001. Reprint of the 1998 edition

work page 2001

[27] [27]

E. Giusti. Minimal surfaces and functions of bounded variation , volume 80 of Monographs in Mathematics . Birkhäuser V erlag, Basel, 1984

work page 1984

[28] [28]

Gonzalez, U

E. Gonzalez, U. Massari, and I. Tamanini. On the regular ity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math., 32:25–37, 1983

work page 1983

[29] [29]

M. Grüter. Boundary regularity for solutions of a parti oning problem. Arch. Rat. Mech. Anal., 97:261–270, 1987

work page 1987

[30] [30]

M. Hayouni. Sur la minimisation de la première valeur pr opre du laplacien. C. R. Acad. Sci. Paris Sér . I Math. , 330(7):551–556, 2000

work page 2000

[31] [31]

Henrot and M

A. Henrot and M. Pierre. V ariation et Optimisation de F ormes, volume 16 of Springer Series in Computational Mathematics. Springer, 2005. Une analyse géométrique

work page 2005

[32] [32]

Henrot (editor)

A. Henrot (editor). Shape optimization and Spectral theory . De Gruyter, 2017

work page 2017

[33] [33]

J. A. Iglesias and G. Mercier. Inﬂuence of dimension on t he convergence of level-sets intotal variation regulariza - tion. Preprint, 2018

work page 2018

[34] [34]

Jerison and O

D. Jerison and O. Savin. Some remarks on stability of con es for the one-phase free boundary problem. Geom. Funct. Anal., 25(4):1240–1257, 2015

work page 2015

[35] [35]

Kinderlehrer and L

D. Kinderlehrer and L. Nirenberg. Regularity in free bo undary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4(2):373–391, 1977

work page 1977

[36] [36]

Lamboley and P

J. Lamboley and P . Sicbaldi. New examples of extremal do mains for the ﬁrst eigenvalue of the laplace-beltrami operator in a riemannian manifold with boundary. International Mathematics Research Notices , (18):8752–8798, 2015

work page 2015

[37] [37]

Mazzoleni, S

D. Mazzoleni, S. Terracini, and B. V elichkov. Regulari ty of the optimal sets for some spectral functionals. Geom. Funct. Anal., 27(2):373–426, 2017

work page 2017

[38] [38]

Morabito and P

F. Morabito and P . Sicbaldi. Delaunay type domains for a n overdetermined elliptic problem in Sn × R and Hn × R. ESAIM: COCV, 22(1):1–28, 2016

work page 2016

[39] [39]

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