An overview on constrained critical points of Dirichlet integrals

Giovanni Franzina; Lorenzo Brasco

arxiv: 1907.00882 · v1 · pith:B5CYIZ5Gnew · submitted 2019-07-01 · 🧮 math.AP · math.SP

An overview on constrained critical points of Dirichlet integrals

Lorenzo Brasco , Giovanni Franzina This is my paper

Pith reviewed 2026-05-25 11:54 UTC · model grok-4.3

classification 🧮 math.AP math.SP

keywords constrained critical pointsDirichlet integralL^q sphereeigenvalue problemLaplacianvariational methodscounterexamplesopen problems

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The pith

Critical values of the Dirichlet integral constrained to the unit L^q sphere generalize the Laplacian eigenvalue problem.

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

The paper examines the problem of locating critical points of the Dirichlet integral when the functions are restricted to the unit sphere in the L^q norm. This formulation is presented as a direct generalization of the classical Dirichlet eigenvalue problem, recovered when the exponent q equals 2. The authors assemble existing results on existence, regularity and qualitative properties of these points for varying q. They also supply counterexamples showing that several features familiar from the linear case do not persist, and they record a list of unresolved questions.

Core claim

We consider a natural generalization of the eigenvalue problem for the Laplacian with homogeneous Dirichlet boundary conditions. This corresponds to look for the critical values of the Dirichlet integral, constrained to the unit L^q sphere. We collect some results, present some counter-examples and compile a list of open problems.

What carries the argument

Critical points of the Dirichlet integral subject to the constraint that the L^q norm equals one.

If this is right

When q equals 2 the constrained problem reduces precisely to the standard Dirichlet eigenvalues.
For q not equal to 2 the first critical value need not be simple and the associated critical points need not be positive.
Explicit counterexamples exist showing failure of properties that hold in the linear case.
A collection of open questions remains concerning multiplicity, regularity, and limits as q varies.

Where Pith is reading between the lines

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

The same constraint technique might be applied to other quadratic or higher-order energies to produce analogous generalized spectra.
Asymptotic analysis of the critical values as q tends to 1 or infinity could connect to limiting problems in BV or L^infty settings.
Domain monotonicity or comparison principles might be tested numerically on simple geometries for selected q.

Load-bearing premise

The premise that the constrained critical-point problem on the L^q sphere forms a sufficiently rich and natural generalization of the classical Dirichlet eigenvalue problem to justify a dedicated overview.

What would settle it

A demonstration that, for every q different from 2, the critical points and their values coincide exactly with those of the linear eigenvalue problem or yield no additional phenomena.

Figures

Figures reproduced from arXiv: 1907.00882 by Giovanni Franzina, Lorenzo Brasco.

**Figure 1.** Figure 1: Proof of Theorem 4.4, case (ii). Case (i) is the simplest one: by taking z0 to be any interior point of Ω, by convexity we have w(x) = hx − z0, ∇U(x)i < 0, for HN−1−a. e. x ∈ ∂Ω. 9We recall that the proof of this result is based on the Courant-Fischer-Weyl min-max formula and the unique continuation principle for eigenfunctions. Both facts hold for the linearized operator −∆ − (q − 1) λ1(Ω; q)U q−1 , thus … view at source ↗

**Figure 2.** Figure 2: Proof of Theorem 4.4, case (iii). Here we used (4.14). Moreover, the normal derivative ∂φ/∂νΩ must have constant sign on ∂Ω. The last two informations, inserted in (4.18), entail that ∂φ ∂νΩ = 0, HN−1−a. e. on ∂Ω. This contradict Hopf’s boundary Lemma. In case (ii), we choose z0 ∈ R 2 \ Ω to be the intersection of the two supporting lines L0 and L1, see [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗

**Figure 3.** Figure 3: A non-convex set verifying the assumption of Remark 4.6. Finally, in case (iii), by assuming for simplicity that L0 and L1 are parallel to the x1 axis, we change the choice (4.15) of w and replace it with the following one w = ∂U ∂x1 . It is not difficult to see that (4.18) still holds11. Then one can proceed as in case (ii) and get the conclusion in this case, as well. Remark 4.6. A result analogous to Th… view at source ↗

**Figure 4.** Figure 4: The set Ωε of Example 4.7. The following example shows that the same phenomena can appear even if the set has a trivial topology. Indeed, observe that the sets Ωε below are contractible. More precisely, they are starshaped. This shows that the simplicity of λ1(Ω; q) for 2 < q < 2 ∗ is linked to the geometry of the underlying set Ω and not simply to its topology. Example 4.7. Let 2 < q < 2 ∗ and 0 < ε < 1, … view at source ↗

**Figure 5.** Figure 5: The graph of the barrier function ψ, neeeded to handle Example 4.7 in the case N = 2. In black, the boundary of the set Ωε. see [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗

read the original abstract

We consider a natural generalization of the eigenvalue problem for the Laplacian with homogeneous Dirichlet boundary conditions. This corresponds to look for the critical values of the Dirichlet integral, constrained to the unit $L^q$ sphere. We collect some results, present some counter-examples and compile a list of open problems.

