Abstract
We derive some new regularity results for the best-Sobolev-constant function defined by
where \(\Omega \) is a bounded and smooth domain of \(\mathbb {R}^{N},\,1<p<N\) and \(p^{\star }:=\frac{Np}{N-p}\). In a previous work, we proved that this function is absolutely continuous and thus its derivative \(\lambda _{q}^{\prime }\) exists at almost all \(q\in [1,p^{\star }]\). In this paper, we prove that \(\lambda _{q}^{\prime }\) exists if, and only if, the functional
is constant on the set \(E_{q}\) of the \(L^{q}\)-normalized extremal functions corresponding to \(\lambda _{q}\). Moreover, we prove that the existence of \(\lambda _{q}^{\prime }\) is also equivalent to the continuity at \(q\) of the function \(s\in [1,p^{\star })\mapsto I_{s}(u_{s})\), where \(u_{s}\) is any function in \(E_{s}\). It follows from these results that \(\lambda _{q}^{\prime }\) exists and is continuous if \(q\in [1,p]\) and \(\Omega \) is a general bounded domain and also if \(q\in (p,p^{\star })\) and \(\Omega \) is a ball. After deriving some estimates for \(I_{q}(u_{q})\), we also prove, under an expected asymptotic behavior of \(u_{q}\), as \(q\rightarrow p^{\star }\), that \(\lambda _{q}\) is \(\alpha \)-Hölder continuous in \([1,p^{\star }]\) for any \(0<\alpha <1\). As a consequence, this Hölder regularity holds for a general bounded domain \(\Omega \) when \(p=2\) and also for a ball when \(p>1\).
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1 Introduction
In this paper, we derive some regularity results for the best-Sobolev-constant function defined by
where \(\Omega \) is a bounded and smooth domain of \(\mathbb {R}^{N}\), \(N\ge 2,\, 1<p<N,\, p^{\star }:=\dfrac{Np}{N-p}\) and
is the Rayleigh quotient associated with the Sobolev immersion \(W_{0} ^{1,p}(\Omega )\hookrightarrow L^{q}(\Omega )\). (Here \(\left\| \cdot \right\| _{s}:=\left( {\int _{\Omega }}\left| \cdot \right| ^{s}\hbox {d}x\right) ^{\frac{1}{s}}\) denotes the usual norm of \(L^{s}(\Omega )\).)
It is well known that the immersion \(W_{0}^{1,p}(\Omega )\hookrightarrow L^{q}(\Omega )\) is continuous if \(1\le q\le p^{\star }\) and compact if \(1\le q<p^{\star }\). Hence, there exists \(u_{q}\in W_{0}^{1,p}(\Omega )\setminus \{0\}\) such that \(\mathcal {R}_{q}(u_{q})=\lambda _{q}\), if \(1\le q<p^{\star }\). Since \(\mathcal {R}_{q}\) is homogeneous the extremal function \(u_{q}\) associated with \(\lambda _{q}\) can be chosen such that \(\left\| u_{q}\right\| _{q}=1\). (From now on \(u_{q}\) will denote any \(L^{q}\)-normalized extremal function corresponding to \(\lambda _{q}\)).
It is straightforward to verify that such a normalized extremal function \(u_{q}\) is a weak solution of the Dirichlet problem
for the \(p\)-Laplacian operator \(\Delta _{p}u:=\mathrm{div }(\left| \nabla u\right| ^{p-2}\nabla u)\). Hence, classical results imply that \(u_{q}\in C^{1,\alpha }(\overline{\Omega })\) for some \(0<\alpha <1\). Thus, \(u_{q}\in W_{0}^{1,p}(\Omega )\cap C^{1,\alpha }(\overline{\Omega })\) satisfies, for each \(q\in [1,p^{\star })\):
Therefore, the infimum in (1) is actually a minimum if \(1\le q<p^{\star }\). In the critical case, \(q=p^{\star }\), one has \(\lambda _{p^{\star }}=S^{p}\), where \(S\) is the well-known Sobolev constant (see [4, 20]) and the minimum is not reached if \(\Omega \) is a proper subset of \(\mathbb {R}^{N}\). We recall that \(S\) is explicitly given by
where \(\Gamma (t)=\int \nolimits _{0}^{\infty }s^{t-1}e^{-s}\hbox {d}s\) is the Gamma Function.
