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Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values

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Pith's one-line read The short-time decay of heat solutions with nonconstant boundary values is governed by the distance to the set where the boundary function is positive, extending the classical asymptotics and driving rigidity results for time-invariant surf

desk verdict Core Varadhan extension for nonnegative data is real and worth knowing; the sign-changing symmetry theorem is false as stated, so the paper needs a clean separation of the two parts. read the letter →

arxiv 2607.18784 v1 pith:HW6STUST submitted 2026-07-21 math.AP

Diego Berti , Rolando Magnanini , Michele Marini This is my paper · ORCID
classification math.AP MSC 35K2035K0535J2535B4035B0635J05
keywords heatequationshort-timeasymptoticsnonconstantboundarydatadistancefunctiontime-invariantsurfacesrigidityviscositysolutionscontent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the classical small-time asymptotics for the heat equation to Cauchy–Dirichlet problems with non-constant boundary data that may vanish or change sign. Its main theorem states that when the initial value is zero, the short-time decay of the solution at a point is governed by the intrinsic distance to the support of the positive part of the boundary data, not by the distance to the whole boundary. The same quantity controls the slow-diffusion limit of solutions to the associated elliptic resolvent problem. The paper also derives curvature-dependent asymptotics for the heat content of spheres touching the boundary, and uses these to prove symmetry and non-existence results for time-invariant surfaces.

What carries the argument

The Hopf–Cole change of variables V_ε = −ε log U_ε converts the elliptic problem into a singularly perturbed Hamilton–Jacobi equation; compactness of the family {V_ε} via uniform gradient bounds, together with stability of viscosity solutions of the eikonal equation, forces the limit to be the distance function d_S. A Laplace-transform lemma transfers the elliptic asymptotics to the parabolic one. For the heat content, explicit radial solutions and comparison arguments give the curvature-dependent leading term. For rigidity, the level-set condition u=c(t) implies the invariant surface is a level set of d_S, and the mean-value property of U_ε turns this into a curvature constraint that, by a

What would settle it

Take the heat equation in a ball with boundary data odd with respect to a hyperplane, e.g. f(x',x_N)=x_N. On the equatorial plane the solution is identically zero for all t, so −4t log|u|=+∞, whereas d_{S_+}=d_{S_-} is finite there; hence the claimed finite asymptotic limit in Corollary 1.2 cannot hold on the equidistant set without extra hypotheses.

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Extended reading notes

Core claim

For a bounded C^{1,1} domain, zero initial data, and non-negative continuous boundary data f, the paper proves lim_{t→0+}(−4t log u(x,t)) = d_S(x)^2 and lim_{ε→0+}(−ε log U^ε(x)) = d_S(x), where S={f>0} and d_S is the intrinsic Riemannian distance to S. The limits hold uniformly on compact subsets of Ω∪S. For sign-changing f, the same asymptotic is governed by the closer of the positive and negative supports, except where the two distances tie.

Load-bearing premise

The proof of the main rigidity theorem for general boundary values assumes the non-negativity of the boundary data, an assumption the theorem statement does not make.

Editorial extensions

If this is right

  • If a surface Γ is time-invariant for the heat equation with non-negative boundary data f, then Γ must be parallel to the distance-level sets of the positivity set S; in particular, Γ cannot touch the boundary at points where f=0.
  • For sign-changing data, the short-time asymptotics selects the dominant sign: a point is governed by whichever of S_+ or S_- is closer; on the equidistant set the behavior is degenerate and can be infinite.
  • The heat-content formula extends the constant-data result to non-constant f, showing that the leading term factors as f(q) times a curvature term involving 1/R−κ_i(q).
  • Two time-invariant surfaces, together with uniform ellipticity and regularity, force the domain to be a ball and the boundary data to be constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The formulas suggest a route to inverse problems: short-time measurements of the heat solution could in principle recover the support S of the positive boundary data, since the exponential decay rate encodes d_S.
  • In the equidistant case d_{S_+}=d_{S_-}, the paper leaves open the question of pointwise convergence; examples with odd symmetric data show that the limit can be +∞ on a whole hyperplane, so a full asymptotic theory would need additional hypotheses on the zeros of f.
  • The sign-changing rigidity theorem as stated appears to require non-negative data; extending it would demand a different mechanism to rule out invariant surfaces touching the boundary when f changes sign.
  • The equidistant set G behaves like a shock in the eikonal limit; a selection principle for the exponential rate would likely require tracking the vanishing order of f at the common boundary points.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the heat equation with variable coefficients in non-divergence form, with zero initial data and continuous Dirichlet data f. The main result (Theorem 1.1) extends Varadhan's short-time asymptotics to nonnegative nonconstant boundary data: if S={f>0}, then -4t log u → d_S^2, with an elliptic analogue for the resolvent. For sign-changing data, Corollary 1.2 gives a formula away from the set where the distances to S_+ and S_- coincide; the paper explicitly leaves the tie-set case open (Remark 3.9). It also proves heat-content asymptotics on spheres tangent to ∂Ω in the Euclidean case (Theorem 1.3) and derives rigidity/non-existence results for time-invariant surfaces (Theorems 1.4–1.6).

