Heat flow, heat content and the isoparametric property

Alessandro Savo

doi:10.1007/s00208-015-1359-9

Mathematische Annalen

December 2016, Volume 366, Issue 3–4, pp 1089–1136 | Cite as

Heat flow, heat content and the isoparametric property

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Alessandro SavoEmail author

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First Online: 11 January 2016

Received: 23 July 2014

Revised: 23 December 2015

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Abstract

Let M be a Riemannian manifold and $\Omega $ a compact domain of M with smooth boundary. We study the solution of the heat equation on $\Omega $ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? This problem is the analogous (for the heat equation) of the classical Serrin’s problem for harmonic domains. In this paper we give an affirmative answer to the above question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem.

Mathematics Subject Classification

58J50 35P15

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Notes

Acknowledgments

We wish to thank Sylvestre Gallot for his interest in this paper and for many enlightening discussions, and Stefano Capparelli for pointing out the role of orthogonal polynomials in the combinatorial part of the paper (Lemma 27). Finally, we are deeply grateful to the referee, who has done a great job in suggesting a number of remarks which really helped to improve the clarity of the exposition.

Appendix: Proof of Lemma 27

This section is based on the paper by Ulrich Tamm [26], and we will follow closely the notation there.

Given a sequence $\{c_0,c_1,c_2,\dots \}$ form the infinite matrix

$$\begin{aligned} A=\begin{pmatrix}c_0&{}c_1&{}c_2&{}\cdots \\ c_1&{}c_2&{}c_3&{}\cdots \\ c_2&{}c_3&{}c_4&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\ddots \\ \end{pmatrix}\end{aligned}$$

(68)

and let $A_n$ be the $n\times n$ sub-matrix in the upper left corner:

$$\begin{aligned} A_1=(c_0), \quad A_2=\begin{pmatrix} c_0&{}c_1\\ c_1&{}c_2\end{pmatrix}, \quad A_3=\begin{pmatrix} c_0&{}c_1&{}c_2\\ c_1&{}c_2&{}c_3\\ c_2&{}c_3&{}c_4\end{pmatrix}, \dots \end{aligned}$$

In this way we obtain a sequence of Hankel matrices. We will also consider shifted sequences, namely for all $k\ge 0$ we set

$$\begin{aligned} A^{(k)}=\begin{pmatrix}c_k&{}c_{k+1}&{}c_{k+2}&{}\cdots \\ c_{k+1}&{}c_{k+2}&{}c_{k+3}&{}\cdots \\ c_{k+2}&{}c_{k+3}&{}c_{k+4}&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\ddots \\ \end{pmatrix}\end{aligned}$$

and let $A^{(k)}_n$ be the $n\times n$ sub-matrix in the upper left corner of $A^{(k)}$; by convention $A^{(0)}=A$. For all $k=0,1,\dots $ we set

$$\begin{aligned} d_n^{(k)}=\det A^{(k)}_n. \end{aligned}$$

By definition, the sequence $\{d_1^{(k)}, d_2^{(k)}, d_3^{(k)}, \dots \}$ is called the Hankel transform of the sequence $\{c_k,c_{k+1}, c_{k+2}\dots \}$.

Our interest is in the sequence

$$\begin{aligned} c_m=\left( {\begin{array}{c}2m+1\\ m\end{array}}\right) , \quad m\ge 0. \end{aligned}$$

(69)

It is just the sequence defined in (54) with offset at $m=0$:

$$\begin{aligned} c_m=a_{m+1}=\{1,3,10,35,126,462,1716, 6735, \dots \}. \end{aligned}$$

Then

$$\begin{aligned} A^{(0)}=\begin{pmatrix}1&{}3&{}10&{}35&{}\cdots \\ 3&{}10&{}35&{}126&{}\cdots \\ 10&{}35&{}126&{}462&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots \\ \end{pmatrix}, \quad \text {and}\quad A^{(1)}=\begin{pmatrix}3&{}10&{}35&{}126&{}\cdots \\ 10&{}35&{}126&{}462&{}\cdots \\ 35&{}126&{}462&{}1716&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\ddots \\ \end{pmatrix}. \end{aligned}$$