We remark that \(\lambda _{q}\) is simple if \(1\le q\le p\) (see [14]), which means that extremal functions associated with \(\lambda _{q}\) are scalar multiple one of the other. This property is still valid in the “super-linear” case \(p<q<p^{\star }\) if \(\Omega \) is a ball (see [1, 15]) or if \(p=2\) and \(2\le q\le 2+\epsilon \) (see [7]). However, for a general bounded \(N\)-dimensional domain, \(\lambda _{q}\) is not known to be simple in the super-linear case. In fact, \(\lambda _{q}\) is not simple if \(\Omega \) is an annulus and \(q\) is close enough to \(p^{\star }\) (see [18]).
Thus, if
denotes the set of the \(L^{q}\)-normalized extremal functions, it follows that \(E_{q}=\left\{ \pm u_{q}\right\} \) whenever \(\lambda _{q}\) is simple. Otherwise, \(E_{q}\) might be larger than \(\left\{ \pm u_{q}\right\} \) and the question of determining the size of \(E_{q} \) for a general domain in the “super-linear”case \(p<q<p^{\star }\) is still an open problem for which our results might be useful.
Up to our knowledge, very little attention has been paid to the behavior of the function \(q\in [1,p^{\star }]\mapsto \lambda _{q}\) in the literature.
In [5] Benci and Cerami considered the function \((q,\mu )\in (2,2^{\star })\times [0,+\infty )\mapsto m(q,\mu )\) where
This minimizing problem is closely associated with Dirichlet problem
Benci and Cerami proved that for each fixed \(\mu \ge 0\) there exists \(q_{\mu }\in (2,2^{\star })\) such that (7) has at least \(\mathrm{cat }\Omega \) positive solutions whenever \(q_{\mu }\le q\le 2^{\star }\) (here \(\mathrm{cat }\Omega \) denotes the Lyusternik–Shnirel’man category of \(\overline{\Omega }\) in itself). In order to reach this remarkable multiplicity result they verified (see [5, Leema 4.1]) that for each \(\mu \ge 0\) fixed the function \(q\in (2,2^{\star })\mapsto m(q,\mu )\) is (left) continuous at \(q=2^{\star }\). We observe that for \(\mu =0\) (and \(p=2\)) this means that the function \(q\in [1,2^{\star }]\mapsto \lambda _{q}\) is (left) continuous at \(q=2^{\star }\). (We remark that Benci and Cerami [5] also proved in the same multiplicity result for all \(\mu \ge \mu _{q}\) where \(\mu _{q}\ge 0\) is obtained from each fixed \(q\in (2,2^{\star })\).)
With respect to \(p>1\), Huang proved in [13, Thm 2.1] that the function \(q\in [1,p^{\star }]\mapsto \lambda _{q}\) is continuous in the open interval \((1,p)\) and lower semi-continuous in the open interval \((p,p^{\star })\).
We have recently started, in [10], a study on the behavior of the function \(q\in [1,p^{\star }]\mapsto \lambda _{q}\) proving that it is decreasing, Lipschitz continuous in \([1,p^{\star }-\epsilon ]\) for each \(\epsilon >0\) and left-continuous at \(q=p^{\star }\), thus obtaining its absolute continuity in \([1,p^{\star }]\). Therefore, for almost all \(q\in [1,p^{\star }]\), the derivative \(\lambda _{q}^{\prime }\) of \(\lambda _{q}\) exists. Using a different method, these results were extended in [2], to allow \(q\in (0,1)\) and \(p\ge N\) as well (so, \(p^{\star }=\infty \) in this case).
In [9] we studied the asymptotic behavior, as \(q\rightarrow p\), of the positive solutions of the “resonant Lane–Emden problem”
and obtained, as a byproduct of our main result, the existence of \(\lambda _{p}^{\prime }\).
In the present paper, we give two characterizations of the existence of \(\lambda _{q}^{\prime }\) and prove a Hölder regularity result for \(\lambda _{q}\).
In Sect. 2, we first prove that the derivative \(\lambda _{q}^{\prime }\) exists if, and only if the functional
is constant on the set \(E_{q}\). It follows that \(I_{q}\) is constant on \(E_{q}\) for almost all \(q\in (p,p^{\star })\) which is a new and nontrivial property that, at least in principle, could be used to prove results on simplicity of \(\lambda _{q}\) for suitable domains or even to give estimates for the number of extremal functions associated with \(\lambda _{q}\).