Significance. If correct, the extended Varadhan formula is a genuine advance: it replaces the usual distance-to-the-boundary by the intrinsic distance to the positivity set of the boundary data, and the viscosity/barrier proof appears coherent. The heat-content formulas with explicit principal-curvature factors are concrete and should be useful in geometric applications. The paper is also honest about its limitations, explicitly flagging the tie-set case in Remark 3.9 and the possibility of self-intersecting invariant surfaces in Remark 4.10. However, one of the advertised symmetry results, Theorem 1.4, is false as stated, because it silently relies on an f≥0 assumption that its proof does not have.

major comments (2)
  1. [§1.5 (Theorem 1.4) and §4.2 (Lemma 4.5)] Theorem 1.4 is stated for arbitrary f∈C(∂Ω), f≠0, but its proof invokes Lemma 4.5, which is proved only for non-negative f. The strong maximum principle step c(t)>0 in Lemma 4.5(i) fails when f changes sign. Concrete counterexample: take Ω=B_1(0) and f(x)=x_N on ∂Ω. By odd symmetry, u(x',x_N,t)=-u(x',-x_N,t), so u≡0 on Γ={x_N=0}∩Ω for all t>0. Let D={x∈Ω:x_N>0}. Then ∂D is connected, ∂D∩Ω=Γ≠∅, the interior cone condition holds, and (1.4) holds with c≡0. The conclusion that D and Ω are concentric balls is false. Thus Theorem 1.4 is false as stated. It should be restricted to f≥0 (as the abstract already suggests) or the conclusion must be weakened; if restricted to f≥0, the proof appears repairable.
  2. [§1.3, Corollary 1.2] Corollary 1.2 claims the limit for x∈Ω∪S, but the displayed formula only covers the cases d_S+(x)<d_S-(x) or d_S+(x)>d_S-(x). On the tie set, no limit is asserted; indeed Remark 3.9(i) gives a symmetric sign-changing datum for which U^ε≡0 on an entire interior surface, so -ε log|U^ε|=+∞ there, while d_S is finite. The corollary should be restated with the explicit hypothesis d_S+(x)≠d_S-(x) and should refer to Remark 3.9 for the open tie-set case. As written, the statement is overbroad and could mislead applications.
minor comments (4)
  1. [Theorem 1.3, Eq. (1.12)] The limit in (1.12) is written as ε→0+, but the formula concerns t→0+; the proof in §3.3 uses Lemma A.2 with t→0+. Please correct the limit symbol.
  2. [Lemma 4.7 heading] The heading reads “Analitycity of invariant surfaces”; should be “Analyticity of invariant surfaces.”
  3. [Remark 4.10 and Theorem 1.6 proof] Minor typos: “charachterization” should be “characterization” in Remark 4.10; “suriective” should be “surjective” in the proof of Theorem 1.6.
  4. [Theorem 3.6 proof] The proof defines F_β(t)=u(x,t)/f, where f is apparently the maximum of the boundary datum. Since f is also the boundary datum, please denote the maximum by, say, \bar f to avoid ambiguity, especially in the case where f vanishes on part of ∂Ω.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main Varadhan-type asymptotics are derived from first principles, and the cited prior work is independent published support rather than a circular input. The sign-changing flaw in Theorem 1.4 is a correctness gap, not circular reasoning.

