Abstract
Let \(\Omega \subset \mathbb {R}^2\) be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to \(\Omega \) with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in Alvino et al. (Commun Pure Appl Math 76:585–603, 2023).. We prove that the equality is achieved only if \(\Omega \) is a disk and the torsion function u is radial.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\beta >0\) and let \(\Omega \subset \mathbb {R}^2\) be an open, bounded and Lipschitz set. We consider the following problem for the Laplace operator:
where \(\nu \) is the outer unit normal to \(\partial \Omega \). A function \(u\in H^1(\Omega )\) is a weak solution to (1) if
Classical arguments, see e.g [1], ensure that there exists a positive and unique weak solution to (1), that we denote by u. So, we can define the Robin torsional rigidity of \(\Omega \) as the \(L^1\)-norm of u:
or, equivalently, as the maximum of the following Rayleigh quotient:
In [2] the authors prove that the Robin torsional rigidity is maximum on balls among bounded and Lipschitz sets of fixed Lebesgue measure and the proof of this Saint-Venant type inequality relays on reflection arguments (see also [3]).
In the recent paper [4], the authors obtain the same result using symmetrization techniques. They establish a Talenti-type comparison result between suitable Lorentz norms of the solution to the following problems:
where \(f\in L^2(\Omega )\), \(f^\sharp \) is the Schwartz rearrangement of f (see Definition 3) and \(\Omega ^\sharp \) is the ball centered at the origin having the same measure as \(\Omega \). Moreover, in the case \(f\equiv 1\), they obtain the following comparison result in any dimension
We observe that, for \(p=1\), inequality (3) is exactly the Saint-Venant inequality proved in [2]. It is still an open problem to establish if, for \(p\in (1,+\infty )\), the ball maximizes the \(L^p\) norm of the torsion function among open, bounded and Lipschitz sets (see [3, Open Problem 1]). A first evidence in this direction is provided in [5], where it is proved that the ball is a critical shape for every \(L^p\) norm in dimension \(n>2\).
On the other hand, in the case \(n=2\), the Open Problem 1 contained in [3] is solved in [4] in the following stronger version:
where \(u^\sharp \) is the Schwartz rearrangement of the solution to (1) and v is the solution to
This kind of results in the Robin boundary setting was generalized to nonlinear case in [6], to anisotropic case in [7], with mixed boundary conditions in [8], in the case of the Hermite operator in [9] and for Riemannian manifolds in [10].
The aim of the present paper is to characterize the equality case in (4), indeed we prove that the Talenti-type comparison is rigid in the planar case.
Theorem 1
Let \(\Omega \subset \mathbb {R}^2\) be an open, bounded and Lipschitz set and let \(\Omega ^\sharp \) be the ball centered at the origin and having the same measure as \(\Omega \). Let u be the solution to (1) and let v be the solution to (5). If \(u^\sharp (x)=v(x)\) for all \(x\in \Omega ^\sharp \), then
Moreover, we have the following extension of Theorem 1.
Theorem 2
Let \(\Omega \subset \mathbb {R}^2\) be an open, bounded and Lipschitz set and let \(\Omega ^\sharp \) be the ball centered at the origin and having the same measure as \(\Omega \). Let u be the solution to (1) and let v be the solution to (5). We denote by R the radius of \(\Omega ^\sharp \).
If \(\displaystyle {\min _\Omega u=\min _{\Omega ^\sharp } v}\) and if there exists \(r\in ]0, R[\) such that \(u^\sharp (x)=v(x)\) for \({\left| x\right| }=r\), then
The idea of the proof is the following. Starting from the proof in [4] of the pointwise comparison (4), we show that the equality \(u^\sharp =v\) implies that the level sets of u are balls on the boundary of which the normal derivative of u is constant. Then, we prove that these balls are concentric, using an argument inspired by [11] (see also [12, Lemma 6]). In the Robin case, the main difficulty is that, contrary to the Dirichlet case, the level sets of the solution may touch the boundary of \(\Omega \).
