A remark on the first p-buckling eigenvalue with an adhesive constraint

1.   Introduction

In this paper we are interested in a fourth order nonlinear eigenvalue problem

with the adhesive constraint

where $p > 1$, $0 < \omega_0 < |\Omega|$ and $\mathcal{O}(u): = \{x \in \Omega \mid u(x)\not = 0 \}$. Here $\Omega$, $\nu$ and $|\Omega|$ denote a smooth bounded domain in $\mathbb{R}^N$, the unit outer normal of $\partial \Omega$ and the Lebesgue measure of $\Omega$, respectively. The eigenvalue problem (1.1) is regarded as a generalization of the buckling eigenvalue problem with the clamped boundary condition

The first eigenvalue of (1.3) with $N = 2$ is called the buckling load of a clamped plate and characterized by

In 1951, Polya and Szegö ^[9] conjectured that the disk minimizes the buckling load $\mu_1$ among domains of given measure. The conjecture has been attracted a great interest and studied by many researchers (e.g., see ^[2,10], [6,Section 3.2] and references therein). In particular, recently Stollenwerk ^[10] considered problem (1.3) with constraint (1.2) to study the Polya-Szegö conjecture for $N = 2, 3$. Although nonlinear eigenvalue problems for the $p$-biharmonic operator of the type

have been well studied in the mathematical literature (e.g., see ^[3,5,7] and references therein), to the best of our knowledge, there is few result on the eigenvalue problem of the type (1.1).

The purpose of this paper is to study the first eigenvalue of (1.1) with constraint (1.2) for $N = 1$:

with

where

Here $I \subset \mathbb{R}$ denotes a bounded open interval and $0 < \omega_0 < |I|$ is a given constant.

One of our motivations is to prove the existence of the first eigenvalue and corresponding eigenfunctions of problem (1.4) with (1.5). To this end, we consider the minimization problem

where

As we prove in Lemma 3.2, solutions to problem $({\rm{P}})$ satisfy problem (1.4) with (1.5) in a weak sense. Thus problem $({\rm{P}})$ gives us the first $p$-buckling eigenvalue and corresponding eigenfunctions.

The second motivation is to show a property of the eigenfunction corresponding to the first $p$-buckling eigenvalue. In 2014, Watanabe ^[12] studied the $p$-elastic curves which are critical points of the $p$-elastic energy

where $\gamma$, $\kappa$ and $s$ respectively denote a planar curve, the curvature of $\gamma$ and the arc length parameter of $\gamma$, and proved the existence of solutions with 'flat core'. Here, we say that $u: I \to \mathbb{R}$ has 'flat core' if the graph $(x, u(x))$ contains a part where the graph is parallel to the $x$-axis (more precisely, see ^[11,12]). In order to state our second motivation precisely, we define

and

We say that $u : I \to \mathbb{R}$ has flat core of adhesion type if the set $\mathcal{I}(u)$ is not connected. Our second motivation is to ask whether constraint (1.5) can induce the eigenfunction corresponding to the first $p$-buckling eigenvalue to have flat core of adhesion type or not.

The main result of this paper is stated as follows:

Theorem 1.1. Let $I \subset \mathbb{R}$ be an open interval. Let $p > 1$ and $0 < \omega_0 < |I|$. Then problem $({\rm{P}})$ possesses a solution $u \in \mathcal{A}_{\omega_0}$. Moreover, $\mathcal{I}(u)$ is connected.

We deduce from Theorem 1.1 that the eigenfunction corresponding to the first $p$-buckling eigenvalue does not have flat core of adhesion type. Due to adhesive constraint (1.5), it is difficult to solve problem $({\rm{P}})$ by the direct method of calculus of variations. To overcome the difficulty, we employ an idea by Alt and Caffarelli ^[1] as in ^[10]. More precisely, considering a penalized problem, once we remove adhesive constraint (1.5) from $({\rm{P}})$. Studying the regularity of the penalized solution $u_\varepsilon$, we prove the relation $|\mathcal{O}(u_\varepsilon)| = \omega_0$ for sufficiently small $\varepsilon > 0$. Then we obtain a minimizer of problem $({\rm{P}})$. We note that, if we employ the same strategy to find the first $p$-buckling eigenvalue for $N \ge 2$, then one of the arising difficulties is the lack of regularity of the penalized solution $u_\varepsilon$.