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

This is a survey that gathers known results, counterexamples, and open problems on L^q-constrained critical points of the Dirichlet integral, without new theorems.

read the letter

The paper frames the search for critical values of the Dirichlet integral on the unit L^q sphere as a direct generalization of the classical Dirichlet eigenvalue problem and then collects results from the literature, adds some counterexamples, and lists open questions. That is the entire contribution. It does a service by pulling the material into one place so that someone entering the subfield does not have to chase every reference separately. The counterexamples and open-problem list are the parts most likely to be used by others. Beyond that there is no new analysis or proof technique. The central claim that the constraint produces a “natural” generalization is presented as motivation rather than a theorem that needs verification. Because the work is explicitly a survey, the main risk is whether the citations are represented accurately and whether the counterexamples are stated with enough detail to be checked; the abstract gives no indication of problems on that score. The paper will be useful to people already working on constrained variational problems or on L^q-eigenvalue questions in domains, and less useful to anyone outside that narrow slice. It is not the sort of manuscript that changes how the broader field thinks about eigenvalues. A journal that regularly publishes focused surveys in calculus of variations could reasonably send it to referees; the authors know the literature and the format is straightforward. I would not bring it to a general reading group or cite it for a new result, but I would keep the open-problem list handy if I started working in this area.

Referee Report

0 major / 3 minor

Summary. The manuscript is an overview that frames the search for critical values of the Dirichlet integral subject to the unit-sphere constraint in L^q (q > 1) as a natural generalization of the classical Dirichlet eigenvalue problem for the Laplacian. It collects existing results from the literature, presents selected counter-examples, and compiles a list of open problems.

Significance. If the compilation of results and counter-examples is accurate and reasonably complete, the paper could serve as a useful reference point for researchers working on variational methods for semilinear elliptic equations and constrained critical-point theory. The explicit listing of open problems is a concrete strength that may help focus future work.

minor comments (3)

The abstract states that the work 'collects some results' but does not indicate the range of q-values covered or the principal theorems being surveyed; a short clarifying sentence would improve readability.
Section headings and the organization of the open-problems list would benefit from explicit cross-references to the results or counter-examples that motivate each open question.
A few citations appear only in the bibliography and are not discussed in the text; either integrate them or move them to a 'further reading' subsection.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for recommending minor revision. The report provides a positive overall assessment but does not list any specific major comments requiring point-by-point replies. We are therefore in a position to address only the general recommendation.

Circularity Check

0 steps flagged

No circularity: survey paper with no derivations

full rationale

The manuscript is framed from the abstract onward as an overview that collects existing results, counter-examples, and open problems around a known generalization of the Dirichlet eigenvalue problem. No new technical derivations, predictions, or load-bearing claims are advanced whose validity could reduce to self-definition, fitted inputs, or self-citation chains. The premise that the constrained problem is a 'natural generalization' is a framing judgment, not a falsifiable derivation step. This is the most common honest non-finding for survey papers.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a survey paper; the central claim rests on the existence and interest of prior literature on the L^q-constrained Dirichlet integral problem. No free parameters, axioms, or invented entities are introduced by the authors.

pith-pipeline@v0.9.0 · 5553 in / 1112 out tokens · 53421 ms · 2026-05-25T11:54:20.334307+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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2008

[1] [1]

Bahri, P.-L

A. Bahri, P.-L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math., 45 (1992), 1205–1215. 37

1992

[2] [2]

Brasco, On principal frequencies and isoperimetric ratios in convex sets, to appear on Ann

L. Brasco, On principal frequencies and isoperimetric ratios in convex sets, to appear on Ann. Fac. Sci. Toulouse Math., available at http://cvgmt.sns.it/paper/3891/ 23

[3] [3]

Brasco, G

L. Brasco, G. De Philippis, G. Franzina, in preparation. 17

[4] [4]

Brasco, G

L. Brasco, G. Franzina, A pathological example in Nonlinear Spectral Theory, Adv. Nonlinear Anal., 8 (2019), 707–714. 13, 17, 18

2019

[5] [5]

Brasco, G

L. Brasco, G. Franzina, Convexity properties of Dirichlet integrals and Picone–type inequalities, Kodai Math. J., 37 (2014), 769–799. 15, 16

2014

[6] [6]

Brasco, E

L. Brasco, E. Parini, M. Squassina, Stability of variational eigenvalues for the fractional p−Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813–1845. 5

2016

[7] [7]

Brasco, B

L. Brasco, B. Ruﬃni, Compact Sobolev embeddings and torsion functions, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire,34 (2017), 817–843 7, 19

2017

[8] [8]

Brezis, L

H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55–64. 15

1986

[9] [9]

J. E. Brothers, W. P. Ziemer, Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384 (1988), 153–179. 21

1988

[10] [10]