full rationale

The derivation of the main asymptotics is self-contained and non-circular. The distance d_S is defined geometrically in Section 2.3 before any asymptotics are proved; Theorem 3.5 establishes V = d_S by viscosity comparison and barrier arguments (Lemmas 3.2-3.4), and Theorem 3.6 transfers the result to u via the Laplace-transform lemma A.1. No quantity appearing in (1.7)-(1.8) is used as an input, and no fitted parameter is renamed as a prediction. The heat-content formulas in Theorem 1.3 are proved by comparison with the classical f-equivalent-1 solution W_epsilon, whose asymptotics are quoted from [30, Theorem 2.3]; the boundary-value dependence f(q) is obtained by sandwiching f between m_delta and M_delta and letting delta to 0, not by assuming the conclusion. The symmetry results invoke [30, Theorem 1.1], [30, Lemma 3.1], and [1,36]; these are prior published theorems with independent proofs, so they are real evidence rather than self-referential circularity. The notable weakness is a non-circular correctness gap: Theorem 1.4 is stated for f in C(partial Omega), f not identically zero, but its proof applies Lemma 4.5, whose hypotheses require f non-negative ('Suppose that f in C(partial Omega) is non-negative...'). For odd boundary data on a ball, the equatorial plane is a time-invariant surface touching partial Omega, contradicting the claimed sphere conclusion. This affects the truth of the theorem, not the circularity of the derivation. Hence score 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its load-bearing premises are standard PDE regularity, comparison principles, explicitly stated geometric assumptions on the domain, and an explicit genericity assumption on the equidistance set G in the sign-changing theorem.

assumptions (8)
  • domain assumption A(x) is uniformly elliptic, differentiable and locally Lipschitz (Section 2.1(a)-(b)).
    Needed for the intrinsic distance, viscosity solutions, and all barrier estimates.
  • domain assumption Ω is a bounded domain with ∂Ω ∈ C^{1,1} for the main Varadhan extension (Theorem 1.1) and for Theorem 1.6.
    Lemmas 3.2-3.5 require C^{1,1} boundary for uniform sphere and barrier arguments.
  • domain assumption f ∈ C(∂Ω) is non-negative with S = {f>0} nonempty (Theorems 1.1, 3.5, 3.6).
    Defines the positivity set; comparison principles and monotonicity of F_β rely on f ≥ 0.
  • standard math Viscosity-solution theory: stability, comparison, and Bernstein gradient estimates for equation (3.1).
    Used without proof in Lemmas 3.2-3.4, citing [10,11,12].
  • standard math Laplace-transform and Tauberian lemmas of Varadhan [37, Theorem 4.7] and Feller [13, Ch. XIII] (Lemmas A.1, A.2).
    Bridges the elliptic resolvent asymptotics to the parabolic solution and heat content.
  • standard math Alexandrov/Soap Bubble theorem A.3 for Wirtinger-type surfaces.
    Used in Theorem 1.5 to conclude spheres from the curvature relation.
  • standard math Large-time spectral expansion of the heat equation and Alikakos L∞ bounds (Section 4.1).
    Used in Theorem 4.1 and Corollary 4.2 to force f constant on ∂Ω.
  • domain assumption For Theorem 1.6: G_reg is a finite union of C^1 (N−1)-surfaces and H^{N−1}(G_sing)=0.
    Explicit genericity hypothesis on the equidistance set; Example 2.4 shows G may have interior in some configurations.

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Pith. "Pith review of Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values." pith.science (2026). https://pith.science/paper/HW6STUST

@misc{pith2026260718784,
  author       = {Pith},
  title        = {Pith review of: Short-time behavior and invariant surfaces for caloric functions with non-constant boundary values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HW6STUST}},
  note         = {Machine review of arXiv:2607.18784}
}
read the original abstract

We consider a Cauchy-Dirichlet problem for a heat equation with variable coefficients in non-divergence form. The initial values are assumed to be homogeneous, while the Dirichlet boundary values are non-constant. In this setting, we derive an asymptotic formula for the short-time behavior of the solution, which extends the celebrated Varadhan formula. We stress the fact that the boundary values are allowed to vanish or change sign. The formula is obtained by combining the original ideas of Varadhan with those of Evans and Ishii, pertaining to the theory of viscosity solutions, and some further remarks. In passing, we prove the elliptic counterpart of the formula. This concerns the slow-diffusion behavior of the solutions of the resolvent equation. Furthermore, for the case of the classical Laplace operator, we prove asymptotic formulas for the heat content and the mean value of the solution of the resolvent equation on spheres touching the boundary of the domain. These extend to the case of non-constant boundary values, certain formulas previously obtained by the second author and S. Sakaguchi. The formulas involve the principal curvatures at the touching point and the Dirichlet boundary values. We then use our formulas to describe time-invariant surfaces for the heat equation, in presence of non-constant Dirichlet boundary values. We give two symmetry results in case of non-negative Dirichlet boundary values and a non-existence result, when those values change sign.

