Maximum Principle for Higher Order Operators in General Domains

We first prove De Giorgi type level estimates for functions in $W^{1,t}(\Omega)$, $\Omega\subset\mathbb{R}^N$, with $t>N\geq 2$. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi's classes as obtained in [Di Benedetto--Trudinger, AIHP (1984)] for functions in $W^{1,2}$. As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.


Introduction
One of the most powerful tools in the study of partial differential equations and nonlinear analysis is without any doubts the Maximum Principle (MP in the sequel). It turns out to be fundamental in obtaining existence, uniqueness and regularity results in the theory of linear elliptic equations, as well as to establish qualitative properties of solutions to nonlinear equations. We mainly refer to [19] for classical results and historical development, where suitable applications also to the parabolic and hyperbolic cases are discussed. Let us merely mention that the roots of MP date back two centuries in the work of Gauss on harmonic functions, up to the ultimate version of Hopf [13], and then further extended in the seminal work of Nirenberg [17], Alexandrov [2] and Serrin [21], within the foundations of modern theory of PDEs. The underlying idea is simple: positivity of a suitable set of derivatives of a function induces positivity of the function itself. This is elementary true for real functions of one variable which vanish at the endpoints of an interval where −u ′′ (x) ≥ 0 and the validity can be extended to second order uniformly elliptic operators for which a prototype is the Laplace operator: (1) −∆u = f, in Ω ⊂ R N , N ≥ 2 u = 0, on ∂Ω for which we have f ≥ 0 =⇒ u ≥ 0 in Ω . Surprisingly, this is no longer true when considering higher order elliptic operators such as the biharmonic operator ∆ 2 : (2) ∆ 2 u = f, in Ω u = ∂u ∂ν = 0, on ∂Ω . Indeed, in this case in general one has This is a well known fact as long as the domain Ω is not a ball, for which the positive Green function was computed by Boggio [5] and which keeps on being positive for slight deformations of the ball [22]. As deeply investigated in [9] and references therein, the lack of the positivity preserving property is due to the appearance of sets carrying small Hausdorff measure (see [12]) where u < 0 and apparently without robust physical motivations. Recently the loss of the MP has been established in [1] also in the case of higher order fractional Laplacians. This paper is a step forward a better understanding of this phenomena and at the same time gives some general principle in order to recover the validity of the MP in the higher order setting. Let us briefly recall some physical interpretation of (1)- (2). Indeed, (1) is modeling, among many other things, a membrane whose profile is u which deflects under the charge load f and clamped along the boundary ∂Ω. This is the case in which tension forces prevale on bending forces which can be neglected because of the "thin" membrane. However, the model does not suite the case of a "thick" plate in which bending forces have to be taken into account. Here higher order derivatives come into play which yield (2). As one expects for (1), and there this is true by the MP, upwards pushing of a plate, clamped along the boundary, should yield upwards bending: this is false for (2) in contrast to some heuristic evidences in applications (see e.g. [14] and references therein). Our point of view here, roughly speaking, is that approaching the boundary, where the bending energy carries some minor effect because of the clamping condition, tensional forces can not be neglected for which the contribute of lower order derivatives may restore the validity of the MP. As a reference example, consider the following simple model: Clearly for γ = 0 one has (2) whence formally as γ → ∞, in a sense one may expect that (3) inherits some properties of (1). As we are going to see, this is the case and for the more accurate model (3) surprisingly the MP holds true, for fairly general domains, provided γ ≥ γ 0 > 0, which is essentially given in terms of Sobolev and Poincaré best constants. Let us state our main result in the case of (3) though it extends to cover the general case of uniformly elliptic operators of any even order, see Corollary 5.1.
As a consequence of Theorem 1.1, the operator ∆ 2 − γ∆, which in addition to (2) contains the contribute of lower order derivatives, turns out to be a more natural extension of (1) to the higher order setting.
Overview. In Section 2 we prove some preliminary estimates which will be the key ingredient to prove in Section 3 a new Harnack type inequality. Indeed, in the higher order case, it is well known how truncation methods fail [9]. Our approach here is to demand some extra integrability on the function entering the Harnack inequality in place of being solution to a PDE, which usually yields Caccioppoli's inequality and the solution belongs to the corresponding De Giorgi class. In [8] the authors prove a Harnack type inequality just for functions with membership in some De Giorgi classes. Here we drop this assumption though we assume more regularity in terms of integrability which however enables us to prove De Giorgi type pointwise level estimates. In Section 5 we apply the results obtained to prove the strong maximum for polyharmonic operators of any order, which contain lower order derivatives, in sufficiently smooth bounded domains which enjoy the interior sphere condition. This is done by a limiting procedure starting from compactly supported functions and then extending the results and estimates to the solutions of higher order PDEs subject to Dirichlet boundary conditions. Those boundary conditions are in a sense the natural ones as the higher order operator in this case does not decouple into powers of a second order operator. In one hand the result we obtain is a first step towards the investigation of qualitative properties of higher order nonlinear PDEs, such as uniqueness, optimal regularity, symmetries and concentration phenomena [4,6,11,15,16,18,20,23]. On the other hand, we are confident the tools introduced here may reveal useful also in different contexts, such as parabolic problems, in the study of the sign of solutions to quasilinear equations and in the higher order fractional Laplacian setting. Notation. In the sequel we will use the following basic definitions: • B(x 0 , r) denotes the ball in R N of center x 0 and radius r; • ω N is the volume of the unit ball in R N ; • d Ω denotes the diameter of the bounded set Ω in R N ; • | · | applied to sets denotes the Lebesgue measure in R N otherwise it is the Euclidean norm in R N with scalar product (· , ·); • Ω satisfies the interior sphere condition if for all x ∈ ∂Ω there exists y ∈ Ω and r 0 > 0 such that B(y, r 0 ) ⊂ Ω and x ∈ ∂B(y, r 0 ); • c and C denote positive constants which may change from line to line and which do not depend on the other quantities involved unless explicitly emphasized; • W m,p (Ω) is the standard Sobolev space endowed with the norm · p m,p = 0≤|α|≤m D α u p p ; • W m,p 0 (Ω) is the completion of smooth compactly supported functions with respect to the norm · m,p ; • the critical Sobolev exponent p * := N p N −mp , 1 < p < N/m.

