A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

It is well known that for higher order elliptic equations the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e. nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function.


Introduction
One of the main obstructions in the development of the theory of higher order elliptic equations is represented by the loss of general maximum principles, see e.g. [8,Chapter 1]. Nevertheless, due to the central role that these technical tolls play in the general theory of second order elliptic equations, in the last century a large part of literature has focused in studying whether the related boundary value problems possibly enjoy the so-called positivity preserving property (PPP in the following). As a matter of example, let us consider the clamped plate problem: where Ω ⊂ R n is a bounded domain and f ∈ L 2 (Ω); we say that the above problem satisfies the PPP if the following implication holds where u is a (weak) solution to (1). The validity of the PPP generally depends either on the choice of the boundary conditions and on the geometry of the domain. For instance, from the seminal works by Boggio [4,5], it is known that problem (1) satisfies the PPP when Ω is a ball in R n , while, in [6], Coffman and Duffin proved that the PPP does not hold when Ω is a two dimensional domain containing a right angle, such as a square or a rectangle, see also Figure 1 below. Things become somehow simpler if in (1), instead of the Dirichlet boundary conditions, we take the Navier boundary conditions, i.e. we consider the hinged plate problem: in Ω u = ∆u = 0 on ∂Ω.
Here, the PPP follows by applying twice the comparison principle for the laplacian under Dirichlet boundary conditions. It is worth noticing that smoothness of the domain cannot be overlooked since it has been shown by Nazarov and Sweers [13] that, also in this case, the PPP may fail for planar domains with an interior corner. We refer to the book [8] for more details and PPP results under different kind of boundary conditions, e.g. Steklov boundary conditions, and to [9,10,11,14,15,16,17,19] for up to date results on the topic.
In the present paper we focus on the less studied partially hinged plate problem which arises in several mathematical models having engineering interest, e.g. models of bridges or footbridges. In particular, a 2-d model for suspension bridges has been proposed in [7]; here the bridge is seen as a thin long rectangular plate Ω ⊂ R 2 hinged at the short edges, see also [3] for further details. More precisely, if, by scaling, we assume that Ω = (0, π) × (− , ) with > 0, the partially hinged problem writes: in Ω u(0, y) = u xx (0, y) = u(π, y) = u xx (π, y) = 0 for y ∈ (− , ) u yy (x, ± ) + σu xx (x, ± ) = u yyy (x, ± ) + (2 − σ)u xxy (x, ± ) = 0 for x ∈ (0, π), where f ∈ L 2 (Ω), σ ∈ [0, 1) is the so-called Poisson ratio and depends on the material by which the plate is made of. It is known that the validity of the PPP for a problem is related to the sign of the associated Green function. Indeed, if G p (q) := G(p, q) denotes the Green function of (2), the (weak) solution to (2) writes u(p) = Ω G p (q)f (q) dq ∀p ∈ Ω and the PPP becomes equivalent to G p (q) 0 ∀(p, q) ∈ Ω .
The proof of the above inequality represents the main result of the present paper. More precisely, we first write the Fourier expansion of G p , i.e. where the (involved) analytic expression of the functions φ m is given explicitly in formula (13) of Section 3. As subsequent step, we develop an accurate analysis of the qualitative properties of the φ m and we show, in particular, that they are strictly decreasing with respect to m ∈ N + . This monotonicity issue is achieved by studying the sign of the derivatives of the φ m ; since they have highly involved analytic expressions, in order to detect their sign, we set up a clever scheme where, step by step, we cancel out the dependence of some variables through optimisation arguments, see Remark 4.1 of Section 4. From the monotonicity of the φ m , through an asymptotic analysis, we also deduce their positivity. These information are essential for the subsequent part of the proof where we study the sign of G p . More precisely, by means of suitable lower bounds, we first show the positivity of G p in a rectangle contained in Ω, far from the hinged edges; then, we obtain the positivity in the remaining parts through suitable iterative procedures which, step by step, stick rectangles where G p is positive up to the boundary, see Section 5 for all details. As already remarked, the validity of the PPP for problem (2) is by no means an obvious fact, recall that it does not hold for problem (1) on rectangular planar domains; furthermore, in general, its validity is not expected for plates having two free edges. In Figure 1 (left) we show a well known example of PPP violated for (1) with Ω = (0, π) × (−π/6, π/6) and with a load f concentrated in (π/3, 0), see also [18]. In Figure 1 (right) we consider the solution to a partially clamped plate problem, i.e. (2) with Dirichlet conditions instead of Navier, with a concentrated load in (π/3, π/6). Numerically, we obtain regions where the PPP fails near the corners.