Thereafter, still in Sect. 2, we prove that the existence of \(\lambda _{q}^{\prime }\) is also equivalent to the continuity of the function \(s\in [1,p^{\star }]\mapsto I_{s}(u_{s})\) at \(s=q\).
As consequence of these characterizations, we conclude that the function \(s\in [1,p^{\star }]\mapsto \lambda _{s}\) is continuously differentiable at \(s=q\) whenever \(\lambda _{q}\) is simple. Thus, for a general bounded \(\Omega \) and \(p>1\) one always has \(\lambda _{s}\in C^{1}([1,p])\). If \(p=2\) one also has \(\lambda _{s}\in C^{1}([1,2+\epsilon ])\), for some \(\epsilon >0\) and if \(p>1\) and \(\Omega \) is a ball then \(\lambda _{s}\in C^{1}([1,p^{\star }))\).
In order to obtain both characterizations of the differentiability, we first prove that
at each point \(q\in [1,p^{\star }]\) where the derivative \(\lambda _{q}^{\prime }\) exists. Thus, \(\lambda _{q}\) satisfies a simple linear and homogeneous ordinary differential equation in the Carathéodory sense (that is, almost everywhere). Then, we combine this fact with the absolute continuity of the function \(q\in [1,p^{\star }]\mapsto \lambda _{q}\) to obtain the following representation formula:
where each extremal function \(u_{s} \) can be arbitrarily chosen in \(E_{s}\) for each \(s\in [1,p^{\star })\).
In Sect. 3, we prove a Hölder regularity at \(p^{\star }\) after deriving some estimates for \(I_{q}(u_{q})\). More precisely, we prove that \(\lambda _{q}\) is \(\alpha \)-Hölder continuous in the interval \([1,p^{\star }]\), for any \(0<\alpha <1\), under the additional hypothesis
for some \(\gamma >0\). This asymptotic behavior holds true, for instance, if \(\Omega \) is a ball and \(p>1\) and also if \(p=2\) and \(\Omega \) is an arbitrary bounded domain. Thus, at least in these two situations, our results guarantee that \(\lambda _{q}\) is \(\alpha \)-Hölder continuous for any \(0<\alpha <1\). However, we believe that (10) holds true for a general bounded and smooth domain, also if \(1<p\not =2\).
2 Characterizations of the differentiability
From now on \(E_{q}\) denotes the set of the \(L^{q}\)-normalized extremal functions, defined in (5), and \(I_{q}\) denotes the functional defined by (8).
Theorem 1
For each \(q\in [1,p^{\star })\), let \(u_{q}\) be arbitrarily chosen in \(E_{q}\). The following estimates hold for each \(q\in (1,p^{\star })\):
Therefore, the function \(q\mapsto \lambda _{q}\) satisfies
at each point \(q\in (1,p^{\star })\) where its derivative \(\lambda _{q}^{\prime }\) exists.
Proof
Let \(q\in (1,p^{\star })\). Since \(\int _{\Omega }\left| u_{q}\right| ^{q}\hbox {d}x=1\) we have
It follows from (3) that
Therefore, the continuity of the function \(q\mapsto \lambda _{q}\) and L’Hôpital’s rule yield
Analogously,
\(\square \)
We remark that Theorem 1 is valid for any choice of the extremal function \(u_{q}\) in \(E_{q}\). Therefore, the following consequence is immediate.
Corollary 2
Let \(q\in (1,p^{\star })\) be such that \(\lambda _{q}^{\prime }\) exists. Then the functional \(I_{q}\) is constant on \(E_{q}\).
In the sequel \(f:[1,p^{\star })\rightarrow \mathbb {R}\) denotes the function defined by
where, for each \(q\in [1,p^{\star })\), \(u_{q}\) is arbitrarily chosen in \(E_{q}\).
Now, we prove the representation formula (9) for the function \(q\in [1,p^{\star }]\mapsto \lambda _{q}\).
