Abstract
In this article, we consider positivity issues for the clamped plate equation with high tension \(\gamma >0\). This equation is given by \(\Delta ^2u - \gamma \Delta u=f\) under clamped boundary conditions. Here, we show that given a positive f, i.e. upwards pushing, we find a \(\gamma _0>0\) such that for all \(\gamma \ge \gamma _0\) the bending u is indeed positive. This \(\gamma _0\) only depends on the domain and the ratio of the \(L^1\) and \(L^\infty \) norm of f. In contrast to a recent result by Cassani and Tarsia, our approach is valid in all dimensions.
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1 Introduction
The Boggio-Hadamard conjecture states, that for a given convex, open, bounded set \(\Omega \subseteq {\mathbb R}^n\), an \(f\ge 0\) and outer normal \(\nu \) of \(\Omega \), a solution u to
is nonnegative, i.e. \(u\ge 0\) (cf. [26, 27]). Problem (1.1) models the bending u of a clamped plate \(\Omega \) under a force f. Hence the problem can be restated as:
The conjecture was substantiated by Boggio’s explicit formula [6] (see also [15, Lemma 2.27] or [28]) for the Greens function of problem (1.1) on the unit disc, because this function is positive, and furthermore by Almansi’s result to calculate the Greens function for certain domains by the Greens function of the unit ball (see [3]).
Several other domains than the disc have been found on which such a positivity preserving property holds (see the references below). Hadamard himself claimed in [27] that such a property for all limaçons is true, which turned out to be wrong in general, though some of them still possess this property (see [15, Fig. 1.2]). Remarkable is that such limaçons are not convex. In [20], Grunau and Robert showed that positivity preserving is preserved under small regular perturbations of the domain in dimensions \(n\ge 3\).
The conformal invariance of the problem was also successfully used to construct domains with a positivity preserving property by, e.g. Dall’Acqua and Sweers in [10] (see also the references therein for more information on such domains).
On the other hand, several counterexamples have been found by now. The first one was by Duffin on an infinite strip [12] and shortly after Garabedian [14] showed that on an elongated ellipse the Greens function changes sign. By now even for uniform forces, i.e. \(f\equiv 1\), counterexamples have been found by Grunau and Sweers in [23] and [24]. We refer to [15, Sect. 1.2] for a comprehensive historical overview to this problem.
Instead of examining (1.1) for positivity, Cassani and Tarsia in [9] examined positivity issues for
with \(\gamma >0\) big enough. The basic motivation is that for \(\gamma \) big enough, the influence of \(\Delta u\) (\(-\Delta u=f\) satisfies positivity preserving via the maximum principle) becomes stronger than that of \(\Delta ^2 u\). In more technical detail Cassani and Tarsia conjectured the existence of a \(\gamma _0=\gamma _0(f,\Omega )\ge 0\), such that \(u\ge 0\) for all \(\gamma \ge \gamma _0\) and provided a proof for dimensions \(n=2,3\), smooth, bounded \(\Omega \) and positive \(f\in L^2(\Omega )\). In this article we provide a different approach, which is valid for all dimensions, see Theorem 1.1.
In dimension \(n=1\), this positivity preserving property is true for all \(\gamma >0\) independent of f. This was shown by Grunau in [19] Proposition 1.
The parameter \(\gamma \) is usually called tension, if it is positive. Several results concerning (1.2) have been achieved, which are usually concerned with vibrations of the plate, i.e. eigenvalue problems. Bickley gave some explicit calculations for the spectrum in the unit disc in [5] already in 1933. Hence the existing literature for these eigenvalue problems is quite vast and is still developing, see, e.g. [8, 4] and the references therein.
Other modifications for (1.1) concerning positivity issues are, e.g. changing the boundary conditions to so called Steklov conditions. This has been examined by, e.g. Bucur and Gazzola in [7].
Different elliptic differential operators of higher order, their respective fundamental solutions and their sign close to a singularity have also been examined by Grunau, Romani and Sweers in [22] in a more systematic approach to understand better the loss of positivity preserving.
Instead of (1.2), we examine the following boundary value problem for positivity preserving. This is obviously equivalent, but (1.3) yields the advantage, that the singularity of the equation is more prominent and hence yields easier access to necessary estimates.
Here \(\Omega \subset \subset {\mathbb R}^n\), \(\partial \Omega \in C^4\), \(f\in L^\infty (\Omega )\) and \(\varepsilon >0\), and the solution \(u_\varepsilon \in W^{4,p}(\Omega ) \text{ for } \text{ all } 1< p < \infty \) (see, e.g. [15, Corollary 2.21] and the references therein for existence, regularity and uniqueness to (1.3)).
