Abstract
In this paper, we study an isospectral problem of a weighted Sturm–Liouville equation with the Dirichlet boundary condition, which lies at the basis of the integrability of the Camassa–Holm equation. We will choose the general setting of the so-called measure differential equations to solve the optimization problem on eigenvalues. It should be noticed that our technique in this paper can be used to deal with other self-adjoint boundary conditions.
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1 Introduction
This paper is concerned with the spectral problem
with the Dirichlet boundary condition
where \( m \in \mathcal {C}^-:= \{f \in {\mathcal C}[0,1]: f(t) \le 0,\ f(t) \not \equiv 0\}.\) It is well known that problem (1.1)–(1.2) has a sequence of simple eigenvalues \(0< \lambda _1(m)< \lambda _2(m)< \cdots < \lambda _k(m) \rightarrow +\infty .\) See [6].
The weighted Sturm–Liouville problem (1.1) lies at the basis of the integrability of a celebrated recent model for shallow water waves, which is the following Camassa–Holm equation:
where u is the fluid velocity in the x-direction. See [2, 3]. To study the integrability of the Camassa–Holm equation (1.3), a key point is to understand the associated spectral problem (1.1) with a certain boundary condition, where \(m=u-u_{xx}\) is the weight function. During the last two decades, there are many important and interesting results obtained for the Camassa–Holm equation. See [1, 2, 5, 7, 8, 14, 16, 17] and the references therein. For results on the inverse spectral problems and the corresponding isospectral problems of the Camassa–Holm equation, we can refer to [9, 11,12,13].
The purpose of this paper is to investigate an isospectral problem as follows:
where \( h \in (0,+\infty ) \) is fixed and
In order to solve the minimization problem (1.4), we will choose the general setting of the so-called measure differential equations (MDEs) to understand the eigenvalues, eigenfunctions, and the minimization. Based on the relationship between ODEs and MDEs, we will obtain the conclusion about the minimization problem (1.4) as follows.
Theorem 1.1
It holds that
Moreover, the infimum I(h) is never attained for any function of \(E_h\).
It should be remarked that our technique in this paper can be used to deal with other self-adjoint boundary conditions, such as the Neumann boundary condition.
This paper is organized as follows. In Section 2, we will recall some basic facts on measure differential equations, which are used in the proof of the main results. In Section 3, based on the relationship between ODEs and MDEs, we will prove Theorem 1.1.
2 Preliminaries
In order to solve the minimization problem (1.4), we will extend in this section the lowest eigenvalues \(\lambda _1(m)\) of (1.1) to the case of MDEs.
Let \({\mathcal C}={\mathcal C}[0,1]\) be the space of continuous functions on [0, 1]. Hence, the dual space of \({\mathcal C}\) is the space of Radon measures on [0, 1]:
where measures \(\nu \in {\mathcal M}_0\) are normalized as \(\nu (0+)=0\). See [4] and also [19] for more details on measures.
For a fixed measure
we study the following second order measure differential equation
with the Dirichlet boundary condition (1.2). Here we are adopting the notation from [20]. It should be remarked that the measure differential equation (2.1) reduces to the ordinary differential equation (1.1) when the measure \(\mu \in {\mathcal M}_0^-\) is continuously differentiable with the derivative \(m(t)= \mu '(t) \in {\mathcal C}^-.\)
As for the lowest eigenvalue for the equation (2.1) with the boundary condition (1.2), we can establish the following minimization characterization by similar arguments as those in [19].
Lemma 2.1
Given \(\mu \in {\mathcal M}_0^-\), we have the following minimization characterization of the lowest eigenvalue \( \lambda _1(\mu )\) for the problems (2.1) and (1.2):
where
Notice that \({\mathcal M}_0\) is a Banach space equipped with the norm
the total variation of \(\mu (t)\) on [0, 1]. Meanwhile, as a dual space, \({\mathcal M}_0\) can be equipped with the weak\(^*\) topology \(w^*.\) We say that \(\mu _n\) is weakly\(^*\) convergent to \(\mu _0\) if and only if it holds that
for any \(f\in {\mathcal C}.\) Generally, a measure can not be a limit of smooth measures in the norm \(\Vert \cdot \Vert _\textbf{V}.\) However, we have the following fact about the weak\(^*\) topology.