The paper is organized as follows. In Section 2 we introduce some notations and we state our main results: the Fourier expansion of G p , together with the qualitative properties of its components, which is given in Theorem 2.1 and the precise statement of the PPP result which is given in Theorem 2.2. The rest of the paper is devoted to the proofs. More precisely, in Section 3 we compute explicitly the Fourier series of the Green function as the limit of the solution to (2) for a specific L 2 forcing term converging to the Dirac delta function. In Section 4 we prove the monotonicity and the positivity of the φ m , while in Section 5 we show the positivity of the Green function. Finally, we collect in the Appendix the proofs of some technical results needed either in Section 4 and in Section 5. Finite element approximated solution u of the clamped (left) and partially clamped (right) plate problems under, respectively, a concentrated load in ( π 3 , 0) and in ( π 3 , π 6 ); the regions where u 0 are grey, while the regions where u < 0 are coloured from blue (less negative) to red (Ω = (0, π) × (−π/6, π/6), σ = 0.2).

Notations and main results
The natural functional space where to set problem (2) is . Note that the condition u = 0 has to be meant in a classical sense because Ω is a planar domain and the energy space H 2 * (Ω) embeds into continuous functions. Furthermore, for σ ∈ [0, 1) fixed, H 2 * (Ω) is a Hilbert space when endowed with the scalar product , which is equivalent to the usual norm in H 2 (Ω), see [7,Lemma 4.1]. Then, we reformulate problem (2) in the following weak sense . Clearly, problem (4) (and consequently (3)) admits a unique solution u ∈ H 2 * (Ω); in the following we shall specify the cases when f ∈ H −2 * (Ω), otherwise we will always assume f ∈ L 2 (Ω). For all p ∈ Ω, the Green function G p of (2) is, by definition, the unique solution to Recalling that H 2 * (Ω) ⊂ C 0 (Ω) the above definition makes sense for all p ∈ Ω. By separating variables, in Section 3 we derive the Fourier expansion of G p and in Section 4 we prove some crucial qualitative properties of its Fourier components. We collect these results in the following: Theorem 2.1. Let σ ∈ [0, 1) and p = (ρ, w) ∈ Ω, furthermore let G p ∈ H 2 * (Ω) ⊂ C 0 (Ω) be the Green function of (2). Then, where the functions φ m (y, w) are given explicitly in formula (13) of Section 3. In particular, the φ m (y, w) are strictly positive and strictly decreasing with respect to m, i.e.
In Figure 2 on the left we provide the plot of φ 1 (y, w) with = π/150 and σ = 0.2; on the right we provide the plot of φ 1 (y, w) and φ 2 (y, w) for = 3π/4. Qualitatively, we have similar plots for any m ∈ N + and they all highlight that the points where the positivity of φ m (y, w) is more difficult to show are (± , ∓ ). This confirms the physical intuition that a concentrated load in w = produces the largest vertical (positive) displacement in y = and the smallest in y = − . We refer to [1] for a detailed analysis about the torsional performances of partially hinged plates under the action of different external forces.
Instead, Figure 2 on the right highlights how the monotonicity issue (with respect m) becomes more difficult to be proved at (± , ± ), where the difference between the φ m reduces. Numerically, we see that this becomes more evident for large . However, Theorem 2.1 assures that the φ m never intersect and preserve their positivity for all > 0.
By exploiting Theorem 2.1 we derive the main result of the paper, namely the positivity of G p . More precisely, we set Ω := (0, π) × [− , ] and we prove Theorem 2.2. Let σ ∈ [0, 1) and p ∈ Ω, furthermore let G p ∈ H 2 * (Ω) ⊂ C 0 (Ω) be the Green function of (2). There holds G p (x, y) > 0 ∀(x, y) ∈ Ω. Therefore, if f ∈ L 2 (Ω) and u is the solution of (2), the following implication holds As explained in the Introduction, the validity of the above implication is not obvious at all; recall that the positivity issue fails on rectangular plates under Dirichlet boundary conditions, see [6] and Figure 1.
The Poisson ratio σ of a material is defined as the ratio between the transversal strain and the longitudinal strain in the direction of the stretching force; for most of materials we have σ ∈ (0, 1/2). Nevertheless, there are materials having negative Poisson ratio, hence the range σ ∈ (−1, 1/2) includes all possible values. Numerical experiments lead us to conjecture that Theorem 2.2 still holds for σ ∈ (−1, 0). In Remark 4.2 of Section 4 we highlight the points where our proof fails when assuming σ negative.