Theorem 1.1
For connected \(\Omega \subset \subset {\mathbb R}^n\), \(\partial \Omega \in C^4\), \(f\in L^\infty (\Omega )\), \(\tau > 0, f \ge 0\) with
there exists \(\varepsilon _0 = \varepsilon _0(\Omega ,\tau ) > 0\) such that
Please note that we do not have any restrictions on the dimension, i.e. \(n\in {\mathbb N}\) arbitrary. Furthermore, our method yields that \(\varepsilon _0\) does not depend on f directly, only on \(\tau \).
The strategy of the proof is as follows: The limiting problem of (1.3) is
which admits a maximum principle, and establishes positivity of \(u_\varepsilon \text{ on } \text{ any } \Omega ' \subset \subset \Omega \text{ for } \varepsilon \) small.
We proceed by contradiction and assume Theorem 1.1 is false. Hence for every \(\varepsilon >0\), we find a nonnegative \(f_\varepsilon \in L^\infty (\Omega )\) satisfying (1.4), such that \(u_\varepsilon \) is not positive in \(\Omega \).
Then, we examine a blow-up of our solutions \(u_\varepsilon \), which is weighted by the supremum of the modulus of the Laplacian at the boundary, i.e. \(\sup _{\partial \Omega } \varepsilon ^2|\Delta u_\varepsilon |\). After a careful analysis (see Sects. 2 and 3), we can show that this blow-up converges in a suitable sense to a solution of \(\Delta ^2 u-\Delta u=0\) on the half-space with Dirichlet boundary conditions (see Sect. 4). With a uniqueness result shown in Appendix A, we explicitly calculate this limit and obtain positivity of the Laplacian of \(u_\varepsilon \) on the boundary for \(\varepsilon \) small. This is crucial, as in the presence of Dirichlet boundary conditions in (1.3) the Laplacian is the second normal derivative of \(u_\varepsilon \) on the boundary, and therefore positivity of the Laplacian on the boundary gives positivity of \(u_\varepsilon \) close to the boundary, see Sect. 4.
Similar strategies of examining a blow-up to the half space and using explicit formulas have been employed by Grunau and Robert in [20] and Grunau, Robert and Sweers in [21] to show lower bounds for the Greens function of a polyharmonic operator. This method was later refined by Pulst in his PhD-thesis [30] to also obtain such estimates, if non-constant lower-order terms are present. If variable coefficients in the principal part of the operator are given by a power of a second-order elliptic linear operator, such estimates were found by the same method by Grunau in [18].
Our blow-up strategy needs careful estimates for the singular problem (1.3). Estimates for these kinds of problems have a long history, see, e.g. [13, 17, 25] and [29]. We are not aware of any specific estimates, which would help in our specific situation. For this reason and for the sake of completeness, we derive them here.
2 Preliminary estimates
We proceed by contradiction and assume we find nonnegative \(f_\varepsilon \in L^\infty (\Omega )\) for \(\varepsilon \downarrow 0\) satisfying (1.4), such that the solution \(u_\varepsilon \) of (1.3) is sign-changing. As (1.3) is homogeneous of degree one, we may assume by scaling
By the Banach–Alaoglu Theorem, we get after passing to subsequence and relabelling \(f_\varepsilon \rightarrow f \text{ weakly}^* \text{ in } L^\infty (\Omega )\) with
In particular, we have \(f \not \equiv 0\).
The limiting problem of (1.3) is thought to be the second-order boundary-value problem
We also consider
These two problems admit some important estimates:
Proposition 2.1
For u and \(u_{0,\varepsilon }\) in (2.3) rsp. (2.4) we find a constant \(c_0=c_0(\Omega ,\tau )>0\), such that
and
for \(\varepsilon >0\) small enough.
Proof
Both problems admit by standard elliptic theory, see [16] Theorem 9.15, unique solutions \(u \text{ respectively } u_{0,\varepsilon } \in W^{2,p}(\Omega ) \hookrightarrow C^{1,\alpha }(\Omega ) \text{ for } \text{ all } 1< p < \infty \text{ and } 1 - (n/p)> \alpha > 0\) with
In particular, the set of all \(u, u_{0,\varepsilon } \text{ for } \text{ any } f, f_\varepsilon \) with (2.2) and (2.1) is compact in \(C^1(\overline{\Omega })\), and in particular
and weakly in \(W^{2,p}(\Omega ) \text{ for } \text{ all } 1< p < \infty \).