Theorem 3
It holds
Proof
Since the function \(q\in [1,p^{\star }]\mapsto \lambda _{q}\) is absolutely continuous and its image is a closed interval \([a,b]\subset (0,\infty )\) we also have that the function \(q\in [1,p^{\star }]\mapsto \log \lambda _{q}\) is absolutely continuous. Therefore,
where the last equality follows from (12). Now, (13) follows after exponentiation. \(\square \)
Since \(\lambda _{p^{\star }}=S^{p}\) we note from (13) that
(We recall that the Sobolev constant \(S\) is explicitly given by (4).)
The following result is an immediate consequence of (13). We leave its proof to the reader.
Corollary 4
Let \(q\in [1,p^{\star })\) be a point where \(f\) is continuous. Then the derivative \(\lambda _{q}^{\prime }\) exists and is given by the continuous expression \(\lambda _{q}^{\prime }=-\lambda _{q}f(q)\).
Proposition 5
Suppose that \(I_{q}\) is constant on \(E_{q}\) for some \(q\in [1,p^{\star })\). Then \(f\) is continuous at \(q\). In particular, \(f\) is continuous at each point \(q\) where \(\lambda _{q}^{\prime }\) exists.
Proof
Let \(q_{n}\rightarrow q\). Then \(\lambda _{q_{n}}\rightarrow \lambda _{q}\) and, up to subsequences, \(u_{q_{n}}\) converges in \(C^{1}(\overline{\Omega })\) to a function \(u\) satisfying (2) and such that \(\left\| u\right\| _{q}=1\). This last claim follows by combining the classical Hölder regularity result (see [17]) with the fact that \(\left\| u_{q_{n}}\right\| _{\infty }\) is bounded from above by a constant which is uniform with respect to \(n\) (see Lemma 11 in the next section).
Therefore, \(u\in E_{q}\) and thus
since \(I_{q}\) is constant on \(E_{q}\). \(\square \)
Corollary 6
The function \(q\mapsto \lambda _{q}\) is continuously differentiable in
-
1.
\([1,p]\) if \(p>1\),
-
2.
\([1,2+\epsilon ]\) (for some \(\epsilon >0\)) if \(p=2\),
-
3.
\((p,p^{\star })\) if \(\Omega \) is ball.
Proof
Since \(I_{q}(u_{q})=I_{q}(\left| u_{q}\right| )\), this corollary follows from the fact that \(\lambda _{q}\) is simple in all of these cases. \(\square \)
Combining corollaries 2 and with Proposition 5 we obtain:
Theorem 7
The following assertions on a point \(q\in [1,p^{\star })\) are equivalent:
-
1.
\(\lambda _{q}^{\prime }\) exists.
-
2.
\(I_{q}\) is constant on \(E_{q}\).
-
3.
The function \(s\in [1,p^{\star }]\mapsto I_{s}(u_{s})\) is continuous at \(s=q\).
We observe that the proof of Corollary also works for \(q=p^{\star }\) if the limit \(\lim \limits _{q\rightarrow p^{\star }}I_{q}(u_{q})\) exists. However, even the verification that \(I_{q}(u_{q})\) is finite at \(q=p^{\star }\) does not seem to be a simple task. One of the difficulties is that \(\left\| u_{q}\right\| _{\infty }\rightarrow \infty \) as \(q\rightarrow p^{\star }\). In fact, otherwise we reach a contradiction by applying a regularity result as in [21] and then making \(q\rightarrow p^{\star }\) in the equation \(-\Delta _{p}u_{q}=\lambda _{q}u_{q}^{q-1}\), thus obtaining a limit function \(u\in W_{0}^{1,p}(\Omega )\) that minimizes the Rayleigh quotient \(\mathcal {R}_{p^{\star }}(\Omega )\). Since \(\Omega \not =\mathbb {R}^{N}\), this is absurd.
In the next section, we derive some estimates for \(I_{q}(u_{q})\) and use them to prove that, under an additional (and natural) hypothesis on the behavior of \(\left\| u_{q}\right\| _{\infty }\) (as \(q\) tends to \(p^{\star }\)) the function \(q\mapsto I_{q}(u_{q})\) belongs to \(L^{r}([0,p^{\star }])\), for all \(r>1\). Unfortunately, the \(L^{r}\) bounds we obtain from these estimates are just linear with respect to \(r\) and thus not enough to prove that \(I_{q} (u_{q})\) stays bounded as \(q\rightarrow p^{\star }\).