As far as the Dirichlet boundary conditions, the starting point for the study of these kinds of problems is the paper by Talenti [13], in which a pointwise comparison is stated between the solution to the following problems:
whenever
\(f\in L^{\frac{2n}{n+2}}(\Omega )\). In particular, he proves in [13] the pointwise inequality:
and, consequently, by integration, the Saint-Venant inequality in the Dirichlet case holds:
conjectured by Saint-Venant in 1856. Moreover, a previous result in this direction is due to Weinberger, that proved in [14] the following result:
We stress that, in the case of Dirichlet boundary conditions, the rigidity result holds and it is proved in [15] (see Remark 6 for the main differences to the Robin case).
Finally, we conclude by a list of generalization of Talenti’s comparison results in different setting with Dirichlet boundary conditions. Extension to the semilinear and nonlinear elliptic case can be found, for instance, in [16], to the anisotropic elliptic operators in [17], to the parabolic case in [18] and to higher order operators in [19, 20]. We also refer the reader to [21, 22] and the references therein for a survey on Talenti’s techniques.
The paper is organized as follows. In Sect. 2 we recall some basic notions about rearrangements of functions and we recall some properties of the Torsion function, while Sect. 3 is dedicated to the proof of Theorems 1 and 2 and to a list of open problems.
2 Notation and Preliminaries
Throughout this article, \({\left| \cdot \right| }\) will denote the Euclidean norm in \(\mathbb {R}^2\), while \(\cdot \) is the standard Euclidean scalar product. By \({\mathcal {H}}^1(\cdot )\), we denote the 1-dimensional Hausdorff measure in \(\mathbb {R}^2\). The perimeter of \(\Omega \) will be denoted by \(P(\Omega )\) and since \(\Omega \) is a bounded, open and Lipschitz set, we have that \(P(\Omega )={\mathcal {H}}^{1}(\partial \Omega )\). Moreover, we denote by \({\left| \Omega \right| }\) the Lebesgue measure of \(\Omega \).
If \(\Omega \) is an open and Lipschitz set, it holds the following coarea formula. Some references for results relative to the sets of finite perimeter and the coarea formula are, for instance, [23, 24].
Theorem 3
(Coarea formula) Let \(f:\Omega \rightarrow \mathbb {R}\) be a Lipschitz function and let \(u:\Omega \rightarrow \mathbb {R}\) be a measurable function. Then,
We recall now some basic definitions and results about rearrangements and we refer to [22] for a general overview.
Definition 1
Let \(u: \Omega \rightarrow \mathbb {R}\) be a measurable function, the distribution function of u is the function \(\mu : [0,+\infty [\, \rightarrow [0, +\infty [\) defined by
Definition 2
Let \(u: \Omega \rightarrow \mathbb {R}\) be a measurable function, the decreasing rearrangement of u, denoted by \(u^*\), is the distribution function of \(\mu \).
Remark 1
We observe that the function \(\mu (\cdot )\) is decreasing and right continuous and the function \(u^*(\cdot )\) is the generalized inverse of the function \(\mu (\cdot )\).
Definition 3
The Schwartz rearrangement of u is the function \(u^\sharp \) whose level sets are balls with the same measure as the level sets of u.
We have the following relation between \(u^\sharp \) and \(u^*\):
and it can be easily checked that the functions u, \(u^*\) e \(u^\sharp \) are equi-distributed, so we have that
Let now u be the solution to (1). For \(t\ge 0\), we introduce the following notations:
and, if v is the solution to (5), using the same notations as above, we set
Because of the invariance of the Laplacian under rotation, we have that v is radial. Moreover, we observe that the solutions u to (1) and v to (5) are both superharmonic and so, by the strong maximum principle, it follows that they achieve their minima on the boundary.
From now on, we denote by
Since we are assuming that the Robin boundary parameter \(\beta \) is strictly positive, we have that \(u_m > 0\) and \(v_m > 0\). Hence, u and v are strictly positive in the interior of \(\Omega \).