From part b) of Proposition 2.1 in [26] we see that, for all n:

$$\begin{aligned} d_n^{(0)}=1, \quad d_n^{(1)}=2n+1, \end{aligned}$$

(70)

and the following expression holds for $k\ge 2$:

$$\begin{aligned} d_n^{(k)}=\Pi _{1\le i\le j\le k}\dfrac{i+j-1+2n}{i+j-1}. \end{aligned}$$

Now introduce an indeterminate x and consider the infinite matrices

$$\begin{aligned} \mathcal{A}(x)=\begin{pmatrix}1&{}x&{}x^2&{}\cdots \\ c_0&{}c_1&{}c_2&{}\cdots \\ c_1&{}c_2&{}c_3&{}\cdots \\ c_2&{}c_3&{}c_4&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\ddots \\ \end{pmatrix}, \quad \mathcal{A}^{(1)}(x)=\begin{pmatrix}1&{}x&{}x^2&{}\cdots \\ c_1&{}c_2&{}c_3&{}\cdots \\ c_2&{}c_3&{}c_4&{}\cdots \\ c_3&{}c_4&{}c_5&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\ddots \\ \end{pmatrix}\end{aligned}$$

and the sequences of polynomials $\{P_n(x)\}, \{P_n^{(1)}(x)\}$ defined by

$$\begin{aligned} \left\{ \begin{aligned}&P_0(x)=1, \quad P_1(x)=\begin{vmatrix} 1&x\\c_0&c_1\end{vmatrix}, \quad P_2(x)=\begin{vmatrix} 1&x&x^2\\c_0&c_1&c_2\\c_1&c_2&c_3\end{vmatrix}, \quad ...\\&P_0^{(1)}(x)=1, \quad P_1^{(1)}(x)=\begin{vmatrix} 1&x\\c_1&c_2\end{vmatrix}, \quad P_2^{(1)}(x)=\begin{vmatrix} 1&x&x^2\\c_1&c_2&c_3\\c_2&c_3&c_4\end{vmatrix}, \quad ...\end{aligned}\right. \end{aligned}$$

Recalling that $c_m=a_{m+1}$ we see that proving Lemma 27 amounts to prove that, for all n:

$$\begin{aligned} P_n(4)=(-1)^n, \quad P_n^{(1)}(4)=(-1)^n(n+1). \end{aligned}$$

(71)

Note that, by (70), the leading coefficient of $P_n(x)$ (resp. $P_n^{(1)}(x)$) is $(-1)^{n}d_n=(-1)^n$ (resp. $(-1)^nd_n^{(1)}=(-1)^n(2n+1)$). Then, the polynomials

$$\begin{aligned} t_n(x)=(-1)^{n}P_n(x), \quad t_n^{(1)}(x)=\dfrac{(-1)^n}{2n+1}P_n^{(1)}(x) \end{aligned}$$

(72)

are of degree n and monic. Taking into account (71) and (72), to prove Lemma 27 it is then sufficient to show the following fact.

Lemma 29

For the polynomials defined in (72) one has, for all n:

$$\begin{aligned} t_n(4)=1,\quad t_n^{(1)}(4)=\dfrac{n+1}{2n+1}. \end{aligned}$$

To prove Lemma 29, we use the theory of orthogonal polynomials. We first observe:

Lemma 30

The polynomials $\{t_n(x)\}$ defined in (72) coincide with the polynomials defined in equations (1.7) and (1.8) in [26]. They are orthogonal with respect to the linear operator T mapping $x^k$ to its moment $c_k$ for all k: i.e.

$$\begin{aligned} T(t_m(x)\cdot t_n(x))=0\quad \text {for all } m\ne n. \end{aligned}$$

Similarly, the polynomials $\{t_n^{(1)}(x)\}$ are orthogonal with respect to the shifted linear operator mapping $x^k$ to $c_{k+1}$ for all k.