Lemma 2.2
([18]). For fixed \(\mu _0 \in {\mathcal M}_0^-,\) there is a sequence of smooth measures \(\{\mu _n \} \subset {\mathcal C}^\infty \cap {\mathcal M}_0^- \) satisfying
and \(\Vert \mu _n\Vert _{\textbf{V}} = \Vert \mu _0\Vert _{\textbf{V}}.\)
Based on the continuous dependence of solutions of initial value problems of measure differential equations in measures [20] and the characterization (2.2), we can prove the continuity of the eigenvalue \( \lambda _1(\mu )\) in measures \(\mu \in {\mathcal M}_0^-\) with respect to \(w^*\) topologies.
Lemma 2.3
The following nonlinear functional is continuous:
Meanwhile, the normalized eigenfunctions \(y_1(t,\mu )\), corresponding to \(\lambda _1(\mu )\) with
are continuous in \(\mu \) with respect to \({w^{*}}\) topologies, i.e.,
and
3 Main results
At first, we will extend problem (1.4) to the measure case. More precisely, let
We will study the following minimization problem on measure differential equations
Using the continuity in Lemma 2.3, we will obtain that the minimal value \(\tilde{I}(h)\) defined in (3.2) can be attained by some measure in \( \tilde{E}_h.\)
Lemma 3.1
Given \(h \in (0,+\infty ),\) there exists some measure \(\mu _h \in \tilde{E}_h\) such that
Proof
Since \(\tilde{I}(h) >0,\) one can take a minimizing sequence \(\{\mu _n\}_{n=1}^{+\infty }\subset \tilde{E}_h\) such that
as \(n \rightarrow +\infty \). Since \(\mu _n \in {\mathcal M}_0^-,\) then \( \Vert \mu _n\Vert _{\textbf{V}} = -\int _{[0,1]} d \mu _n(t) \le C\) for all \(n \ge 1\) for some constant \( C>0\). According to the Banach-Alaoglu theorem, there is a subsequence \(\{\mu _{n_k}\}_{k=1}^{+\infty } \subset \{\mu _n\}_{n=1}^{+\infty }\) such that \( \mu _{n_k} \rightarrow \mu _h \) in \(( {\mathcal M}_0, w^*)\) for some \(\mu _h \in {\mathcal M}_0\). Then, we have
Furthermore, it holds that \(\mu _h \in {\mathcal M}_0^-\) and \( \lambda _1(\mu _h) = \lim _{k\rightarrow +\infty } \lambda _1(\mu _{n_k}) = h\) by Lemma 2.3, which implies that \(\mu _h \in \tilde{E}_h.\) \(\square \)
In order to describe the minimizing measures, we denote the (unit) Dirac measure \(\delta _{a}\) located at \(a\in (0,1)\) as follows:
Notice that \(- r \delta _a \in {\mathcal M}_0^-\) for \(r >0.\) To find the solution of problem (3.2), we solve the following measure differential equation
with the initial condition
Denote
The solution \((y(t),y^\bullet (t))\) of equation (3.3) is
with
To obtain eigenvalues of problems (2.1)–(1.2), we only need to study the solution of the equaiton (2.1) with the initial value \((y_0,z_0)=(0,1)\). In this case,
Denote
By the above formulas, we get that
Now \(\lambda \in {\mathbb C}\) is an eigenvalue of (2.1)–(1.2) if and only if \(\lambda \) satisfies
which implies that
Especially, when \(a = 1/2,\) we have that
Moreover, we have the following properties of the function \(\lambda _{1}( -r \delta _{a})\) by considering \(a\in (0,1)\) as a variable.