This paper is organized as follows: In Section 2, we collect notations and inequalities which are used in this paper; In Section 3, we define a penalized problem and prove the existence and the regularity of the penalized solutions; In Section 4, we prove Theorem 1.1.

2.   Preliminary

In this section, we collect function spaces and inequalities used in this paper.

The space $W^{2, p}_0(I)$ is the closure of $C^{\infty}_{\rm c}(I)$ in $W^{2, p}(I)$. In this paper, we employ $\| v \|_{2, p}: = \| v''\|_{L^p(I)}$ as the norm in $W^{2, p}_0(I)$. Here we note that the norm $\| \cdot \|_{2, p}$ is equivalent to the standard $W^{2, p}$ norm. Indeed, by the Poincaŕe inequality we find a positive constant $C$ such that

This clearly implies that the norm $\| \cdot \|_{2, p}$ is equivalent to the standard $W^{2, p}$ norm.

In order to treat $L^p$ norms, we employ the following inequality (see ^[8]):

which expresses the convexity of the function $x \mapsto |x|^p$ for $p \ge 1$.

3.   Penalized problem

In this section we consider a penalized problem. We define the function $f_\varepsilon$ and the functional $E_\varepsilon$ by

for $\varepsilon > 0$. Then the penalized problem corresponding to $({\rm{P}})$ is written as follows$:$

To begin with, we prove the existence of solutions of penalized problem $({\rm P}_\varepsilon)$.

Lemma 3.1. Problem $({\rm P}_\varepsilon)$ possesses a nontrivial solution for each $\varepsilon > 0$.

Proof. Let $\{u_k\}_{k \in \mathbb{N}} \subset W^{2, p}_0(I)$ be a minimizing sequence for $E_\varepsilon$, i.e.,

We note that $E_\varepsilon$ is nonnegative. Extracting a subsequence, we find a constant $C > 0$ such that

where we denote by $\{u_k \}$ this subsequence, for short. Since $E_\varepsilon$ is homogeneous of degree $0$, we are able to normalize the minimizing sequence $\{u_k\}_{k \in \mathbb{N}}$ as follows:

Then problem $({\rm P}_\varepsilon)$ is reduced into

where

By (3.1), (3.2), (3.3) and (3.4) we have

Thus we find a function $u_\varepsilon \in W^{2, p}_0(I)$ such that

up to a subsequence. Since the embedding $W^{2, p}_0(I) \subset C^{1, \alpha}(\bar{I})$ is compact for each $\alpha \in (0, 1-1/p)$, it follows from (3.5) that

This together with $\{ u_k \} \subset \mathcal{A}$ implies that

and then $u_\varepsilon \in \mathcal{A}$. Moreover, this clearly implies that $u_\varepsilon$ is nontrivial.

Next we show that $u_\varepsilon \in \mathcal{A}$ is the desired minimizer of $E_\varepsilon$. First it follows from (3.5) that

We can prove the relation

along the same line as in [10,Theorem 2.1]. Indeed, since $f_\varepsilon$ is non-decreasing, it suffices to prove the relation

By the Banach–Alaoglu theorem we find a function $\rho \in L^\infty(I)$ with $0 \le \rho(x)\le 1$ for a.e. $x \in I$ such that

for all $\varphi \in L^1(I)$ up to a subsequence, where

Then we observe from (3.6) and (3.10) that

where $g^+: = \max\{ g, 0 \}$ and $g^-: = \max\{-g, 0\}$. This together with $0 \le \rho \le 1$ implies that $\rho = 1$ a.e. in $\mathcal{O}(u_\varepsilon)$. Thus we obtain (3.8) as follows:

Combining (3.7) with (3.8), we obtain

Therefore Lemma 3.1 follows.