Green function computation
The aim of this section is to provide the Fourier expansion of the Green function G p , namely of the solution to (5). This is done by developing a suitable limit approach where, in principle, δ p is replaced by a suitable L 2 function converging to it.
We notice that Φ m,w,η is given by the convolution of the H 3 (R) function (1+m|y|)e −m|y| 4m 3 and the L 2 (R) , hence Φ m,w,η ∈ C 3 (R) and all the above constants are well-defined. Then we prove where the constants c i and Φ m,w,η are defined in (8) and (9). Furthermore, the above series converges in H 2 * (Ω) and in C 0 (Ω). Proof. First we set Then, for M 1 we define where, for each 1 m M , ϕ m = ϕ p m,α,η (y) is the unique solution to the problem: and some computations yield that the ϕ p m,α,η are as given in (10), see [2, Theorem 5.1] for details. The proof of Lemma 3.2 follows by showing that y) is the unique solution of (7). Let v ∈ H 2 * (Ω), it is readily checked that, for M 1 fixed, u p,M α,η satisfies Since f p α,η ∈ L 2 (Ω), a well-known result for Fourier series yields π 0 (f p α,η (x, y)) 2 dx = π 2 +∞ m=1 (f p m,α,η (y)) 2 . Hence, by direct computation we infer that From the above inequality we deduce that f p,M α,η is a Cauchy sequence in L 2 (Ω) and, in turn, that By this, a direct computation yields v M 2 and, in turn, we conclude that For what remarked, the proof of the statement follows by passing to the limit in (12).
Proof. Let u p α,η ∈ H 2 * (Ω) be the unique solution to (7); by expanding in Fourier series, we have that Passing to the limit above and thanks to Lemma 3.1, we infer that On the other hand, from Lemma 3.2 we know that the ϕ p m,α,η write as in (10). Now, as η → 0, a direct inspection reveals that: and Therefore, The above limits inserted in (10) yield where φ m is as given in (13), which proves (15) for all p ∈ Ω. Let now p ∈ Ω and let G p be the corresponding solution to (5). It is readily seen that δ pn → δ p in H −2 * (Ω) for all {p n } ⊂ Ω : p n → p; then, arguing as in Lemma 3.1, it follows that G pn → G p in H 2 * (Ω) and, consequently, in C 0 (Ω). By this we infer that (15) extends continuously to all p ∈ Ω.
The convergence of the series (15) in H 2 * (Ω) and in C 0 (Ω) can be easily checked by exploiting the monotonicity property (6) (see Section 4.1 for the proof). Indeed, we have by which the convergence in C 0 (Ω) follows at once. The convergence in H 2 * (Ω) follows from similar estimates.

Proof of Theorem 2.1
The first part of the statement, namely the Fourier expansion of the Green function, has already been derived in the previous section, see Proposition 3.3. Here we focus on the sign and monotonicity properties of the functions φ m (y, w).

4.1.
Proof of the monotonicity issue in (6). We rewrite the functions φ m (y, w) in a more convenient way; to this aim we introduce the functions ζ, η, ψ, ξ : where ζ, η, ψ, ξ are given in (14). See the proof of Lemma 6.2 in the Appendix for the explicit form of the above functions. Putting into (13) z = m > 0, y = k with k ∈ [−1, 1] and w = s with s ∈ [−1, 1], each φ m (y, w) rewrites as the three variable function: where and h(s, k, z) := (1 + z|k − s|)e −z|k−s| . It is readily seen that the monotonicity issue (6) follows by showing that the function φ(s, k, z) is decreasing with respect to z > 0 for all k, s ∈ [−1, 1], i.e.
Since h z (s, k, z) = −(k − s) 2 ze −z|k−s| 0, for all z > 0 and k, s ∈ [−1, 1], a sufficient condition for the validity of the above inequality is: The proof of this inequality will be the goal of this section. To this aim we compute and In view of the elementary implication: for all W, Q : (0, +∞) → R continuous functions, it follows that a sufficient condition for (18) to hold is We consider The maps [−1, 1] s → W (s, k, z) ± Q(s, k, z) are concave parabolas for all z > 0 and k ∈ [−1, 1] fixed. Indeed, we have Furthermore, there holds: The first condition assures that the abscissa s of the parabolas vertex satisfies, respectively, s > 1 or s < −1, implying that the maximum is achieved, respectively, at s = 1 or at s = −1; condition (24) implies the negativity of such maxima proving (20) and, in turn, (18). We postpone the (long) proofs of (22), (23) and (24), respectively, to Sections 4.3, 4.4 and 4.5 below.