Proof

We closely follow the exposition in [26], page 3, with a slight change of notation to suit our needs. For all $n\ge 1$, consider the following equation in the unknowns $\alpha _{n,0},\dots ,\alpha _{n,n-1}$:

$$\begin{aligned} \begin{pmatrix}c_0&{}c_1&{}c_2&{}\cdots &{}c_{n-1}\\ c_1&{}c_2&{}c_3&{}\cdots &{}c_{n}\\ c_2&{}c_3&{}c_4&{}\cdots &{}c_{n+1}\\ \ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ c_{n-1}&{}c_n&{}c_{n+1}&{}\cdots &{}c_{2n-2}\\ \end{pmatrix}\cdot \begin{pmatrix} \alpha _{n,0}\\ \alpha _{n,1}\\ \alpha _{n,2}\\ \vdots \\ \alpha _{n,n-1}\end{pmatrix}=\begin{pmatrix} -c_n\\ -c_{n+1}\\ -c_{n+2}\\ \vdots \\ -c_{2n-1}\end{pmatrix}. \end{aligned}$$

(73)

In our case, we know that $\det A_n=1$ for all n, hence $A_n$ is non-singular and (73) has a unique solution. As in (1.8) of [26], define, for $n\ge 1$, the monic polynomials

$$\begin{aligned} \tau _n(x)\doteq x^n+\alpha _{n,n-1}x^{n-1}+\alpha _{n,n-2}x^{n-2}+\dots +\alpha _{n,1}x+\alpha _{n,0}. \end{aligned}$$

(74)

Now, by Cramer’s rule, $\alpha _{n,k}$ is the determinant of the matrix obtained by replacing the $(k+1)$-st column of $A_n$ by the column in the right-hand side of (73); in turn, this determinant is also equal to $(-1)^n$ times the coefficient of $x^k$ in $P_n(x)$. This means that

$$\begin{aligned} \tau _n(x)=t_n(x) \end{aligned}$$

as asserted. The orthogonality of the sequence $\{t_n(x)\}$ with respect to the linear mapping T has been proved in a book by Bressoud (as cited in [26]); however it is not difficult to verify that, from definition (72) of $t_n(x)$, one has $T(x^m\cdot t_n(x))=0$ for all $m=0,\dots ,n-1$, which easily implies this fact. A similar argument applies to the polynomials $t_n^{(1)}(x)$ associated to the shifted sequence. $\square $

Proof of Lemma 29 Orthogonal polynomials satisfy a three-term recursive relation: following [26], pages 5–6 and equation (1.19), one can write:

$$\begin{aligned} \left\{ \begin{aligned}&t_n(x)=(x-\alpha _n)t_{n-1}(x)-\beta _{n-1}t_{n-2}(x), \quad t_0(x)=1, t_1(x)=x-\alpha _1\\&t_n^{(1)}(x)=(x-\alpha _n^{(1)})t_{n-1}^{(1)}(x)-\beta _{n-1}^{(1)}t_{n-2}^{(1)}(x) \quad t_0^{(1)}(x)=1, t_1^{(1)}(x)=x-\alpha _1^{(1)}\end{aligned}\right. \end{aligned}$$

(75)

for suitable numerical sequences $\alpha _n\doteq \alpha _n^{(0)},\beta _n\doteq \beta _n^{(0)}, \alpha _n^{(1)}, \beta _{n-1}^{(1)}$. In fact, for arbitrary k, the coefficients $\alpha _n^{(k)}, \beta _{n-1}^{(k)}$ can be computed in terms of the coefficients $q_n^{(k)},e_n^{(k)}$ in the continued fraction expansion of $1-xF(x)$, where $F(x)=\sum _{m=0}^{\infty }c_{m+k}x^m$ (see (1.14), (1.19) and (1.20) and (1.21) in [26]):

$$\begin{aligned} \alpha _1=q_1\quad \text {and, for } n\ge 1: \quad \alpha _{n+1}^{(k)}=q_{n+1}^{(k)}+e_{n}^{(k)},\quad \beta _n^{(k)}=q_n^{(k)}\cdot e_n^{(k)}, \end{aligned}$$