As \(f, f_\varepsilon \ge 0, f, f_\varepsilon \not \equiv 0 \text{ and } \Omega \) is connected, we get by the strong maximum principle, see [16] Theorem 8.19, that
By Hopf’s maximum principle, see [16] Lemma 3.4, and by compactness of the \(u, u_{0,\varepsilon } \text{ in } C^1(\overline{\Omega })\) we find a constant \(c_0 = c_0(\Omega ,\tau ) > 0\) such that (2.5) and (2.6) both hold for \(\varepsilon >0\) small enough.
We put \(v_\varepsilon := u_\varepsilon - u_{0,\varepsilon } \in W^{2,p}(\Omega ) \cap C^{1,\alpha }(\Omega ) \text{ with } v_\varepsilon = 0 \text{ on } \partial \Omega \). Subtracting in (1.3), we see
that is \(\varepsilon ^2 \Delta u_\varepsilon - v_\varepsilon \) is harmonic in \(\Omega \).
Equation (2.3) is indeed the limiting problem of (1.3) in the sense of the following proposition.
Proposition 2.2
For \(u_\varepsilon , u, v_\varepsilon = u_\varepsilon - u_{0,\varepsilon }\) as in (1.3), (2.1), (2.4), we have
Proof
Multiplying (2.10) by \(v_\varepsilon \) and integrating by parts, we get
Replacing \(\varepsilon ^2 \Delta ^2 u_\varepsilon \text{ by } \Delta v_\varepsilon \) in the second term with (2.10), we continue
in particular \(\parallel \nabla v_\varepsilon \parallel _{L^2(\Omega )} \le \parallel \nabla u_{0,\varepsilon } \parallel _{L^2(\Omega )} \le C(\Omega )\) by (2.7), hence
Passing to a subsequence, we get
Multiplying (2.10) by some \(\eta \in C^\infty _0(\Omega )\) and passing to a subsequence, we get
and \(v \in W^{1,2}_0(\Omega )\) is harmonic in \(\Omega \), hence \(v = 0 \text{ and } u_\varepsilon \rightarrow u \text{ weakly } \text{ in } W^{1,2}_0(\Omega )\). Returning to (2.12), we improve now to
which is (2.11). \(\square \)
The following proposition shows that the Laplacian cannot be bounded throughout \(\Omega \).
Proposition 2.3
For \(u_\varepsilon , \tau \) as in (1.3), (2.1), (2.2), we have for any \(\Omega ' \subset \subset \Omega \) that
Proof
We see for any \(\eta \in C^\infty _0(\Omega )\) with the previous Proposition 2.2
and the homogeneous boundary conditions in (1.3) that
Choosing \(\eta \in C^\infty _0(\Omega ) \text{ with } 0 \le \eta \le 1 \text{ and } \eta \equiv 1 \text{ in } \Omega '\), we get
Letting \(\eta \nearrow \chi _\Omega \), we get from (2.2) that
and the proposition follows.
3 The Laplacian on the boundary
In this section, we investigate the values of the Laplacian on the boundary and put
With subscripts ±, we denote the positive, respectively, negative part, i.e.
Furthermore, we set
The quantity \(M_\varepsilon \) will be crucial throughout the exposition. Our goal is to show that it has the same asymptotic as \(\varepsilon \) itself, i.e. we find constants \(c_0,C>0\) such that
for \(\varepsilon >0\) small. A first step in this direction is Proposition 3.3, which will later be improved to our desired result in Proposition 4.2 and (4.19).
With the maximum principle, we get the following estimates.
Proposition 3.1
For \(u_\varepsilon , f_\varepsilon , u_{0,\varepsilon }, v_\varepsilon = u_\varepsilon - u_{0,\varepsilon }, {\ell ^{\pm }_{\varepsilon }}\) as in (1.3), (2.1), (2.4), (3.1), we have
Proof
As \(v_\varepsilon = u_\varepsilon - u_{0,\varepsilon } = 0 \text{ on } \partial \Omega \) by (1.3) and (2.4), we get (3.4) from (2.10).