From now on, we set

Moreover, we define

and

Lemma 3.2. Let $u_\varepsilon$ be a solution of $({\rm P}_\varepsilon)$. Then $(u_\varepsilon, \Lambda_\varepsilon)$ satisfies the following eigenvalue problem in the weak sense$\colon$

Proof. It suffices to prove that $(u_\varepsilon, \Lambda_\varepsilon) \in W^{2, p}_0(I) \times \mathbb{R}$ satisfies

Fix $\varphi \in W^{2, p}_0(\mathcal{I}(u_\varepsilon))$ arbitrarily. Since $u_\varepsilon+\delta \varphi_{\varepsilon} \in W^{2, p}_0(\mathcal{I}(u_\varepsilon))$, we deduce from the minimality of $u_\varepsilon$ that

Moreover, it follows from $|\mathcal{O}(u_\varepsilon+ \delta \varphi_{\varepsilon})| = |\mathcal{O}(u_\varepsilon)|$ that

for sufficiently small $\delta$, and then

By a direct calculation we have

Recalling that $\|u'_\varepsilon\|_{L^p(I)} = 1$ and $\|u^{\prime\prime}_\varepsilon\|^p_{L^p(I)} = \Lambda_\varepsilon$, we obtain

This together with (3.13) implies (3.12). Therefore Lemma 3.2 follows.

We prove the regularity of the minimizer $u_\varepsilon$. To begin with, we show some properties of the support of $u_\varepsilon.$

Lemma 3.3. Let $u_\varepsilon$ be a solution of $({\rm P}_\varepsilon)$. Then

Proof. Assume that (3.14) does not hold. Then we find $\varepsilon_* > 0$ such that

Then there exist $x_0 \in I$ and $0 < r < 1$ such that

where $B(y, \rho): = \{ x \in I \mid |x-y| < \rho\}$. Fix $v \in C_{\rm c}^{\infty}(B(0, 1))$ arbitrarily. We define $v_r : B(x_0, r)\to\mathbb{R}$ by $v_r(x) : = v(x_0 + rx).$ Since

and

we observe from (3.15) and (3.17) that

This together with the minimality of $u_{\varepsilon_*}$ implies that

Recalling the definition of $\Lambda_\varepsilon$, we deduce from (3.18) that

Thanks to (3.16), we reduce (3.19) into

Since

recalling that $\|u_\varepsilon^\prime\|_{L^p(I)}^p = 1$ and $\|u_\varepsilon^{\prime\prime}\|_{L^p(I)}^p = \Lambda_\varepsilon$ for all $\varepsilon > 0$, we observe from (3.20) that

On the other hand, combining $\|u'_\varepsilon\|_{L^p(I)}^p = 1$ and $\|u''_\varepsilon\|_{L^p(I)}^p = \Lambda_\varepsilon$ with Poincaré's inequality, we find a constant $C > 0$ being independent of $r$ such that

This together with (3.21) implies that

Taking $0 < r < 1$ small enough, we lead a contradiction. Thus Lemma 3.3 follows.

Lemma 3.3 implies that the 'size' of the support of minimizer $u_\varepsilon$ is uniformly bounded from below with respect to $\varepsilon > 0$. Next we prove that the support of $u_\varepsilon$ is connected.

Lemma 3.4. Let $u_\varepsilon$ be a solution of $({\rm P}_\varepsilon)$. Then $\mathcal{I}(u_\varepsilon)$ is connected for all $\varepsilon > 0$.