(76)

For our sequence $c_m=\left( {\begin{array}{c}2m+1\\ m\end{array}}\right) $ the corresponding coefficients have been computed in the equation (2.6) of Corollary 2.1 of [26].

$$\begin{aligned} q_n^{(k)}=\dfrac{(2n+2k)(2n+2k+1)}{(2n+k-1)(2n+k)},\quad e_n^{(k)}=\dfrac{(2n-1)(2n)}{(2n+k)(2n+k+1)}. \end{aligned}$$

In particular,

$$\begin{aligned} \left\{ \begin{aligned}&q_n=q_n^{(0)}=\dfrac{2n+1}{2n-1}\\&e_n=e_n^{(0)}=\dfrac{2n-1}{2n+1}\end{aligned}\right. \quad \text {and}\quad \left\{ \begin{aligned}&q_n^{(1)}=\dfrac{(2n+2)(2n+3)}{(2n)(2n+1)}\\&e_n^{(1)}=\dfrac{(2n-1)(2n)}{(2n+1)(2n+2)}.\end{aligned}\right. \end{aligned}$$

From (76) we obtain $\alpha _1=q_1=3$, $\alpha _1^{(1)}= q_1^{(1)}=\frac{10}{3}$ and for $n\ge 2$ (after some calculations):

$$\begin{aligned} \left\{ \begin{aligned}&\alpha _n=2\\&\beta _{n-1}=1\end{aligned}\right. , \quad \text {and}\quad \left\{ \begin{aligned}&\alpha _n^{(1)}=2+\frac{4}{(2n+1)(2n-1)}\\&\beta _{n-1}^{(1)}=\dfrac{(2n-3)(2n+1)}{(2n-1)^2}\end{aligned}\right. \end{aligned}$$

(77)

From (75) and (77) one has the following recursive scheme for the sequence $\{t_n(x)\}$:

$$\begin{aligned} \left\{ \begin{aligned}&t_n(x)=(x-2)t_{n-1}(x)-t_{n-2}(x)\\&t_0(x)=1\\ {}&t_1(x)=x-3\end{aligned}\right. \end{aligned}$$

Setting $X_n=t_n(4)$ we obtain

$$\begin{aligned} \left\{ \begin{aligned}&X_n=2X_{n-1}-X_{n-2}\\&X_0=1\\ {}&X_1=1\end{aligned}\right. \end{aligned}$$

hence $X_n=1$ for all n. That is:

$$\begin{aligned} t_n(4)=1 \end{aligned}$$

for all n. This shows the first relation in the Lemma.

About the sequence $\{t_n^{(1)}(x)\}$ we know:

$$\begin{aligned} \left\{ \begin{aligned}&t_n^{(1)}(x)=(x-\alpha _n^{(1)})t_{n-1}^{(1)}(x)-\beta _{n-1}^{(1)}t_{n-2}^{(1)}(x)\\&t_0^{(1)}(x)=1\\ {}&t_1^{(1)}(x)=x-\frac{10}{3}\end{aligned}\right. \end{aligned}$$

where $\alpha _n^{(1)}$ and $\beta _{n-1}^{(1)}$ are as in (77). We are interested in the sequence $ X_n=t_n^{(1)}(4), $ which obeys the recursive law:

$$\begin{aligned} \left\{ \begin{aligned}&X_n=(4-\alpha _n^{(1)})X_{n-1}-\beta _{n-1}^{(1)}X_{n-2}\\&X_0=1\\ {}&X_1=\frac{2}{3}\end{aligned}\right. \end{aligned}$$