3 Hölder regularity
In order to obtain the absolute continuity of the function \(\lambda _{q}\) in [10] we first prove that this function is Lipschitz continuous in each closed interval of the form \([1,p^{\star }-\epsilon ]\subset [1,p^{\star }]\). Of course, this fact guarantees that \(\lambda _{q}\) is Hölder continuous (with any exponent \(\alpha \in (0,1)\)) in \([1,p^{\star }-\epsilon ]\). However, as pointed out in the end of the previous section, the precise behavior of \(\lambda _{q}^{\prime }\) (or equivalently of \(I_{q}(u_{q})\)) as \(q\rightarrow p^{\star }\) seems difficult to determine, even when \(\Omega \) is a ball.
In this section we prove a Hölder regularity result for the function \(\lambda _{q}\) in \([1,p^{\star }] \) by estimating \(\big \Vert \lambda _{q}^{\prime }\big \Vert _{L^{r}([0,p^{\star }])}\) for any \(r>1\). Taking into account (12), we first need to estimate \(I_{q}(u_{q})\).
Lemma 8
The following estimates hold for each \(0\not \equiv u\in W_{0}^{1,p}\left( \Omega \right) \):
where \(\left| \Omega \right| :=\int \nolimits _{\Omega }\hbox {d}x\).
Proof
The continuous function \(\varphi :[0,\infty )\rightarrow \mathbb {R}\) defined by \(\varphi (\xi )=\xi \log \xi \), if \(\xi >0\) and \(\varphi (0)=0\) is strictly convex. Hence, for each \(u\not \equiv 0\) Jensen’s inequality yields
Thus,
(Note that the equality in the Jensen’s inequality occurs only if \(u^{q} \equiv \int _{\Omega }\left| u\right| ^{q}\hbox {d}x\).)
Now, we prove the upper bound in (14) following [8]. Let \(t\in [q,p^{\star }]\). It follows from Hölder inequality that
with
Thus,
and
Since \(g(q)=0 \), we obtain
But
and thus,
implying that
\(\square \)
Remark 9
We emphasize that the estimates in (14) become simpler to handle if \(\left| \Omega \right| \le 1=\left\| u\right\| _{q}\). In fact, under these conditions one has
We also note that a simple scaling argument gives
where \(\Omega _{1}:=\left\{ x\in \mathbb {R}^{N}:x\left| \Omega \right| ^{\frac{1}{N}}\in \Omega \right\} \) is such that \(\left| \Omega _{1}\right| =1\) and \(\lambda _{q}(D)\) is defined as in (1) with \(\Omega =D\).
It is also worth mentioning that \(\frac{\log \left\| u_{q}\right\| _{p^{\star }}^{p^{\star }}}{p^{\star }-q}\) becomes an indeterminate form of the type \(0/0\) as \(q\rightarrow p^{\star }\), since
Indeed, since \(1=\left\| u_{q}\right\| _{q}^{q}\le \left\| u_{q}\right\| _{p^{\star }}^{q}\left| \Omega \right| ^{1-\frac{q}{p\star }}\) one has
On the other hand, the inequality
yields
Lemma 10
For each \(\beta >0\) and all \(1\le q<p^{\star }\) one has
Proof
The first inequality in (18) follows from (15) since
Since \(\log (x)\le \beta ^{-1}x^{\beta }\) for all \(x\ge 1\) and \(1=\left\| u_{q}\right\| _{q}\le \left\| u_{q}\right\| _{\infty }\left| \Omega \right| ^{\frac{1}{q}}\) the second inequality follows. \(\square \)
Lemma 11
The following estimate holds
where \(C\) is a positive constant which does not depend on \(q\).
Proof
By taking \(\sigma =q\) in Lemma 5 of Ercole [10] we obtain
where
Hence, (19) follows with
\(\square \)
Now, we are in position to prove a Hölder regularity result for the function \(q\mapsto \lambda _{q}\) by assuming that
for some constant \(\gamma >0\). Before proceeding, let us give some motivations for the assumption (20).
In [16] Knaap and Peletier proved the following asymptotic behavior for the case where \(\Omega \) is the ball centered at the origin:
being \(A_{N,p,\Omega }\) a positive constant given explicitly in terms of \(N\), \(p\) and the volume of \(\Omega \). Of course, (21) implies (20) with \(\gamma =\frac{p}{p-1}\).