Figures

Figures reproduced from arXiv: 2607.18784 by the authors.

Figure 1
Figure 1. The set G′ is contained in G. Left: Ω is convex and the sets S + and S − accumulate at the point A. Right: Ω is not convex. The distance depends on the shape of Ω [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Works this paper leans on

39 extracted references · 3 linked inside Pith

  1. [1]

    Aleksandrov,Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ

    A .D. Aleksandrov,Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ. 13(1958), 5–8 (English translation: Trans. Amer. Math. Soc.21(1962), 412–415)

    1958

  2. [2]

    N. D. Alikakos,An application of the invariance principle to reaction-diffusion equations,J. Differential Equations 33 (1979) 201 – 225

    1979

  3. [3]

    Berti and R

    D. Berti and R. Magnanini,Short-time behavior for game-theoretic p-caloric functions, Jour- nal de Math´ ematiques Pures et Appliqu´ es 126 (2019), 249–272

    2019

  4. [4]

    Berti and R

    D. Berti and R. Magnanini,Asymptotics for the resolvent equation associated to the game- theoretic p-laplacian, Applicable Analysis 98 (2019), 1827–1842

    2019

  5. [5]

    Berti and R

    D. Berti and R. Magnanini,Small diffusion and short-time asymptotics for Pucci operators, Applicable Analysis 101 (2022), 3716–3732

    2022

  6. [6]

    Courant and D

    R. Courant and D. Hilbert, Methods of Mathematical Physics, v. I. J. Wiley & Sons, New York, 1989

    1989

  7. [7]

    Capuzzo-Dolcetta and P

    I. Capuzzo-Dolcetta and P. L. Lions,Hamilton-Jacobi equations with state constraints,Trans. Amer. Math. Soc. 318 (1990), 643–683

    1990

  8. [8]

    Cavallina and A

    L. Cavallina and A. Pinamonti,A discrete-time overdetermined problem for the heat equation, arXiv:2604.20430, (2026)

    arXiv 2026

Show all 39 references
  1. [9]

    Ciraolo, R

    G. Ciraolo, R. Magnanini, and S. Sakaguchi,Symmetry of minimizers with a level surface parallel to the boundary, J. Eur. Math. Soc. (JEMS) 17 (2015), 2789–2804

    2015

  2. [10]

    M. G. Crandall, H. Ishii, and P. L. Lions,User’s guide to viscosity solutions of second order partial differential equations,Bull. Amer. Math. Soc. 27 (1992), 1–67

    1992

  3. [11]

    M. G. Crandall and P. L. Lions,Viscosity solutions of Hamilton-Jacobi equations,Trans. Amer. Math. Soc. 277 (1983), 1–42

    1983

  4. [12]

    L. C. Evans and H. Ishii,A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities,Ann. Inst. Henri Poincar´ e 2 (1985), 1–20

    1985

  5. [13]

    Feller, An Introduction to Probability Theory and Its Applications, Vol

    W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II. 2nd ed. John Wiley & Sons, New York, 1971

    1971

  6. [14]

    W. H. Fleming,Exit probabilities and optimal stochastic control,Appl. Math. Op. 4 (1978), 329–346

    1978

  7. [15]

    W. H. Fleming,Logarithmic transformations and stochastic control,in Advances in Filtering and Stochastic Control (Fleming and Gorostiza Eds.), Springer, New York, 1983

    1983

  8. [16]

    M. I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems. Springer- Verlag, New York 1984

    1984

  9. [17]

    P. R. Garabedian, Partial Differential Equations. J. Wiley & Sons, New York, 1964

    1964

  10. [18]

    Hasebe, J

    T. Hasebe, J. Masamune, T. Oka, K. Sakai, and T. Yamada,Construction of signed distance functions with an elliptic equation, arXiv:2401.17665, (2024)

    2024 arXiv

  11. [19]

    Hasebe, J

    T. Hasebe, J. Masamune, H. Teramoto, and T. Yamada,PDE methods for extracting normal vector fields and distance functions of shapes, arXiv:2506.19323, (2025)