By induction on n, let us show that $X_n=\frac{n+1}{2n+1}$. In fact this holds for $n=1$. Assume it to be true for all $k\le n-1$. By (77):

$$\begin{aligned} 4-\alpha _n^{(1)}=\dfrac{8n^2-6}{(2n-1)(2n+1)}, \quad \beta _{n-1}^{(1)}=\dfrac{(2n-3)(2n+1)}{(2n-1)^2} \end{aligned}$$

hence:

$$\begin{aligned} X_n&=(4-\alpha _n^{(1)})X_{n-1}-\beta _{n-1}^{(1)}X_{n-2}\\&=\dfrac{8n^2-6}{(2n-1)(2n+1)}\cdot \frac{n}{2n-1}-\dfrac{(2n-3)(2n+1)}{(2n-1)^2}\cdot \frac{n-1}{2n-3}\\&=\frac{n+1}{2n+1} \end{aligned}$$

which shows the claim. With this, the proof of Lemma 27 is complete.

Remark We point out that the expression of $\alpha _{n+1}^{(k)}$ in the statement of Corollary 2.3 in [26] is wrong; this is due to an incorrect algebraic manipulation of $q_{n+1}^{(k)}+e_n^{(k)}$ in the proof of Corollary 2.3 there. The correct value of $\alpha _{n+1}^{(1)}$ shown in (77) is directly computed from the expressions of $q_n^{(1)}$ and $e_n^{(1)}$ taken from Corollary 2.1, equation (2.6) of [26].