Since v is radial, positive and decreasing along the radius then, for \(0\le t\le v_m\),
while, for \(v_m<t<v_M\), we have that \(V_t\) is a ball concentric to \(\Omega ^\sharp \) and strictly contained in it.
In the next remarks, we collect some general and useful results.
Remark 2
By the weak formulation (2) and the isoperimetric inequality, we have that
and, as a consequence,
Moreover, from (10) follows that
Remark 3
We observe that \(\phi \), the distribution function of v, is absolutely continuous. Indeed, in [11, Lemma 2.3], is proved that the absolutely continuity of \(\phi \) is equivalent to the following condition:
which is verified by v, as its gradient never vanishes on the level sets \(V_t\).
The starting point of the proof of our main results is the following Lemma, proved in [4]. For the convenience of exposition, we report here the proof.
Lemma 4
Let u be a solution to (1) and let v be a solution to (5). Then, for almost every \(t >0\), we have
and
Proof
Let \(t >0\) and \(h >0\). Let us choose the following test function in the weak formulation (2)
Then, we have
Dividing (15) by h, using coarea formula (7) and letting h go to 0, we have that for a.e. \(t>0\)
where
Using the isoperimetric inequality, for a.e. \(t\in [0, u_M )\) we have
and, so, (13) follows. Finally, we notice that, if v is the solution to (5), then all the inequalities above are equalities, and, consequently, we have (14). \(\square \)
Remark 4
By integrating (14), it is possible to write the explicit expression of v, that is
Remark 5
Integrating (13) and (14) between 0 and t and integrating by parts, it is proved in [4] that
Finally, we observe that the pointwise comparison (4) easily follows from (20).
3 Proof of the Main Results
Proof of Theorem 1
First of all, let us observe that, from the fact that we are assuming that \(u^\sharp =v\), we have
We integrate now (13) and (14) from 0 to t and, since \(u^*\) is the generalized inverse of \(\mu \) (Remark 1), we perform the following change of variables \(\mu (t)=s\) and \(\phi (t)=s\). So, we get
From \(u^\sharp =v\), we have \(u^*=v^*\) and, so, combining (22) and (23), we get
where the last equality follows from (21). Therefore, all the inequalities in (24) are equalities and, consequently, equality holds in (13).
We now divide the proof in two steps.
Step 1 Let us prove that every level set \(\{u>t\}\) is a ball.
Equality in (13) implies the equality in (17), i.e.
that means that almost every level set is a ball. On the other hand, for all \(t\in [u_m, u_M)\), there exists a sequence \(\{t_k\}\) such that
-
1.
\(t_k\rightarrow t\);
-
2.
\(t_k>t_{k+1}\);
-
3.
\(\{u>t_k\}\) is a ball for all k.
Since \(\{u>t\}=\cup _k\{u>t_k\}\) can be written as an increasing union of balls, then we have that \(\{u>t\}\) is a ball for all t and, from the fact that \(\Omega =\{u>u_m\}\), we obtain that \(\Omega =x_0+\Omega ^\sharp \). From now on, we can assume without loss of generality that \(x_0=0\).
Step 2 Let us prove that the level sets are concentric balls.
Equality in (13) implies also equality in (18), i.e.
This means that, as we have equality in the Hölder inequality, for almost every t, the function
is constant, in particular
and by continuity we can infer that this is true for all t. By the way, we observe that for all \(x\in \partial U_t\),
where \(\nu _t\) is the unit outer normal to \(\partial U_t\).
From equality (13), we have also that
and, consequently, we can deduce from Remark 3 that also \(\mu \) is absolutely continuous. If we denote by
we can observe that the function \(\mu (t)\) is locally Lipschitz in \((u_m, u_M)\), and, so, the function
is also locally Lipschitz. Moreover, since \(\{u>t\}\subseteq \{u>s\}\) for \(t>s\), we have
and, consequently, x(t) is locally Lipschitz.
Let us assume now by contradiction that x(t) is not constant. This means that there exists \(t_0\in (u_m,u_M)\) such that
Let us set \(z:=y/{\left| y\right| }\) and
We have that, for all \(t\in (u_m,u_M)\),
and
On the other hand, from (27), we obtain
and, consequently,
Moreover, by (25) we have
and, so, we have \(|y|=0\), that is absurd.