    2025 arXiv

  12. [20]

    Henry, Geometric Theory of Semilinear Parabolic Equations

    D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin, 1981

    1981

  13. [21]

    Honda and J

    S. Honda and J. Lu,Gel’fand’s inverse problem under Ricci curvature bounds, arXiv:2602.14527, (2026). 28 D. BERTI, R. MAGNANINI, AND M. MARINI

    2026

  14. [22]

    M. S. Klamkin,A physical characterization of a sphere, (Problem 64-5 ∗) SIAM Review 6 (1964), 61; also in Problems in Applied Mathematics. Selection from SIAM Review. Edited by M.S. Klamkin. SIAM, Philadelphia, PA. 1990

    1964

  15. [23]

    Kuznetsov,Mean value properties of solutions to the Helmotz and modified Helmotz equa- tions, J

    N. Kuznetsov,Mean value properties of solutions to the Helmotz and modified Helmotz equa- tions, J. Mathematical Sciences, 257 (2021), 673–683

    2021

  16. [24]

    O. A. Ladyˇ zenskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. Transl. Math. Mono. 23, AMS, Providence, R. I., 1968

    1968

  17. [25]

    Littman, G

    W. Littman, G. Stampacchia, and H. F. Weinberger,Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa17 (1963), 43–77

    1963

  18. [26]

    Magnanini and M

    R. Magnanini and M. Marini,Characterization of ellipsoids as K-dense sets, Proc. Roy. Soc. Edin. 146 (2016), 213–223

    2016

  19. [27]

    Magnanini and M

    R. Magnanini and M. Marini,The Matzoh Ball Soup problem: a complete characterization, Nonlinear Anal. 131 (2016), 170–181

    2016

  20. [28]

    Magnanini, D

    R. Magnanini, D. Peralta-Salas, and S. Sakaguchi,Stationary isothermic surfaces in Eu- clidean 3-space, Math. Ann. 364 (2016), 97–124

    2016

  21. [29]

    Magnanini, J

    R. Magnanini, J. Prajapat, and S. Sakaguchi,Stationary isothermic surfaces and uniformly dense domains, Trans. A.M.S. 385 (2006), 4821-4841

    2006

  22. [30]

    Magnanini and S

    R. Magnanini and S. Sakaguchi,Matzoh ball soup: heat conductors with a stationary isother- mic surface, Ann. Math. 156 (2002), 931-946

    2002

  23. [31]

    Magnanini and S

    R. Magnanini and S. Sakaguchi,Interaction between degenerate diffusion and shape of do- main, Proc. Roy. Soc. Edin. 137A (2007), 373–388

    2007

  24. [32]

    Magnanini and S

    R. Magnanini and S. Sakaguchi,Stationary isothermic surfaces for unbounded domains, In- diana Univ. Math. Journ. 56, no. 6 (2007), 2723–2738

    2007

  25. [33]

    Magnanini and S

    R. Magnanini and S. Sakaguchi,Nonlinear diffusion with a bounded stationary level surface, Annales de l’Institut Henri Poincar´ e, Analyse non lin´ eaire 27 (2010), 937–952

    2010

  26. [34]

    Magnanini and S

    R. Magnanini and S. Sakaguchi,Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space, J. Differential Equations 248 (2010), 1112–1119

    2010

  27. [35]

    O. A. Oleinik,On the Dirichlet problem for equations of elliptic type, (Russ.), Math. Sb. N. S. 24 (66), (1949), 3–14

    1949

  28. [36]

    R. C. Reilly,Mean curvature, the Laplacian, and soup bubbles, Amer. Math. Monthly 89 (1982), 180–198

    1982

  29. [37]

    S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20 (1967), 431–455

    1967

  30. [38]

    Vessella, Unique Continuation Properties for Partial Differential Equations — introduction to the stability estimates for inverse problems

    S. Vessella, Unique Continuation Properties for Partial Differential Equations — introduction to the stability estimates for inverse problems. Birkh¨ auser/Springer, Cham 2025

    2025

  31. [39]

    G. Peano

    L. Zalcman,Some inverse problems of potential theory, Contemp. Math. 63 (1987), 337–350. Dipartimento di Matematica “G. Peano”, Universit`a di Torino, Torino (Italy). Email address:diego.berti@unito.it Dipartimento di Matematica ed Informatica “U. Dini”, Universit`a di Firenze...

    1987

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