Adding (2.4), we see
in particular
Since \(v_\varepsilon \ge -\varepsilon ^2 f_\varepsilon - {\ell ^{+}_{\varepsilon }}_{,+} \text{ on } \partial \Omega _0\), as \(v_\varepsilon = 0 \text{ on } \partial \Omega \) by above, we get from the mean-value estimate for superharmonic functions or by Alexandroff’s maximum principle, as \(v_\varepsilon \in W^{2,n}(\Omega )\), see [16] Theorem 9.1, that \(v_\varepsilon \ge -\varepsilon ^2 f_\varepsilon - {\ell ^{+}_{\varepsilon }}_{,+} \text{ in } \Omega _0\), hence \(\Omega _0 = \emptyset \), and the left estimate in (3.5) follows. The right estimate is obtained by symmetry observing that \(f_\varepsilon \ge 0\).
Next for \(x \in \overline{\Omega } \text{ with } u_\varepsilon (x) = \min _{\overline{\Omega }} u_\varepsilon \) and assuming that this minimum is negative, we see \(x \in \Omega \), as \(u_\varepsilon = 0 \text{ on } \partial \Omega \) by (1.3), hence \(\Delta u_\varepsilon (x) \ge 0\), as \(u_\varepsilon \in W^{4,p}(\Omega ) \hookrightarrow C^2(\Omega ) \text{ for } 2 - (n/p) > 0\). Then, we get with (3.4) and (2.9) that
which is (3.6). \(\square \)
Using the fourth-order equation, we get estimates for the Laplacian.
Proposition 3.2
For \(u_\varepsilon , f_\varepsilon , u_{0,\varepsilon }, v_\varepsilon = u_\varepsilon - u_{0,\varepsilon }, {\ell ^{\pm }_{\varepsilon }}\) as in (1.3), (2.1), (2.4), (3.1), we have
Proof
We have with (1.3) that
and get
Since \(\Delta u_\varepsilon \ge -\parallel f_{\varepsilon ,+} \parallel _{L^\infty (\Omega )} - \varepsilon ^{-2} {\ell ^{-}_{\varepsilon }}_{,-} \text{ on } \partial \Omega _0\), as \(\Delta u_\varepsilon \ge \varepsilon ^{-2} {\ell ^{-}_{\varepsilon }}\ge -\varepsilon ^{-2} {\ell ^{-}_{\varepsilon }}_{,-} \text{ on } \partial \Omega \) with (3.1), we get from the mean-value estimate for superharmonic functions or by Alexandroff’s maximum principle, as \(\Delta u_\varepsilon \in W^{2,n}(\Omega )\), see [16] Theorem 9.1, that \(\Delta u_\varepsilon \ge -\parallel f_{\varepsilon ,+} \parallel _{L^\infty (\Omega )} - \varepsilon ^{-2} {\ell ^{-}_{\varepsilon }}_{,-} \text{ in } \Omega _0\), hence \(\Omega _0 = \emptyset \), and the left estimate in (3.7) follows. The right estimate is obtained by symmetry observing that \(f_{\varepsilon } \ge 0\). \(\square \)
Here, we can give a preliminary asymptotic estimate for \({ M_\varepsilon }\). Actually, we will improve this asymptotic later in Proposition 4.2 and (4.19). Anyway we present this estimate at this stage to get more compact bounds already now.
Proposition 3.3
For \(u_\varepsilon , { M_\varepsilon }, {\ell ^{\pm }_{\varepsilon }}\) as in (1.3), (2.1), (3.1), (3.3), we have
in particular \({ M_\varepsilon }\ge {\ell ^{+}_{\varepsilon }}= {\ell ^{+}_{\varepsilon }}_{,+} > 0 \text{ for } \varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Proof
Combining Proposition 3.2 (3.7) and Proposition 2.3, we get for any \(\Omega ' \subset \subset \Omega \) that
hence, as \(\mathcal{L}^n(\Omega \setminus \Omega ')\) can be made arbitrarily small, that
which yields the assertion. \(\square \)
With the above asymptotic, we can already bound \(v_\varepsilon \), and we can prove that u is positive on large parts of \(\Omega \).