Proof. Suppose not, we find $\varepsilon_* > 0$ such that $\mathcal{I}(u_{\varepsilon_*})$ is not connected. Then there exist an open interval $I_1$ and an open set $I_2$ such that $I_1 \cap I_2 = \emptyset$ and

We define $U_i$ by

Then it holds that $U_i \in W^{2, p}_0(I)$ for $i = 1, 2$. If $|I_1| \ge \omega_0$, then we deduce from (3.22) that

This clearly contradicts to the minimality of $u_\varepsilon$. If $|I_1| < \omega_0$, then Lemma 3.3 implies that

Since it follows from $|I_1| < \omega_0$ that $f_\varepsilon(U_1) = 0$, we observe from (3.24) that $\Lambda_\varepsilon < E(U_1)$, and then

On the other hand, it follows from (3.23) that

Plugging (3.25) into (3.26), we have

where we used (3.22) again in the last equality of (3.27). Then (3.27) implies that

Since $f_\varepsilon(|\mathcal{O}(U_2)|)\leq f_\varepsilon(|\mathcal{O}(u_\varepsilon)|)$, we deduce form (3.28) that

This contradicts to the minimality of $u_\varepsilon$. Therefore Lemma 3.4 follows.

Lemma 3.5. There exists a constant $\Lambda_{\rm max}$ such that

Proof. Let $r_0 : = \omega_0/4$ and set $x_0 \in I$ such that $B(x_0, r_0) \subset I$, where $B(y, \rho): = \{ x \in I \mid |x-y| < \rho\}$. We consider the problem

Along the same line as in the proof of Lemma 3.1, we find a solution $\varphi_0 \in W^{2, p}_0(B(x_0, r_0))$ of problem (3.29) satisfying the following:

Since $\varphi_0 \in W^{2, p}_0(B(x_0, r_0))$, we can extend $\varphi_0$ as a function in $W^{2, p}_0(I)$. Recalling that

we deduce from Lemma 3.3 that

This together with (3.30) that

Therefore Lemma 3.5 follows.

Here we employ the idea in [4,Proof of Theorem 3.9]:

Lemma 3.6. Let $a_1, a_2 \in I$ with $a_1 < a_2$. Fix $\eta \in C_{\rm c}^{\infty}((a_1, a_2))$ and set

Then $\varphi_1 \in W^{2, p}_0((a_1, a_2))$ and there exist $C_1, C_2, C_3 > 0$ depending only on $a_1$ and $a_2$ such that

Proof. By (3.31), (3.32) and (3.33) we have $\varphi_1(a_1) = \varphi_1(a_2) = 0$. Since

it follows from (3.32) and (3.33) that $\varphi'_1(a_1) = \varphi'_1(a_2) = 0$. Thus we see that $\varphi_1 \in W^{2, p}_0((a_1, a_2))$. Moreover, we have

Similarly we obtain

Thus Lemma 3.6 follows.

Theorem 3.7. Let $u_\varepsilon\in W^{2, p}_0(I)$ be a solution to $({\rm P}_\varepsilon)$. Then there exists a constant $M > 0$ such that

Proof. Fix $\varepsilon > 0$ arbitrarily. Since $u_\varepsilon \in W^{2, p}_0(I)$ is a solution to $({\rm P}_\varepsilon)$, by Lemma 3.2 we have

where the constant $\Lambda_\varepsilon$ is defined by (3.11). By Lemmas 3.3 and 3.4 we find $a^\varepsilon_1, a^\varepsilon_2 \in I$ such that

Fix $\eta \in C^\infty_{\rm c}(\mathcal{I}(u_\varepsilon))$ arbitrarily. Taking $(a^\varepsilon_1, a^\varepsilon_2)$ as $(a_1, a_2)$ in Lemma 3.6, we observe from (3.35) that the constants $C_1$, $C_2$ and $C_3$ in Lemma 3.6 depends only on $\omega_0$ and $I$. Taking $\varphi = \varphi_1$ in (3.34), where $\varphi_1$ is the function defined in Lemma 3.6, we have

This together with Lemma 3.6 implies that

By Hölder's inequality we have

Similarly we obtain

Plugging (3.37) and (3.38) into (3.36) we see that

where the constant $C > 0$ depends only on $\omega_0$ and $I$. This together with Lemma 3.5 implies that

where $C > 0$ depends only on $\omega_0$ and $I$. Using the fact that $(L^1(\mathcal{I}(u_\varepsilon)))^* = L^{\infty}(\mathcal{I}(u_\varepsilon))$ and Riesz's representation theorem, we deduce from (3.39) that

Therefore Theorem 3.7 follows.