Thus, we have proved that u is radially symmetric and, since
u is decreasing along the radii and \(u=u^\sharp \). \(\square \)
Remark 6
In the proof of Theorem 1.1 the main difference from the proof of the rigidity result in the Dirichlet case contained in [15] is Step 2. Indeed, in [15], the authors use the steepest descent lines method, which relays on the fact that \({\left| \nabla u\right| }\) is constant on the level set of u, which is not a priori true in the Robin case.
Proof of Theorem 2
Let us set \(s=\pi r^2\). The assumption \(u^\sharp (x)=v(x)\) for \({\left| x\right| }=r\), implies
Arguing now as in the proof of Theorem 1, we have
where in the last equality we have used the hypothesis \(u_m=v_m\). So, we have equality in (23) and, consequently, in (13) for \({\overline{t}}:=u^*(s)\). As before, this implies that
-
\(\{u> {\overline{t}}\}\) is a ball;
-
\(\mu ({\overline{t}})=\phi ({\overline{t}})\);
-
the function g defined in (16) is constant on \(\partial U_{{\overline{t}}}\).
Let us observe that, for all \(\tau >v_m\)
while, for v it holds
where the first equality follows from the fact that \(\forall t> v_m\)
If we multiply (13) and (14) by t and we integrate from 0 to \({\overline{t}}\), we get
where in the last inequality we use (29), and we get
where in the last equality we use (30). Therefore, combining (31) and (32), we have that
and, integrating by parts and recalling that \(\mu ({\overline{t}})=\phi ({\overline{t}})\), we get
On the other hand, since (20) holds for all \(t\ge 0\), we have
and this implies that equality holds in (13) for all \(t\in [0, {\overline{t}}]\). Now, arguing as in Theorem 1, we recover \(\Omega =\Omega ^\sharp + x_0\) and \(u(\cdot +x_0)=u^\sharp (\cdot )\) in \(\{ r\le {\left| x\right| }\le R\}\). Finally, for the uniqueness of the solution to problem (5), once we have that \(\Omega \) is a ball, it follows that \(u=v\) for all \(x\in \Omega \). \(\square \)
As a particular case of the above result, if we take \(r=0\), we have
Corollary 1
Let \(\Omega \subset \mathbb {R}^2\) be an open, bounded and Lipschitz set and let \(\Omega ^\sharp \) be the ball, centered at the origin, having the same measure of \(\Omega \). Let u be the solution to (1) and let v be the solution to (5). If \(u_m=v_m\), and \(u_M=v_M\), then
Open Problem 1
Below we present a list of open problems and work in progress.
-
Generalize the results contained in Theorems 1 and 2 to higher dimension. In order to do that, one should prove (4) in \(\mathbb {R}^n\) for \(n\ge 3\) (we address to Open Problem 1 in [4]).
-
Generalize the results contained in Theorems 1 and 2 under weaker assumptions.
-
Generalize the previous results to the p-Torsion or to the anisotropic Torsion.