Proposition 3.4
For \(u_\varepsilon , u_{0,\varepsilon }, v_\varepsilon = u_\varepsilon - u_{0,\varepsilon }, { M_\varepsilon }\) as in (1.3), (2.1), (2.2), (2.4), (3.3), we have
Proof
Combining Proposition 3.1 (3.5) and Proposition 3.3, we get observing (2.1) that
\(\square \)
Proposition 3.5
For \(u_\varepsilon , {\ell ^{\pm }_{\varepsilon }}\) as in (1.3), (2.1), (3.1), we have
for some \(C = C(\Omega ,\tau ) < \infty \text{ and } \varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Proof
From (2.6), we see for \(x \in \Omega \text{ with } c_0 d(x,\partial \Omega ) > {\ell ^{+}_{\varepsilon }}_{,+} + \varepsilon ^2 \parallel f_\varepsilon \parallel _{L^\infty (\Omega )}\) by (3.5) that
By Proposition 3.3 and (2.1) clearly
for \(\varepsilon \) small, and (3.8) follows for \(C = 2 c_0^{-1} < \infty \). \(\square \)
4 Blow-up
In this section, we consider a blow-up of our solutions \(u_\varepsilon \) by translating and rescaling with \(x_{0,\varepsilon } \in {\mathbb R}^n\). We will have to choose \(x_{0,\varepsilon }\) differently in different steps in the proof of Theorem 1.1. In, e.g. the proof of Claim 1 in the proof of Theorem 1.1, we choose \(x_{0,\varepsilon }\in \partial \Omega \) such that \(\max _{x\in \partial \Omega }|\varepsilon ^2 \Delta u_\varepsilon (x)|\) is attained, while in the proof of Claim 2, we choose \(x_{0,\varepsilon }\in \partial \Omega \) to attain \({\ell ^{-}_{\varepsilon }}\). This will not cause a problem, because the constants yielded by these claims do not depend on \(\varepsilon \). We put
Then
and by (1.3) that
and with (2.1) and Proposition 3.3 that
where \(o(1) \rightarrow 0\) depending on \(\Omega \text{ and } \tau \). We extend \(u_\varepsilon \text{ respectively } \tilde{u}_\varepsilon \) by putting \(0 \text{ outside } \Omega \text{ respectively } \text{ outside } \tilde{\Omega }_\varepsilon \). By the homogeneous boundary conditions in (1.3) and (4.3), we see \(u_\varepsilon , \tilde{u}_\varepsilon \in W^{2,2}_{loc}({\mathbb R}^n)\).
We also have to stretch \(\tilde{u}_\varepsilon \) to get a nontrivial limit, and it turns out that reaching bounded values of the Laplacian of \(\tilde{u}_\varepsilon \) on the boundary is the right measure for stretching.
Proposition 4.1
For \(\tilde{u}_\varepsilon , \tilde{f}_\varepsilon , u_\varepsilon , f_\varepsilon , { M_\varepsilon }\) as in (4.1), (1.3), (2.1), (3.3) and \(x_{0,\varepsilon } \in \partial \Omega \), we get for any subsequence with after rotating \(\tilde{\Omega }_\varepsilon \) such that
after passing to a subsequence
as \(\varepsilon \rightarrow 0\) after flattening the boundary of \(\partial \tilde{\Omega }_\varepsilon \). Further
Proof
We get from Proposition 3.1 (3.6) that
and also outside \(\tilde{\Omega }_\varepsilon \), as \(\tilde{u}_\varepsilon = 0\) there. Next by Proposition 3.2 (3.7), Proposition 3.3, (2.1) and (4.2) that
Then, \(\tilde{u}_\varepsilon + { M_\varepsilon }\ge 0 \text{ in } {\mathbb R}^n\), and we can apply the Harnack inequality, see [16] Theorems 8.17 and 8.18, and get observing that \(\tilde{u}_\varepsilon (0) = 0\), as \(x_{0,\varepsilon } \in \partial \Omega \), that
hence
for \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Next by Friedrichs’s Theorem in the interior, see [16] Theorem 8.8, [16] Exercise 8.2, (4.9) and (4.10) that
for \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \), which yields the first convergence in (4.6) after passing to a subsequence.
Proceeding from (4.3), we get from fourth-order \(L^p-\)estimates, see [1, 2] Sect. 10, after flattening the boundary of \(\partial \tilde{\Omega }_\varepsilon \), as \(\partial \Omega \in C^4\), with (4.4) and (4.10) that
and \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \). After passing to a subsequence, we obtain with (4.5) the second convergence in (4.6).
Finally (4.7) follows from (4.3), (4.4), (4.8), (4.9), when recalling that \(\tilde{u}_\varepsilon = 0 \text{ in } {\mathbb R}^n \setminus \Omega _\varepsilon \).
Actually by fourth-order higher-order \(L^p-\)estimates, see [1, 2] Sect. 10, we get that the blow-up \(\tilde{u}_\infty \) is smooth on \(\overline{{\mathbb R}^n_+}\).
Now we are able to give a lower bound for \({ M_\varepsilon }\) which improves the asymptotic in Proposition 3.3.