In the case where \(p=2\), the asymptotic behavior (21) for a ball had already been proved by Atkinson and Peletier in [3]. In [6], still with \(p=2\) and for a ball, Brezis and Peletier gave another proof of (21) and, among other important results, they conjectured that a similar asymptotic behavior should be true for a nonspherical bounded and smooth domain \(\Omega \) (keeping \(p=2\)). This conjecture was then proved, independently, by Rey in [19] and by Han in [12]. The regular part of the Green function of \(\Omega \) (associated with the Laplacian) played an essential role in the proofs presented in [12, 19]. Indeed, the proofs rely on the fact that, as \(q\) goes to \(2^{\star }\), the maximum points of \(u_{q}\) concentrate at a point \(x_{0}\in \Omega \) which is a critical point of the Robin function of \(\Omega \) (the diagonal of the regular part of the Green function). In the case where \(\Omega \) is a ball \(x_{0}\) is its center.
However, to the best of our knowledge in the case \(1<p\not =2\), the only result specifically related to the asymptotic behavior (21) is that by Knaap and Peletier in [16] for a ball. For a general bounded and smooth domain \(\Omega \), Garcia Azorero and Peral Alonso showed in [11] that \(u_{q}\) converges in the sense of measure to a multiple of the Dirac Delta Function concentrated at a point \(x_{0}\in \Omega \). Thus, they reproduced for \(1<p\not =2\) the concentration property known in the case \(p=2\). After [11] some works have dealt with the convergence of the family \(\left\{ u_{q}\right\} \) in the measures sense but, as far as we are aware, the generalization of (21) for a nonspherical bounded domains remains open if \(1<p\not =2\).
In the sequel, we show that the asymptotic behavior (20) combined with (15) imply that \(\lambda _{q}\in W^{1,r}([1,p^{\star }])\) for any \(r\ge 1\) and hence that \(\lambda _{q}\in C^{0,\alpha }([1,p^{\star }])\) for any \(0<\alpha <1\) (by taking \(\alpha :=1-\dfrac{1}{r}\)). Thus, for a ball and \(p>1\) and for a general bounded domain and \(p=2\) we get more regularity at \(p^{\star }\) than just absolute continuity.
We remark that \(\lambda _{q}\in W^{1,1}([1,p^{\star }])\) for a general bounded domain \(\Omega \) (because \(\lambda _{q}\) is absolutely continuous in the closed interval \([1,p^{\star }]\)).
Theorem 12
Assume that (20) happens for some \(\gamma >0\). The function \(q\mapsto \lambda _{q}\) belongs to the Sobolev space \(W^{1,r} ([1,p^{\star }])\) for any \(r>1\). Moreover,
Proof
By taking into account (16) we can assume, without loss of generality, that \(\left| \Omega \right| \le 1\). Hence,
for any \(\beta >0\), according to (18).
It follows from (12) that
Therefore, we just need to prove that \(\left\| I_{q}(u_{q})\right\| _{L^{r}([1,p^{\star }])}\) is finite.
The hypothesis (20) implies that there exists \(\overline{q} \in (1,p^{\star })\) such that
for some positive constant \(c\) (which may depend on \(\gamma \)).
Combining this with (23) yields
Thus, by taking \(\beta :=\dfrac{\gamma }{2r}\) we obtain
where \(k:=c^{(\gamma /2)}2(p^{\star }-\overline{q})^{\frac{1}{2}}\).
On the other hand, it follows from Lemma 11 that
Thus, we conclude that \(\lambda _{q}^{\prime }\in L^{r}([1,p^{\star }])\) and that
for positive constants \(C_{1}\) and \(C_{2}\) which do not depend on \(r\). Hence we obtain (22). \(\square \)
The following corollary follows immediately.
Corollary 13
Suppose that \(p>1\) and \(\Omega \) is a ball or that \(p=2\) and \(\Omega \) is a general bounded and smooth domain. Then \(\lambda _{q}\in C^{0,\alpha }([1,p^{\star }])\) for any \(0<\alpha <1\).
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Ercole, G. Regularity results for the best-Sobolev-constant function. Annali di Matematica 194, 1381–1392 (2015). https://doi.org/10.1007/s10231-014-0425-3
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DOI: https://doi.org/10.1007/s10231-014-0425-3