References

1.
Berenstein, C.A.: An inverse spectral theorem and its applications to the Pompeiu problem. J. Anal. Math. 37, 128–144 (1980)MathSciNetCrossRefMATHGoogle Scholar
2.
Berenstein, C.A.: On an overdetermined Neumann problem. Geom Semin. Univ. Stud, Bologna, pp 21–26 (1986)Google Scholar
3.
Berenstein, C.A., Shahshahani, M.: Harmonic analysis and the Pompeiu problem. Am. J. Math. 105, 1217–1229 (1983)MathSciNetCrossRefMATHGoogle Scholar
4.
Berenstein, C., Yang, P.: An inverse Neumann problem. J. Reine Angew. Math. 382, 1–21 (1987)MathSciNetMATHGoogle Scholar
5.
Caffarelli, L.: The regularity of free boundaries in higher dimensions. Acta Math. 139, 155–184 (1977)MathSciNetCrossRefMATHGoogle Scholar
6.
Capparelli, S., Maroscia, P.: On two sequences of orthogonal polynomials related to Jordan blocks. Mediterr. J. Math. 10(4), 1609–1630 (2013)MathSciNetCrossRefMATHGoogle Scholar
7.
Cartan, E.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Annali di Mat. 17, 177–191 (1938)MathSciNetCrossRefMATHGoogle Scholar
8.
Chakalov, L.: Sur un probleme de D. Pompeiu. Annuaire Univ. Sofia Phys. Math. Livre I 40, 1–44 (1944)Google Scholar
9.
El Soufi, A., Ilias, S.: Domain deformations and eigenvalues of the Dirichlet Laplacian in Riemannian manifolds. Ill. J. Math. 51, 645–666 (2007)MATHGoogle Scholar
10.
Garabedian, P.R., Schiffer, M.: Variational problems in the theory of elliptic partial differential equations. J. Rat. Mech. Anal. 2, 137–171 (1953)MathSciNetMATHGoogle Scholar
11.
Kinderleher, D., Nirenberg, L.: Regularity in free boundary problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4, 373–391 (1977)MathSciNetMATHGoogle Scholar
12.
Kumaresan, S., Prajapat, J.: Serrin’s result for hyperbolic space and sphere. Duke Math. J. 91(1), 17–28 (1998)MathSciNetCrossRefMATHGoogle Scholar
13.
Liu, G.: Symmetry theorems for the overdetermined eigenvalue problems. J. Differ. Equ. 233, 585–600 (2007)MathSciNetCrossRefMATHGoogle Scholar
14.
Magnanini, R., Sakaguchi, S.: Heat conductors with a stationary isothermic surface. Ann. Math. Second Ser. 156(3), 931–946 (2002)MathSciNetCrossRefMATHGoogle Scholar
15.
Molzon, R.: Symmetry and overdetermined boundary problems. Forum Math. 3, 143–156 (1991)MathSciNetCrossRefMATHGoogle Scholar
16.
Münzner, H.F.: Isoparametrische hyperflachen in spharen I, II. Math. Ann. 251, 57–71 (1980) and 256 (1981), 215–232Google Scholar
17.
Münzner, H.F.: Isoparametrische hyperflachen in spharen I, II. Math. Ann. 256, 215–232 (1981)Google Scholar
18.
Pacard, F., Sicbaldi, P.: Extremal domains for the first eigenvalue of the Laplace–Beltrami operator. Ann. Institut Fourier 59(2), 515–542 (2009)MathSciNetCrossRefMATHGoogle Scholar
19.
Raulot, S., Savo, A.: On the first eigenvalue of the Dirichlet-to-Neumann operator on forms. J. Funct. Anal. 262, 889–914 (2012)MathSciNetCrossRefMATHGoogle Scholar
20.
Ros, A., Sicbaldi, P.: Geometry and topology of some overdetermined elliptic problems. J. Differ. Equ. 255, 951–977 (2013)MathSciNetCrossRefMATHGoogle Scholar
21.
Savo, A.: Uniform estimates and the whole asymptotic series of the heat content on manifolds. Geom. Dedicata 73, 181–214 (1998)MathSciNetCrossRefMATHGoogle Scholar
22.
Savo, A.: Asymptotics of the heat flow for domains with smooth boundary. Commun. Anal. Geom. 12(3), 671–702 (2004)MathSciNetCrossRefMATHGoogle Scholar
23.
Schiffer, M.: Hadamard’s formula and variation of domain functions. Am. J. Math. 68, 417–448 (1946)MathSciNetCrossRefMATHGoogle Scholar
24.
Serrin, J.: A symmetry problem in potential theory. Arch. Rat. Mech. Anal. 43, 304–318 (1971)MathSciNetCrossRefMATHGoogle Scholar
25.
Shklover, V.: Schiffer problem and isoparametric hypersurfaces. Revista Mat. Iberoamericana 16(3), 529–569 (2000)MathSciNetCrossRefMATHGoogle Scholar
26.
Tamm, U.: Some aspects of Hankel matrices in coding theory and combinatorics. Electron. J. Combin. 8, # A1 (2001)Google Scholar
27.
Tang, Z., Yan, W.: Isoparametric foliation and Yau conjecture on the first eigenvalue. J. Differ. Geom. 94, 521–540 (2013)MathSciNetMATHGoogle Scholar
28.
Thorbergsson, G.: A survey on isoparametric hypersurfaces and their generalizations. In: Handbook of Differential Geometry, vol. 1. North-Holland, Amsterdam, pp 963–995 (2000)Google Scholar
29.
Wang, Q.M.: Isoparametric functions on Riemannian manifolds, I. Math. Ann. 277, 639–646 (1987)MathSciNetCrossRefMATHGoogle Scholar
30.
Weinberger, H.F.: Remark on the preceeding paper by Serrin. Arch. Rat. Mech. Anal. 43, 319–320 (1971)MathSciNetCrossRefMATHGoogle Scholar
31.
Williams, S.A.: A partial solution of the Pompeiu problem. Math. Ann. 223, 183–190 (1976)MathSciNetCrossRefMATHGoogle Scholar
32.
Williams, S.A.: Analyticity of the boundary for Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J. 30, 357–369 (1981)MathSciNetCrossRefMATHGoogle Scholar
33.
van den Berg, M., Gilkey, P.: Heat content asymptotics for a Riemannian manifold with boundary. J. Funct. Anal. 120, 48–71 (1994)MathSciNetCrossRefMATHGoogle Scholar
34.
van den Berg, M., Gilkey, P.: The heat equation with inhomogeneous Dirichlet boundary conditions. Commun. Anal. Geom. 7(2), 279–294 (1999)MathSciNetCrossRefMATHGoogle Scholar

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1.Dipartimento SBAI, Sezione di MatematicaSapienza Università di RomaRomeItaly

Heat flow, heat content and the isoparametric property

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Acknowledgments

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