Proposition 4.2
For \(u_\varepsilon , { M_\varepsilon }\) as in (1.3), (2.1), (3.3), we have
for some \(c_0 = c_0(\Omega ,\tau ) > 0 \text{ and } \varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Proof
We see for \(x_\varepsilon \in \Omega \text{ with } |x_\varepsilon - x_{0,\varepsilon }| = d(x_\varepsilon ,\partial \Omega ) = \varepsilon \text{ for } \varepsilon \) small by (2.6), Proposition 3.4 and by the local boundedness of \(\tilde{u}_\varepsilon \) in Proposition 4.1, or more directly by (4.10), for \(\tilde{x}_\varepsilon := (x_\varepsilon - x_{0,\varepsilon }) / \varepsilon \in \overline{B_1(0)}\) that
hence
for \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \). \(\square \)
The blow-up for \(u_{0,\varepsilon }\) is rather elementary by the strong convergence in \(C^{1,\alpha }(\Omega )\) in (2.8). As in (4.1), we put
Proposition 4.3
For \(\tilde{u}_{0,\varepsilon }, u_\varepsilon , u_{0,\varepsilon }, f_\varepsilon , { M_\varepsilon }\) as in (4.12), (1.3), (2.1), (3.3), and with (4.5), we have after passing to a subsequence such that
for some \(c_0 = c_0(\Omega ,\tau ) > 0\) for the linear function \(\tilde{u}_{0,\infty }: (y,t) \rightarrow \beta t\) that
after flattening the boundary of \(\partial \tilde{\Omega }_\varepsilon \). Further
in particular \(\tilde{u}_\infty - \tilde{u}_{0,\infty } \in L^\infty ({\mathbb R}^n_+)\).
Proof
From (2.7) and, as \(u_{0,\varepsilon } = 0 \text{ on } \partial \Omega \) by (2.4), we get by Taylor’s expansion for any \(x \in {\mathbb R}^n_+ \text{ and } \text{ any } 0< \alpha < 1\) that
As \(u_{0,\varepsilon } = 0 \text{ on } \partial \Omega \) by (2.4), we get with (2.8), (4.5) and \(\partial \Omega \in C^4\) after passing to a subsequence with \(x_{0,\varepsilon } \rightarrow x_0 \in \partial \Omega \) that
in particular with (2.5) that
and \(\nabla u_{0,\varepsilon }(x_{0,\varepsilon }) / |\nabla u_{0,\varepsilon }(x_{0,\varepsilon })| \rightarrow e_n\). By Proposition 4.2 we extract a subsequence, such that \(\varepsilon /M_\varepsilon \) converges for \(\varepsilon \downarrow 0\). Then, (2.5) yields
which is (4.13). Furthermore, Proposition 4.2 yields
Together, we get
and the proposed convergence follows from the Taylor expansion (4.15).
Further we get with Proposition 3.4 that
which is (4.14).
By our investigation of half space solutions in Appendix A, Proposition A.3 applied to \(\tilde{u}_\infty \text{ and } \tilde{u}_{0,\infty }\) with Proposition 4.1 (4.7) and Proposition 4.3 (4.14) determines \(\tilde{u}_\infty \) uniquely as the one-dimensional solution and immediately yields the following Proposition.
Proposition 4.4
For \(\tilde{u}_\infty \) as in Proposition 4.1 and \(\beta \) as in Proposition 4.3, we have
and
\(\square \)
Now we are able to conclude the proof of Theorem 1.1.
Proof of Theorem 1.1
We consider \(u_\varepsilon , f_\varepsilon , u_{0,\varepsilon }, {\ell ^{\pm }_{\varepsilon }}, { M_\varepsilon }\) as in (1.3), (2.1), (2.4), (3.1), (3.3) and \(\tilde{u}_\varepsilon , \tilde{f}_\varepsilon , \tilde{u}_{0,\varepsilon }\) as in (4.1), (4.12) with their blow-ups \(\tilde{u}_\infty , \tilde{u}_{0,\infty } \text{ and } \beta \) obtained in the Propositions 4.1 and 4.3. We prove various claims.
Claim 1
hence with Proposition 4.2 that
for some \(c_0 = c_0(\Omega ,\tau ) > 0, C = C(\Omega ,\tau ) < \infty \text{ and } \varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Proof
If on contrary \(\varepsilon / { M_\varepsilon }\rightarrow 0\) for a subsequence \(\varepsilon \rightarrow 0\), then we get from Proposition 4.3 and (2.7) that \(\beta = 0\), hence with Proposition 4.4 (4.16) after passing to this subsequence we have that \(\tilde{u}_\infty \equiv 0\).