Lemma 3.2
It holds that
Proof
By (3.5), it is easy to check that \(\lambda _{1}( -r \delta _{a})\) is strictly decreasing in \(a \in (0,1/2]\) and \(\lambda _{1}( -r \delta _{1-a}) = \lambda _{1}( -r \delta _{a})\) for all \(a \in (0,1).\) \(\square \)
Lemma 3.3
For fixed \(\mu \in {\mathcal M}_0^-\) with \( -\int _{[0,1]} d \mu (t) = r,\) there exists \(a \in (0,1)\) such that
Proof
Assuming \(\mu \in {\mathcal M}_0^- \) with \( -\int _{[0,1]} d \mu (t) = r,\) we have \(\Vert \mu \Vert _{\textbf{V}}=r\) since \(\mu (t)\) is decreasing. Taking an eigenfunction y(t) corresponding to \( \lambda _1(\mu )\), it holds that
Notice that there exists \(a\in (0,1)\) such that
Then, the denominator in (3.6) can be estimated as
Now (3.6) and (3.7) imply that \( \lambda _1(\mu )\) satisfies
in which the last inequality holds by the minimization characterization (2.2). \(\square \)
As for the minimizer of the minimization problem (3.2), we have the following conclusion.
Lemma 3.4
Let \(\mu _h \in \tilde{E}_h \) be a minimizer as in Lemma 3.1 and \( \Vert \mu _h\Vert _{\textbf{V}} = r_h.\) It holds that
Furthermore, we have \(\mu _h = -r_h \delta _{1/2}.\)
Proof
Because of Lemmas 3.2 and 3.3, we have
It remains to show this is in fact an equality. Otherwise, let us suppose the strict inequality
Then there exists some \(0< r < r_h\) such that
Hence, it holds that \( -r \delta _{{1/2}} \in \tilde{E}_h.\) However, \( \Vert -r \delta _{{1/2}}\Vert _{\textbf{V}} = r < r_h = \Vert \mu _h\Vert _{\textbf{V}},\) which contradicts the assumption that \( \mu _h\) is a minimizer.
Furthermore, assume that \(y(t; \mu _h)\) is an eigenfunction with respect to \( \lambda _1(\mu _h)\) and \( |y(a; \mu _h)| =\max _{t\in [0,1]} |y(t; \mu _h) | \) for some \(a\in (0,1)\). So, we have
which implies the identity
since \( \lambda _1(\mu _h) = \lambda _1(-r_h \delta _{1/2}).\)
Hence, \(y(t; \mu _h)\) is also an eigenfunction with respect to \( \lambda _1(-r_h \delta _{{1/2}})\) and then \(\mu _h = -r_h \delta _{1/2}.\) \(\square \)
Now, we can solve the minimization problem (3.2) on measure differential equations.
Theorem 3.1
It holds that
Moreover, the minimal value \(\tilde{I}(h)\) can be attained and only attained by \( \mu = - \tilde{I}(h) \delta _{1/2}.\)
Proof
It follows from Lemma 3.4 that the minimizer \( \mu _h = - r_h \delta _{1/2} \) with \( r_h = \frac{\coth \frac{1}{4}}{h}.\) So, we have that
\(\square \)
Finally, we will show the main conclusion of this paper.
Proof of Theorem 1.1
Notice that \(m \in E_h \) induces an absolutely continuous measure
Furthermore, we have \(-\int _{[0,1]}d \mu _m = -\int _{[0,1]} m(s)\,\textrm{d}s\) and \( \lambda _1(\mu _m) = \lambda _1(m)=h\). Hence, it holds that
which yields
On the other hand, we know that the minimizer \(\mu _h =- \frac{\coth \frac{1}{4}}{h}\delta _{1/2}\in \tilde{E}_h\) and
By Lemma 2.2, there exists a sequence of smooth measures \(\{ \mu _n \}\subset {\mathcal C}^\infty \cap {\mathcal M}_0^- \) satisfying
and \(\Vert \mu _n\Vert _{\textbf{V}} = \Vert \mu '_n\Vert _1 = r_h.\) Let \(\varphi _n = \frac{ \lambda _1(\mu _n)}{ \lambda _1(\mu _h)} \mu _n.\) Then \( \lambda _1(\varphi _n) = \lambda _1(\mu _h) = h.\)
Therefore, by Lemma 2.3, we obtain
We have thus proved the equality \(\tilde{I}(h) = I(h)\). Hence, the proof is complete by Theorem 3.1. \(\square \)
Remark 3.1
Notice that for every \(h > 0\), one has \( m(t) \in E_h\) if and only if \(h m(t) \in E_1.\) Hence, it follows that for every \(h > 0,\)
References
Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183, 215–239 (2007)
Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Camassa, R., Holm, D., Hyman, J.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)