Preliminaries
Let Ω ⊂ R N , N ≥ 2, be an open bounded set with sufficiently smooth boundary. The following holds true Lemma 2.1. Let u ∈ W 1,t (Ω), t > N and 1 < s < N. Then there exists c(s, t) > 0 such that for all k ∈ R, x 0 ∈ Ω and ρ ∈ (0, r) where 0 < r < dist(x 0 , ∂Ω) the following holds Proof. Consider a standard cut-off function Θ ∈ C ∞ 0 (R N ) given by As W 1,t (Ω) ֒→ W 1,s (Ω), one has from Sobolev's emedding and Hölder's inequality namely that x 0 lies in the interior of Ω, is crucial to extend to the whole R N the function (u − k)Θ. Therefore when x 0 approaches ∂Ω, necessarily r = r(x 0 ) tends to zero. Lemma 2.2. Let u ∈ W 1,t (Ω), t > N ≥ 2 and 1 < s < N. Let l, k ∈ R such that l > k, x 0 ∈ Ω and r < dist(x 0 , ∂Ω). Then for all ρ ∈ (0, r) one has Proof. Let x 0 ∈ Ω, r < dist(x 0 , ∂Ω) and for simplicity let us write Let q > 2 for which one has Let now p > 1 and 2 < q < 2p and estimate by Hölder's inequality where in the last inequality we have used Lemma 2.1 with s * = 2q(p−1) 2p−q . Combine (8) and (9) to get (10) In what follows we will use the following result from [3] in order to prove a version of the well known Poincaré inequality.
where c = c(N) depends only on the dimension N.
Then the following holds where c = c(N) is the constant in (11).
Proof. Apply Theorem 2.1 to the function |u| p , to get