On the other hand, choosing \(x_{0,\varepsilon } \in \partial \Omega \) in such a way that
we get from the convergence in Proposition 4.1 (4.6) that
hence \(\Delta \tilde{u}_\infty (0) \ne 0 \text{ and } \tilde{u}_\infty \not \equiv 0\). This is a contradiction, and the claim follows.
Claim 2
for some \(c_1 = c_1(\Omega ,\tau ) > 0 \text{ and } \varepsilon \) small depending on \(\Omega \text{ and } \tau \) and
Proof
(4.19) implies with Proposition 4.3 (4.13) that
which immediately gives (4.23) by Proposition 4.4 (4.18).
Next we choose \(x_{0,\varepsilon } \in \partial \Omega \) in such a way that
and get as in (4.20) from the convergence in Proposition 4.1 (4.6) and Proposition 4.4 (4.17) that
and (4.21) follows. Clearly (4.21) implies with (3.1) that
for some \(c_1 > 0 \text{ and } \varepsilon \) small, which is (4.22).
Further (4.21) implies that \({\ell ^{+}_{\varepsilon }}\ge {\ell ^{-}_{\varepsilon }}> 0 \text{ and } {\ell ^{-}_{\varepsilon }}_{,-} = 0\) for \(\varepsilon \) small, hence with (3.3) that
for \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Claim 3
for some \(C = C(\Omega ,\tau ) < \infty \text{ and } \varepsilon \) small depending on \(\Omega \text{ and } \tau \).
This follows directly from Proposition 3.5, (4.19) and (4.24).
Claim 4
for some \(c_2 = c_2(\Omega ,\tau ) > 0 \text{ and } \varepsilon \) small depending on \(\Omega \text{ and } \tau \).
By the homogeneous boundary conditions in (1.3), we have \(\Delta u_\varepsilon (x_{0,\varepsilon }) = \partial _{\nu \nu } u_\varepsilon (x_{0,\varepsilon })\), hence with (4.22), Proposition 4.1 (4.6) and the embedding \(W^{4,p} \hookrightarrow C^3 \text{ for } 1 - (n/p) > 0\) that
for \(0< t< c_1 / (2 { C_{n} }) < 1 \text{ and } \varepsilon \) small. As \(x_{0,\varepsilon } \in \partial \Omega \) can be chosen arbitrarily, we get for \(c_2 = c_1 / (4 { C_{n} })\) that
and the claim follows by rescaling in (4.1).
Claim 5
for \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \).
Here for any \(x_\varepsilon \in \Omega \text{ with } c_2 \varepsilon \le d(x_\varepsilon ,\partial \Omega ) \le C \varepsilon \text{ for } \varepsilon \) small, we select \(x_{0,\varepsilon } \in \partial \Omega \text{ with } d(x_\varepsilon ,\partial \Omega ) = |x_\varepsilon - x_{0,\varepsilon }|\) and get for \(\tilde{x}_\varepsilon := \varepsilon ^{-1} (x_\varepsilon - x_{0,\varepsilon }) \in \tilde{\Omega }_\varepsilon \) that
Passing to subsequence, we get \(\tilde{x}_\varepsilon \rightarrow \tilde{x} \text{ with } \tilde{x} \in \overline{{\mathbb R}^n_+}\) and
hence \(\tilde{x} \in {\mathbb R}^n_+\). Then by Proposition 4.1 (4.6) and (4.23) that
and we conclude that \(u_\varepsilon (x_\varepsilon ) > 0 \text{ for } \varepsilon \) small, and the claim follows.
Combining (4.25), (4.26) and (4.27), we get
for \(\varepsilon \) small depending on \(\Omega \text{ and } \tau \), which proves Theorem 1.1.
References
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Appendices
Appendix
Uniqueness of a bi-Laplace equation on the half space
In this section, we show the following uniqueness theorem.
Theorem A.1
Let \(v:\overline{{\mathbb R}^{n}_+}\rightarrow {\mathbb R}\) be a smooth solution to
Then, \(v= 0\).
The proof is based on an energy type estimate and on the one-dimensional case. We start with the one-dimensional case and show the following lemma:
Lemma A.1
Let \(v\in C^4_{loc}([0,\infty [)\) satisfy
Then, \(v=0\).