Carter, M., van Brunt, B.: The Lebesgue-Stieltjes Integral: A Practical Introduction. Springer, New York (2000)
Chu, J., Meng, G., Zhang, M.: Continuity and minimization of spectrum related with the periodic Camassa–Holm equation. J. Differential Equations 265, 1678–1695 (2018)
Constantin, A.: On the spectral problem for the periodic Camassa–Holm equation. J. Math. Anal. Appl. 210, 215–230 (1997)
Constantin, A.: On the Cauchy problem for the periodic Camassa–Holm equation. J. Differential Equations 141, 218–235 (1997)
Constantin, A.: Quasi-periodicity with respect to time of spatially periodic finite-gap solutions of the Camassa–Holm equation. Bull. Sci. Math. 122, 487–494 (1998)
Constantin, A.: On the inverse spectral problem for the Camassa–Holm equation. J. Funct. Anal. 155, 352–363 (1998)
Constantin, A., McKean, H.P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)
Eckhardt, J.: The inverse spectral transform for the conservative Camassa–Holm flow with decaying initial data. Arch. Ration. Mech. Anal. 224, 21–52 (2017)
Eckhardt, J., Kostenko, A.: An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Comm. Math. Phys. 329, 893–918 (2014)
Eckhardt, J., Teschl, G.: On the isospectral problem of the dispersionless Camassa–Holm equation. Adv. Math. 235, 469–495 (2013)
Fu, Y., Liu, Y., Qu, C.: On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations. J. Funct. Anal. 262, 3125–3158 (2012)
Halas, Z., Tvrdý, M.: Continuous dependence of solutions of generalized linear differential equations on a parameter. Funct. Differential Equations 16, 299–313 (2009)
Holden, H., Raynaud, X.: Periodic conservative solutions of the Camassa–Holm equation. Ann. Inst. Fourier (Grenoble) 58, 945–988 (2008)
McKean, H.P.: Breakdown of the Camassa–Holm equation. Comm. Pure Appl. Math. 56, 998–1015 (2003)
Meng, G.: Extremal problems for eigenvalues of measure differential equations. Proc. Amer. Math. Soc. 143, 1991–2002 (2015)
Meng, G., Yan, P.: Optimal lower bound for the first eigenvalue of the fourth order equation. J. Differential Equations 261, 3149–3168 (2016)
Meng, G., Zhang, M.: Dependence of solutions and eigenvalues of measure differential equations on measures. J. Differential Equations 254, 2196–2232 (2013)
Mingarelli, A.B.: Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions. Lecture Notes in Mathematics, vol. 989. Springer, New York (1983)
Qi, J., Chen, S.: Extremal norms of the potentials recovered from inverse Dirichlet problems. Inverse Probl. 32, 035007, 13 pp. (2016)
Schwabik, Š: Generalized Ordinary Differential Equations. World Scientific, Singapore (1992)
Yan, P., Zhang, M.: Continuity in weak topology and extremal problems of eigenvalues of the \(p\)-Laplacian. Trans. Amer. Math. Soc. 363, 2003–2028 (2011)
Zhang, M.: Minimization of the zeroth Neumann eigenvalues with integrable potentials. Ann. Inst. H. Poincaré C Anal. Non Linéaire 29, 501–523 (2012)
Zhu, H., Shi, Y.: Dependence of eigenvalues on the boundary conditions of Sturm-Liouville problems with one singular endpoint. J. Differential Equations 263, 5582–5609 (2017)
Acknowledgements
The third author is supported by the National Natural Science Foundation of China (Grant Nos. 12071456 and 12271509) and the Fundamental Research Funds for the Central Universities.
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Dou, Y., Han, J. & Meng, G. On the isospectral problem of the Camassa–Holm equation. Arch. Math. 121, 67–76 (2023). https://doi.org/10.1007/s00013-023-01870-1
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DOI: https://doi.org/10.1007/s00013-023-01870-1