A Harnack type inequality
Next we derive a De Giorgi type level estimate (see [3,10]) for functions u ∈ W 1,t , t > N ≥ 2 which will be the key ingredient in establishing a new Harnack type inequality. Let us emphasize that in De Giorgi's theorem [7], level estimates hold for u ∈ W 1,2 which is a solution to a uniformly elliptic second order equation with bounded and measurable coefficients. As a consequence, Caccioppoli's inequality holds and u ∈ W 1,2 belongs to the corresponding so-called De Giorgi class. Later, Di Benedetto and Trudinger relaxed the framework and in [8] they merely assume u ∈ W 1,2 belonging to some De Giorgi class. Here, we further improve the setting, without requiring any of those previous assumptions, though demanding for some augmented integrability which turns out to be necessary, as it is well known, functions in W 1,N (Ω), Ω ⊂ R N , may not be bounded.
be open and bounded set with sufficiently smooth boundary ∂Ω. For all k ∈ R, y ∈ Ω, r > 0 such that r < dist(y, ∂Ω) the following holds and where c = c(t, ξ, η, µ, p) is a positive constant.
From (16) we are done provided we prove that for all k ∈ R and r < dist(y, ∂Ω) there exists d > 0 such that which in turn by (15) yields Next we proceed by using the iterative scheme from the proof of De Giorgi's theorem. For m ∈ N set where the parameter d > 0 has to be chosen in the sequel and k 0 = k. The idea is to exploit the inequality (16) with r = r m and ρ = r m+1 where the sequence {r m } m∈N is decreasing so that B(r m+1 ) ⊂ B(r m ). On the other hand {k m } m∈N is increasing, and we set in (16) l = k m+1 e k = k m . With this choice we obtain from (16) the following inequality Now multiply (19) by 2 µ(m+1) , µ > 0 and set to obtain form (19) Let us choose µ > 0 to avoid the dependence on m in the first factor in the right hand side of (21), namely and thus (21) becomes so that for all m ∈ N one has At this point we choose d > 0 such that and by induction on m ∈ N we have Finally by (20) we obtain Φ(k m , r m ) ≤ 1 2 µm Φ(k 0 , r) and the proof is complete by letting m → ∞.
Next we prove the following Harnack type inequality If I 1 = ∅ then we prove for all k ∈ I 1 the following Indeed, by Theorem 3.1 we have for all k ∈ I 1 Since k ∈ I 1 one has and apply Lemma 2.3 to the function (u(x) − k) + to get In the case I 2 = ∅, for all k ∈ I 2 set h = −k and v(x) = −u(x). Thus h ∈ (−M, −m) and the following holds Therefore, the function v enjoys (24), namely As a consequence, for all k ∈ I 2 we get Next we distinguish three cases, precisely: i) I 1 = ∅ and I 2 = ∅. In this case any k ∈ (m, M) belongs toI 1 , for which (24) which holds for all k ∈ I 1 , it holds for k = m as well; ii) I 1 = ∅ e I 2 = ∅. IIn this case any k ∈ (m, M) belongs toI 2 , and thus (27) which holds for all k ∈ I 2 , in particular holds for k = M; iii) I 1 = ∅ e I 2 = ∅. In this case we consider inf I 1 and sup I 2 and it is standard to prove there exists a unique k 0 = inf I 1 = sup I 2 which enjoys both (24) and (27) and the Theorem follows. x max and x min be respectively a local maximum and local minimum for u ∈ W 1,t (Ω), t > N. Then, there exists h ∈ N, h = h(Ω, x max , x min ) such that , with c = c(N) provided by Thorem 3.2 and where in particular h depends only on dist(x max , ∂Ω) and dist(x min , ∂Ω).

Now inequality (30) in particular holds for
x ∈ B x 1 , r 2 ∩ B x 0 , r 2 and thus By applying iteratively Theorem 3.2 we end up with which completes the proof.
Applying to this sequence the reasoning carried out in the proof of Theorem 3.3 where x 0 = x max , we get We would get a contradiction if the above series converge. Actually as we are going to see this is not the case. Consider B(x n , rn 2 ) and B(x n+1 , r n+1 2 ) and let C ∈ B(x n , rn 2 )∩B(x n+1 , r n+1 2 ) and D its projection on the segment with endpoints A = x n and B = x n+1 . Set AD = ρ n , DB = ρ n+1 , so that considering the triangle ADC and CDB one has We can apply Kummer's test to the series with general terms a n = r n 4 + ρ n a and b n = r n 4 − ρ n a , a > 0, from which since a n a n+1 = b n+1 b n , for all n ∈ N, and ∞ n=0 1 b n = +∞ we obtain ∞ n=0 a n = +∞. From a n < r a n we have ∞ n=0 r a n = +∞ .