Proof
Since the differential equation is ordinary and linear, the solution space of the equation itself is of dimension 4. By inserting the following functions, we see that they constitute a basis of the solution space
The following functions are therefore a basis of the solution space including the initial conditions at \(t=0\):
Hence v has to be of the form
Inserting the exponential function for \(\cosh \) and \(\sinh \) yields
Since the solution is bounded, we have \(A=-B\). Again the boundedness then yields \(B=0\), to rule out linear growth. Hence \(v=0\).
For our next step, we introduce a bit of notation for half spaces
Next we show an energy type estimate.
Lemma A.2
Let v satisfy
Then, there exists a constant \(C=C(n)>0\), such that for all \(R>1\) we have
Proof
Let \(\eta \in C^\infty _0(B_{2R}(0))\) with \(0\le \eta \le 1\), \(\eta =1\) on \(B_R(0)\) and for any \(k\ge 1\)
i.e. \(\eta \) is a cut-off function for the ball \(B_R(0)\). Then, \(v \eta ^4\) and its first derivative are zero on \(\partial B_{2R}^+\). Therefore, partial integration and the differential equation itself yield
By the previous identity and Young’s inequality with an \(\varepsilon >0\) we get
Now choosing \(\varepsilon \) small enough, we can absorb the terms with \(\varepsilon \) as a prefactor into the left hand side
Together with \(0\le \eta \le 1\), the estimates on the derivatives of the cut-off function (A.1) yield
Since we have chosen \(R>1\), we obtain
Now we can show our main result Theorem A.1, by iterating Lemma A.2:
Proof of Theorem A.1
Since v is bounded and the differential equation is linear and elliptic, we can employ Schauder-type estimates (see [1, Thm. 6.2] and [11, Sect. 4]) to obtain
Let \(x=(y,t)\in \overline{{\mathbb R}^{n}_+}\), such that \(y\in {\mathbb R}^{n-1}\) and \(t\ge 0\). By \(\partial _y^k v\), we denote any partial derivative of v of order k only after horizontal directions, i.e. indices in \(\{1,\ldots ,n-1\}\). Then, for every \(k\in {\mathbb N}\) the function \(\partial ^k_{y} v:\overline{{\mathbb R}^n_+}\rightarrow {\mathbb R}\) still solves the differential equation, satisfies the Dirichlet boundary conditions and by (A.2) is again bounded. Now we can iteratively apply Lemma A.2 for \(k\ge \ell \in {\mathbb N}\) and obtain
By choosing \(k=\ell \), (A.2) yields
If \(k>2n\), this yields for \(R\rightarrow \infty \):
Hence \(\partial ^k_{y} v=0\). This implies
Therefore \(\partial ^{k-1}_{y}v(y,t)\) is independent of \(y\in {\mathbb R}^{n-1}\) and \(t\mapsto \partial ^{k-1}_{y}v(y,t)\) satisfies the assumptions of Lemma A.1. Hence, we also have
Especially we have
for all \(i=1,\ldots , n-1\). Hence again \(\partial ^{k-2}_{y}v(y,t)\) is independent of y and therefore again satisfies the assumptions of Lemma A.1. Iterating this final process yields
which is the desired conclusion. \(\square \)
Remark: The proof above works, because both \(\Delta ^2 \) and \(-\Delta \) are monotone operators which are added correctly. If we would destroy this monotonicity by, e.g. examining \(\Delta ^2 + \Delta \), Theorem A.1 is not true anymore. For example, a bounded nontrivial solution to \(\Delta ^2 v + \Delta v = 0\), is \((y,t) \mapsto 1 - \cos t\). \(\square \)
Proposition A.3
Let \(u:\overline{{\mathbb R}^{n}_+}\rightarrow {\mathbb R}\) be a smooth solution of the fourth-order boundary-value problem
Furthermore, let u for some \(\beta \in {\mathbb R}\) and the corresponding linear function \(u_0: (y,t) \mapsto \beta t\) satisfy
Then, u is one dimensional, that is
in particular
and
Proof
We put
and see \(v \in C^\infty _{loc}(\overline{{\mathbb R}^n_+})\) and
Then, the uniqueness in Proposition A.1 gives \(v \equiv 0\), which is (A.4), and by direct calculation
which is (A.5). For \(\beta > 0\), we get by the strict convexity of the exponential function
which is (A.6).
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Eichmann, S., Schätzle, R.M. Positivity for the clamped plate equation under high tension. Annali di Matematica 201, 2001–2020 (2022). https://doi.org/10.1007/s10231-022-01188-9
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DOI: https://doi.org/10.1007/s10231-022-01188-9