Remark 4.1. It's worth pointing out that the proofs of (23) and (24) are achieved by repeating several times the scheme outlined above, i.e. we first put in evidence an expression of the type: W cosh(ωz) + Q sinh(ωz), for suitable functions W and Q, and then, in order to show that this expression is always negative, we exploit (19) and we come to study the sign of W ± Q. As in (21), the functions W ± Q can always be seen as parabolas with respect to one of the variables: we locate the maximum point of these parabolas and we estimate the sign of the maximum in a suitable interval. The advantage of this procedure is that, at each step, we obtain a reduction of the number of variables. Indeed, we start with the three variables functions W and Q in (21) and we reduce to two or one variables functions, see e.g. (30) below. (23), all steps in the proof of the monotonicity issue (6) hold for all σ ∈ (−1, 1). Our numerical experiments suggest that (20) is still satisfied when σ ∈ (−1, 0) but the vertex of the parabolas s → W (s, k, z) ± Q(s, k, z) in (21), differently to what happens for σ ∈ [0, 1), may belong to the interval [−1, 1]. Therefore, to extend the proof to the case σ ∈ (−1, 0), condition (24) should be modified accordingly.
Proof. Since see Lemma 6.2 in the Appendix for the explicit form of the above functions, the second term of (28) is given by the sum of an even and an odd function with respect to k. Hence, to obtain (28) it is enough to prove that We rewrite (29) as where W(k, z) := k 2 z 2 s(z) + kz t(z) + u(z), Q(k, z) := k 2 z 2 p(z) + kz q(z) + r(z) and By (19) We prove the validity of (32) and (33) here below; this concludes the proof of Lemma 4.4.
Proof of (32). By (31), s(z) + p(z) < 0 for all z > 0, hence χ + (k, z) is a concave parabola with respect to k. Therefore, χ + (k, z) < 0 if the ordinate of its vertex is negative, namely if Through many computations we obtain We have Hence, recalling (26), the first term in the definition of µ is positive. Moreover, by estimating sinh(2z) > 2z for z > 0, we have By Lemma 6.1 in the Appendix we know that 2F (z)−F (z) we have µ 1 (z) > 0 for all z > 0. On the other hand, we have implying (34). This assures χ + (k, z) < 0 for all k ∈ [−1, 1] and for all z > 0.
We prove that the parabola has a point of maximum for k < −1, i.e. that To this aim we study By Lemma 6.1 in the Appendix we have that the last term above is negative; about the remaining terms we distinguish the cases z ∈ (0, 1] and z > 1.
In view of (35), to obtain χ − (k, z) < 0 for all z > 0 and k ∈ [−1, 1] it is enough to study the sign of we conclude that χ(−1, z) < 0 and the thesis. By (19) this follows by showing that W (z) ± Q(z) < 0 for all z > 0, namely that These inequalities hold true for all z > 0 thanks to Lemma 6.1 in the Appendix.
Lemma 4.5. Given F (z), F (z) as in (14) and F (z), F (z), ζ(k, z), η(k, z) as in (16), we have see Lemma 6.2 in the Appendix for the explicit form of the above functions. The second term of (36) is given by the sum of an even and an odd function with respect to k; hence, to obtain (36) it is enough to prove that for all k ∈ [−1, 1] and z > 0. We rewrite (37) as cosh(kz)V(k, z) + sinh(kz)P(k, z) where V(k, z) := k 2 a(z) + k b(z) + c(z), P(k, z) := k 2 d(z) + k e(z) + f (z), and a(z) Then, by (19), we obtain the thesis if Ξ(k, z) ± := V(k, z) ± P(k, z) < 0 for all k ∈ [−1, 1] and z > 0, i.e. if We prove the validity of (39) and (40) here below. This concludes the proof of Lemma 4.5.
Proof of (39). We consider Since, from (26), we have by (19) we infer that a(z) + d(z) < 0; hence the map k → Ξ + (k, z) is a concave parabola for all z > 0. Now we prove that the parabola has the abscissa vertex at k = k with k > 1; this follows by showing that .
We have by (19), we infer that a(z) − d(z) < 0 and the map k → Ξ − (k, z) is a concave parabola for all z > 0. Now we prove that the abscissa k of the parabola vertex satisfies k < −1, namely that Through (19), (43) follows if this condition is guaranteed for all z > 0 by Lemma 6.1 in the Appendix. Hence, by (43), Ξ − (k, z) achieves its maximum at k = −1; we prove that To this aim, we consider where ς(z) and ς(z) are as defined in the proof of (39). We have already proved that ς(z) < 0 for all z > 0, by (42) we also get ς(z) − ς(z) = e z ς(z) < 0 ∀z > 0 .