Towards the Positivity Preserving Property
Next we apply the results so far obtained to prove the strong maximum principle for the biharmonic operator perturbed by the Laplacian for compactly supported data. As we are going to see, here it comes for the first time the restriction on the Euclidean dimension N < 4 and the fact that we deal with the solution to a PDE. Precisely, this Section is devoted to prove the following where γ > 0, f ∈ L 2 (Ω), f ≥ 0 in Ω and |{x : f (x) > 0}| > 0. Moreover, f (x) = 0 on Ω \ Ω 1 , with Ω 1 a bounded subset of Ω such that dist(∂Ω 1 , ∂Ω) > 0. Then, there exists γ 0 > 0 such that for all γ > γ 0 the solution to (32) satisfies u(x) > 0, for all x ∈ Ω.
Assuming the hypotheses of Theorem 4.1 we have the following preliminary lemmas: Lemma 4.1. The following holds true Proof. By multiplying (32) by u and integrating by parts In order to apply the Harnack inequality established in Section 3 we next estimate first order derivatives of the solution to (32). Though from one side elliptic regularity yields enough summability, on the other side we need estimates which are uniform with respect to the parameter γ, and for this reason we restrict ourself to dimensions N < 4.
for any t > 2 when N = 2 and for t = 6 when N = 3.
Proof. Since u = ∇u = 0 su ∂Ω, one has By Sobolev's embedding and from (34), when N = 3 and t = 6 we have Similarly when N = 2 and t ≥ 1 we obtain Proof of Theorem 4.1. Let x max be an absolute maximum point for u in Ω 1 and x min a local minimum for u in Ω. Set From (31), Theorem 3.2 and Lemma 4.2 we have .
The thesis follows as γ is large enough. If sup Peforming the change of variable x = sy, with s > 0, v k (y) = w k (sy), g(y) = f (sy), y min = x min s , y max = xmax s , we obtain where Ω s = {y : y = x/s, x ∈ Ω}. Next apply (35) to the solution of (37) to get With respect to the original variables it reads as follows Let us now observe that thanks to the interior sphere condition, the number h of balls covering he path from y max to y min does not depend on the parameter s. The same happens for the parameter k. Thus we choose the parameter s such that namely the thesis of the Theorem follows for all , and thus γ 0 is the right hand side of (41) with optimal constant c. When γ = γ 0 we just get the weak inequality u ≥ 0.

The validity of the strong maximum principle for higher order elliptic operators
In this Section we first prove Theorem 1.1 for which we have to remove the restriction to compactly supported data of Theorem 4.1. Then, we will extend the result obtained to polyharmonic operators and to more general uniformly elliptic operators of any even order with constant coefficients.
Proof of Theorem 1.1. Consider the following family of sets {Ω m } m∈N such that for all m ∈ N satisfy: and set where converges pointwise on Ω. Moreover, notice that g m ∈ L 2 (Ω). Next consider the following problems where by construction g m (x) = 0 for x ∈ Ω \ Ω m and thus by Theorem 4.1 there exists γ m > 0 such that for all γ > γ m , one has u m (x) > 0, for all x ∈ Ω, m ∈ N.
It is crucial here that by (40) Therefore, for all γ > γ ∞ and for all m ∈ N one has Finally, we prove that the function and thus by (44) we conclude that for all γ > γ ∞ and for all x ∈ Ω one has v(x) > 0 . By uniqueness of the solution to the Dirichlet problem (32) the Theorem follows. Hence, it remains to show that v m → v ∈ W 4,2 ∩ H 2 0 (Ω) which is a solution to (46). Set By Lebesgue's dominated convergence f m → f in L 2 (Ω) and notice that v m solves the following Thus for all m, l ∈ N we have and multiplying by v m − v l and integrating by parts we get which together with the equation (47) yields Thus {v m } is a Cauchy sequence in W 4,2 (Ω) which converges to v ∈ W 4,2 (Ω), the solution to (46).
What we have seen so far naturally extends to polyharmonic operators of any order and more in general to uniformly elliptic operators of any even order as established in the following (Ω), m ≥ 2 be the solution to the following equation where f ∈ L 2 (Ω), We conclude by the Sobolev embedding theorem as follows: • If N ≤ 2(m − 1) one has ∇u ∈ L t (Ω), for all t ≥ 1 and in particular for t > N and ∇u L t (Ω) ≤ c u W m,2 (Ω) ≤ f L 2 (Ω) ; • If N = 2m − 1 one has ∇u ∈ L t (Ω) with t = 4m − 2 and ∇u L t (Ω) ≤ c u W m,2 (Ω) ≤ c f L 2 (Ω) .