Courant, D

R. Courant, D. Hilbert, Methods of mathematical physics. Vol. I. Wiley Classics Library. A Wiley- Interscience Publication. John Wiley & Sons, Inc., New York, 1989 Methods of Mathematical Physics, 28

1989

[11] [11]

Damascelli, M

L. Damascelli, M. Grossi, F. Pacella, Qualitative properties of positive solutions of semilinear ellip- tic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire,16 (1999), 631–652. 26, 30

1999

[12] [12]

E. N. Dancer, On the inﬂuence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann., 285 (1989), 647–669. 35

1989

[13] [13]

E. N. Dancer, The eﬀect of domain shape on the number of positive solutions of certain nonlinear equations, J. Diﬀ. Eq., 74 (1988), 120–156. 30, 35

1988

[14] [14]

Dr´ abek, R

P. Dr´ abek, R. Man´ asevich, On the closed solution to some nonhomogeneous eigenvalue problems with p−Laplacian, Diﬀerential Integral Equations, 12 (1999), 773–788. 5, 20 CONSTRAINED CRITICAL POINTS OF DIRICHLET INTEGRALS 41

1999

[15] [15]

Ercole, Sign-deﬁniteness of q−eigenfunctions for a super-linear p−Laplacian eigenvalue problem, Arch

G. Ercole, Sign-deﬁniteness of q−eigenfunctions for a super-linear p−Laplacian eigenvalue problem, Arch. Math., 103 (2014), 189–194. 36

2014

[16] [16]

L. C. Evans, R. Gariepy, Measure theory and ﬁne properties of functions . Studies in Advanced Math- ematics. CRC Press, Boca Raton, FL, 1992. 27

1992

[17] [17]

Franzina, P

G. Franzina, P. D. Lamberti, Existence and uniqueness for a p−Laplacian nonlinear eigenvalue prob- lem, Electron. J. Diﬀerential Equations, 26 (2010), pp. 1-10. 4, 6, 9, 21

2010

[18] [18]

Gidas, W

B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209–243. 22

1979

[19] [19]

Grisvard, Elliptic Problems in Nonsmooth Domains , Monographs and Studies in Mathematics, 24

P. Grisvard, Elliptic Problems in Nonsmooth Domains , Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. 27

1985

[20] [20]

Hardt, M

R. Hardt, M. Hoﬀmann-Ostenhof, T. Hoﬀmann-Ostenhof, N. Nadirashvili, Critical sets of solutions to elliptic equations, J. Diﬀerential Geom., 51 (1999), 359–373. 40

1999

[21] [21]

Henrot, Extremum problems for eigenvalues of elliptic operators

A. Henrot, Extremum problems for eigenvalues of elliptic operators . Frontiers in Mathematics. Birkhauser Verlag, Basel, 2006. 2

2006

[22] [22]

Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Dis- crete Contin

B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Dis- crete Contin. Dynam. Systems, 6 (2000), 683–690. 15, 20

2000

[23] [23]

M. K. Kwong, Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339–363. 23

1992

[24] [24]

Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2, Manuscripta Math., 84 (1994), 13–19

C.-S. Lin, Uniqueness of least energy solutions to a semilinear elliptic equation in R2, Manuscripta Math., 84 (1994), 13–19. 23, 26, 30, 36

1994

[25] [25]

Magnanini, An introduction to the study of critical points of solutions of elliptic and parabolic equations, Rend

R. Magnanini, An introduction to the study of critical points of solutions of elliptic and parabolic equations, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 121–166. 40

2016

[26] [26]

Maz’ya, Sobolev spaces, Sobolev spaces with applications to elliptic partial diﬀerential equations

V. Maz’ya, Sobolev spaces, Sobolev spaces with applications to elliptic partial diﬀerential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. 7

2011

[27] [27]

M¨ uller, On the behavior of the solutions of the diﬀerential equation ∆ U =F (x,U ) in the neigh- borhood of a point, Comm

C. M¨ uller, On the behavior of the solutions of the diﬀerential equation ∆ U =F (x,U ) in the neigh- borhood of a point, Comm. Pure Appl. Math., 7 (1954), 505–515. 28

1954

[28] [28]

A. I. Nazarov, The one-dimensional character of an extremum point of the Friedrichs inequality in spherical and plane layers, J. Math. Sci., 102 (2000), 4473–4486. 30

2000

[29] [29]

ˆOtani, On certain second order ordinary diﬀerential equations associated with Sobolev-Poincar´ e– type inequalities, Nonlinear Anal., 8 (1984), 1255–1270

M. ˆOtani, On certain second order ordinary diﬀerential equations associated with Sobolev-Poincar´ e– type inequalities, Nonlinear Anal., 8 (1984), 1255–1270. 5, 6, 20

1984

[30] [30]

Ulisse Dini

M. Struwe, Variational methods. Applications to nonlinear partial diﬀerential equations and Hamil- tonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. 5, 7, 14 (L. Brasco) Dipartimento di Matematica e Informatica Universit`a degli Studi di...

2008