Proof of Theorem 2.2
The proof is achieved by showing the positivity of the Green function G p (q) for p and q belonging to suitable rectangles or union of rectangles covering Ω. By Theorem 2.1, we know that In this section we will omit the dependence of φ m from y and w, implying that all relations we write hold true for all y, w ∈ [− , ]; for this reason and for brevity, in all the proofs of this section we often write G(x, ρ) instead of G p (q) = G(x, y, ρ, w). We start by showing the positivity of G p (q) for p or q far from the hinged edges of Ω.
Our next aim is to show the positivity issue for both p and q near the same hinged edge, i.e. near x = 0 and ρ = 0 or near x = π and ρ = π. The proof is based on a multi step procedure; the first step is given by the following: Proof. We fix N 3 and we rewrite G p (q) as follows Then, we exploit the elementary inequality sin(mx) sin(mρ) > sin(x) sin(ρ) ∀x, ρ ∈ 0, π N + 1 , ∀m = 2, . . . , N (see Lemma 6.3 in the Appendix for a proof) and Theorem 2.1-(6) to get On the other hand, through | sin(mρ)| < m sin(ρ) for all ρ ∈ (0, π) and Theorem 2.1-(6), we get By combining (45) and (46) we infer Next we denote by x N the unique solution to the equation the above definition makes sense for all N 1 since the map N → C N is positive, strictly decreasing and 0 < C N < 1. We prove that When N = 3 we have x 3 ≈ 0.25 < π 5 and (49) follows. We complete the proof of (49) by showing that To this purpose we write some estimates on the numerical series; it is easy to see that To tackle (50) we use the estimate: Combining (51) and (52), (50) follows by noticing that 1 N < 3 N + 2 for all N 4. Finally, in view of (49) the statement readily comes from the positivity of the r.h.s. of (47).
It remains to study the sign of the Green function for p and q near to opposite hinged edges, i.e. near x = 0 and ρ = π or near x = π and ρ = 0; as for Proposition 5.4 the proof is based on a multi step procedure. At first we prove the following: Proof. We set ρ = π − ρ with ρ ∈ 0, π N +1 and we exploit (59). For N 3, odd integer, we rewrite the Green function as By Theorem 2.1 we know that φ m > 0 and is strictly decreasing with respect to m ∈ N + for all y, w ∈ [− , ]; hence we have On the other hand, since | sin(mρ)| < m sin(ρ) for all ρ ∈ (0, π) and through the monotonicity of the φ m , we get From (61)-(62), for all N 3 odd, we infer Next we denote by x N the unique solution to the equation The above definition makes sense for all N 3, odd, since the map N → C N is positive, strictly decreasing and 0 < C N < 1. We prove that To this aim we note that Recalling (52), (64) follows by noticing that 2 N + 1 3 N + 3 sin π N + 3 for all N 3.
In the next Lemma we extend the validity of Lemma 5.5 to all ρ ∈ (3π/4, π).
Proof. The case N = 3 is included in the statement of Lemma 5.5. When N 5, odd, as explained in the proof of Lemma 5.5, we set ρ = π − ρ and we exploit (59), getting through Lemma 5.5 that , ∀ρ ∈ 0, π N + 1 with π N +1 < π 4 . Moreover, with C N as in (63), we estimate in which we used Lemma 6.5 of the Appendix with N − 2 (instead of N) and | sin(mx)| < m sin(x) for all x ∈ (0, π). For N 5, in the following we denote by ρ N the unique solution to the equation sin(ρ N ) = C N −2 , with C N as in (63). Clearly, ρ N = x N and by (64) we obtain (67) ρ N < π N + 1 .
Fix now N 7, odd, from (69) we clearly have .., N − 2, odd and the thesis follows by noticing that 0, By Lemma 5.6 we derive the following: Proof. From Lemma 5.6 we have Passing to the limit N → +∞ we obtain To complete the proof we exploit the trigonometric relations (59). Then, for x ∈ 3 4 π, π and ρ ∈ 0, π 4 π we set x = π − x, ρ = π − ρ and we study the positivity of already guaranteed by (70).
Proof of Theorem 2.2 completed. The proof follows by combining the statements of Propositions 5.1, 5.4 and 5.7.
Finally, we obtain (71) thanks to the elementary implication Before stating and prove the second inequality we need the following lemma.