4.   Proof of Theorem 1.1

In order to prove Theorem 1.1, it suffices to show the following:

Theorem 4.1. Let $u_\varepsilon$ be a solution to $({\rm P}_\varepsilon)$ obtained by Lemma 3.1. Then there exists a constant $\varepsilon_0 > 0$ such that

Proof. Assume that (4.1) does not hold. Then, by Lemma 3.3 we find a sequence $\{ \varepsilon_j\}_{j\in \mathbb{N}}$ such that $\varepsilon_j \to 0$ as $j \to \infty$ and

Fix $j\in \mathbb{N}$ arbitrarily, and set $\varepsilon_j = \varepsilon$ for short. By (4.2) we find $x_{\varepsilon}\in \mathcal{I}(u_{\varepsilon})$ such that

where

Let $\eta \in C_{\rm c}^{\infty}(B(x_\varepsilon, 2 r_\varepsilon))$ be a cut-off function with

where $C$ is a positive constant. Let $v_{\varepsilon} : = u_{\varepsilon} - \eta u_{\varepsilon}$. Then we see that $v_\varepsilon\in W^{2, p}_0(I)$ and

Recalling that

by the minimality of $u_{\varepsilon}$ we have $E_\varepsilon(u_\varepsilon) \le E_\varepsilon(v_\varepsilon)$, i.e.,

Since it follows from (3.1) that

plugging (4.6) into (4.5), we obtain

From (4.4) we see that

Similarly we have

Combining (4.7) with (4.8) and (4.9), we find

Since it follows from (2.1) that

and

we reduce (4.10) into

By Theorem 3.7 we have

where $y \in \partial \mathcal{I}(u_\varepsilon)$, and the constant $C$ is independent of $\varepsilon$. Moreover, we deduce from (4.12) that

where $y \in \partial \mathcal{I}(u_\varepsilon)$. It follows from (4.3), (4.12) and (4.13) that

where $C > 0$ is independent of $\varepsilon$. Similarly, we infer from (4.3), (4.12), (4.13) and Theorem 3.7 that

where $C > 0$ is independent of $\varepsilon$. Plugging (4.14) and (4.15) into (4.11), we obtain

where $C > 0$ is independent of $\varepsilon$. Since $\| u'_\varepsilon\|_{L^p(I)} = 1$, by (2.1) we have

Taking $x_\varepsilon \in \mathcal{I}(u_\varepsilon)$ sufficiently close to $\partial \mathcal{I}(u_\varepsilon)$, we see that

This together with (4.16) implies that

Letting $\varepsilon \to 0$, we lead a contradiction. Therefore Theorem 4.1 follows.

We are in a position to prove Theorem 1.1:

proof of Theorem 1.1. By Lemma 3.1 we see that there exists a solution $u_\varepsilon$ of $({\rm P}_\varepsilon)$ for each $\varepsilon > 0$. Thanks to Theorem 4.1, we find $\varepsilon_0 > 0$ such that

This implies that $u_\varepsilon$ is a solution of problem $({\rm{P}})$, providing that $\varepsilon > 0$ is small enough. Moreover, it follows from Lemma 3.2 that $(u_\varepsilon, \Lambda_\varepsilon)$ satisfies problem (1.4) in a weak sense. Finally, we deduce from Lemma 3.4 that $\mathcal{I}(u_\varepsilon)$ is connected. Therefore Theorem 1.1 follows.

Acknowledgments

The second author was partially supported by the Grant-in-Aid for Scientific Research (S) (No. 19H05599). Moreover, the authors would like to thank the referee for useful comments.

Conflict of interest

The authors declare no conflict of interest.