References
Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Applied Mathematical Sciences, vol. 78. Springer, New York (2008)
Bucur, D., Giacomini, A.: A variational approach to the isoperimetric inequality for the Robin eigenvalue problem. Arch. Ration. Mech. Anal. 198(3), 927–961 (2010). https://doi.org/10.1007/s00205-010-0298-6
Bucur, D., Giacomini, A.: The Saint-Venant inequality for the Laplace operator with Robin boundary conditions. Milan J. Math. 83(2), 327–343 (2015). https://doi.org/10.1007/s00032-015-0243-0
Alvino, A., Nitsch, C., Trombetti, C.: A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions. Commun. Pure Appl. Math. 76, 585–603 (2023). https://doi.org/10.1002/cpa.22090
Sannipoli, R.: Some properties of the torsion function with Robin boundary conditions. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 33(1), 23–37 (2022). https://doi.org/10.4171/rlm/962
Amato, V., Gentile, A., Masiello, A.L.: Comparison results for solutions to \(p\)-Laplace equations with Robin boundary conditions. Ann. Mat. Pura Appl. (4) 201(3), 1189–1212 (2022). https://doi.org/10.1007/s10231-021-01153-y
Sannipoli, R.: Comparison results for solutions to the anisotropic Laplacian with Robin boundary conditions. Nonlinear Anal. 214, 112615–112621 (2022). https://doi.org/10.1016/j.na.2021.112615
Alvino, A., Chiacchio, F., Nitsch, C., Trombetti, C.: Sharp estimates for solutions to elliptic problems with mixed boundary conditions. J. Math. Pures Appl. 152, 251–261 (2021). https://doi.org/10.1016/j.matpur.2020.12.003
Chiacchio, F., Gavitone, N., Nitsch, C., Trombetti, C.: Sharp estimates for the Gaussian torsional rigidity with Robin boundary conditions. Potential Anal. (2022). https://doi.org/10.1007/s11118-022-09995-8
Daguang, C., Li, H., Wei, Y.: Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds. arXiv preprint (2021). arXiv:2110.06005
Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384, 153–179 (1988)
Serra, J.: Radial symmetry of solutions to diffusion equations with discontinuous nonlinearities. J. Differ. Equ. 254(4), 1893–1902 (2013). https://doi.org/10.1016/j.jde.2012.11.015
Talenti, G.: Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4), 697–718 (1976)
Weinberger, H.F.: Symmetrization in uniformly elliptic problems. In: Studies in Mathematical Analysis and Related Topics, pp. 424–428. Stanford University Press, Stanford (1962)
Alvino, A., Lions, P.-L., Trombetti, G.: A remark on comparison results via symmetrization. Proc. R. Soc. Edinb. Sect. A 102(1–2), 37–48 (1986). https://doi.org/10.1017/S0308210500014475
Talenti, G.: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl. 4(120), 160–184 (1979). https://doi.org/10.1007/BF02411942
Alvino, A., Ferone, V., Trombetti, G., Lions, P.-L.: Convex symmetrization and applications. Ann. Inst. H. Poincaré C Anal. Non Linéaire 14(2), 275–293 (1997). https://doi.org/10.1016/S0294-1449(97)80147-3
Alvino, A., Lions, P.L., Trombetti, G.: Comparison results for elliptic and parabolic equations via Schwarz symmetrization. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(2), 37–65 (1990). https://doi.org/10.1016/S0294-1449(16)30303-1
Ashbaugh, M.S., Benguria, R.D.: On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions. In: Differential Equations and Mathematical Physics (Birmingham, AL, 1994), pp. 17–27. International Press, Boston (1995)
Talenti, G.: Inequalities in rearrangement invariant function spaces. In: Nonlinear Analysis, Function Spaces and Applications, vol. 5 (Prague, 1994), pp. 177–230. Prometheus, Prague (1994)
Kawohl, B.: Rearrangements and convexity of level sets in PDE. In: Lecture Notes in Mathematics, vol. 1150, p. 136. Springer, Berlin (1985). https://doi.org/10.1007/BFb0075060
Kesavan, S.: Symmetrization & Applications. Series in Analysis, vol. 3, p. 148. World Scientific Publishing Co. Pte. Ltd., Hackensack (2006). https://doi.org/10.1142/9789812773937
Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge Studies in Advanced Mathematics, vol. 135, p. 454. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139108133. An introduction to geometric measure theory
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, p. 434. The Clarendon Press/Oxford University Press, New York (2000)
Funding
Open Access funding enabled and organized by Projekt DEAL. The author Gloria Paoli is supported by the Alexander von Humboldt Foundation with an Alexander von Humboldt research fellowship. The authors A. L. Masiello and G. Paoli are supported by GNAMPA of INdAM.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
There is no conflict of interest to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Masiello, A.L., Paoli, G. A Rigidity Result for the Robin Torsion Problem. J Geom Anal 33, 149 (2023). https://doi.org/10.1007/s12220-023-01202-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01202-3