A k uniform maximum principle when 0 is an eigenvalue

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1 A k uniform maximum principle when 0 is an eigenvalue Genni Fragnelli and Dimitri Mugnai Abstract. In this paper we consider linear operators of the form L + λi between suitable functions spaces, when 0 is an eigenvalue of L with constant eigenfunctions. We introduce a new notion of quasi uniform maximum principle, named k uniform maximum principle, which holds for λ belonging to certain neighborhoods of 0 depending on k R +. Our approach is based on a L L 2 estimate, which let us prove some generalization of known results for elliptic and parabolic problems with Neumann or periodic boundary conditions. Mathematics Subject Classification (2000). Primary 35B50; Secondary 35J25, 35J40, 35K10, 34C25. Keywords. Maximum principle, antimaximum principle, k uniform maximum principle, polyharmonic operators, periodic solutions. 1. Introduction and abstract setting In the recent paper [6], Campos, Mawhin and Ortega showed a very general maximum and antimaximum principle for linear differential equations whose prototypes were given by linear ODE s with periodic boundary conditions and the linear damped wave equation (or telegraph equation) in one spatial dimension with double periodic boundary conditions. In that paper the abstract setting relies on a L L 1 estimate for solution datum of the form u L (Ω) K f L 1 (Ω), which is common and natural for ODE s and for the wave equation in 1D. On the other hand, for the classical theory of elliptic problems like u + λu = f(x) in Ω, (1.1) Bu = 0 on Ω,

2 2 Genni Fragnelli and Dimitri Mugnai where Ω is a bounded domain of R N, N 1, and B denotes Dirichlet or Neumann boundary conditions, a more natural setting would be an L 2 L 2 estimate of the form u L 2 (Ω) K f L 2 (Ω), since data belonging to L 1 are not the good ones to perform a standard variational approach (we refer to [3] for a well established theory for this case). In this classical framework many results have been established for problem (1.1), the typical maximum principle sounding as: (MP): if λ < λ 1, the first eigenvalue of under the corresponding boundary condition, then for any f 0 the associated solution u is nonpositive in Ω. On the other hand, a related stronger version, namely the strong maximum principle, holds: (SMP): if in addition f 0, then u is strictly negative in Ω. However, it is now well know that jumping after λ 1 changes the situation a lot: indeed, Clément and Peletier in [7] were the first to show the following antimaximum principle: (AMP): for any f 0 in L p (Ω), p > N, there exists δ = δ(f) > 0 such that if u solves (1.1) with λ (λ 1, λ 1 + δ) under Dirichlet or Neumann boundary conditions, then u 0 in Ω. They also showed that under Neumann boundary conditions it is possible to take δ independent of f, thus showing a uniform antimaximum principle (UAMP), only when N = 1. Refinements of the (UAMP) are established for higher order ODE s with periodic boundary conditions in [5], for general second order PDE s with Neumann or Robin boundary conditions in [15] (where it is proved that (UAMP) holds only if N = 1), in [8]-[9]-[17] for polyharmonic operators in low dimensions (essentially for all those dimensions for which the natural Sobolev space containing weak solutions are embedded in C 0 (Ω)), while in [23] it is showed that requiring a right hand side f L 2 (Ω) is equivalent to assume N = 1 for the validity of an antimaximum principle in presence of a second order elliptic problem under Dirichlet boundary conditions. On the other hand, the result of [6] seems to be much more general, since the authors show that it is possible to state maximum and antimaximum principles in a unitary way. Roughly speaking, having in mind Neumann boundary conditions, so that λ 1 = 0, they start with the following definition of maximum principle, which is actually formulated therein for data f belonging to L 1 (Ω), but which we rephrase here for functions in L 2 (Ω). Definition 1.1. Given λ R \ 0}, we say that the operator L + λi satisfies a maximum principle if for every f L 2 (Ω) the equation Lu + λu = f, u Dom(L) C 0 (1.2)

3 A k uniform maximum principle 3 has a unique solution with λu 0 for any f 0. Moreover, the maximum principle is said to be strong if λu(x) > 0 for any x Ω whenever f 0 and f(x) > 0 in a subset of Ω with positive measure. Thus it is clear that the authors are actually dealing with a classical maximum principle when λ < λ 1 = 0 and with a classical antimaximum principle when λ > 0; more precisely, we remark that for λ > 0 their definition includes a (UAMP) tout court. Without going into the detailed description of L, but thinking for instance to Lu as u with Neumann boundary conditions, the main result in [6] is the following Theorem 1.2 ([6]). There exist λ and λ + such that λ < 0 < λ + + and L + λi has a maximum principle if and only if λ [λ, 0) (0, λ + ]. Moreover the maximum principle is strong if λ (λ, 0) (0, λ + ). As already said, an L L 1 estimate is the main ingredient of their proofs; for this reason, having in mind weak solutions to (1.1), this setting is natural for all those problems in which L 1 is contained in the dual of the Sobolev space where weak solutions are sought. In this context, the easy problem u + λu = f in Ω, u ν = 0 on Ω, (1.3) where Ω is a bounded domain of R 2 or R 3, cannot be handled by Theorem 1.2 if f L 2 (Ω), which is the most reasonable assumption since L 2 (Ω) (H 1 (Ω)), while L 1 (Ω) (H 1 (Ω)) ; indeed, we remark that the inclusion L 1 (Ω) (H 1 (Ω)) actually holds only in dimension 1, and this fact was used in [6] to considered (1.3) for N = 1 as a special case of polyharmonic problems. On the other hand, classical regularity results for elliptic PDE s guarantee that if f L 2 (Ω) and Ω is a bit regular, say of class C 2 just for simplicity, then the corresponding solution u of (1.3) with λ 0 belongs to H 2 (Ω), and there exists C = C(Ω) > 0 such that u H 2 (Ω) C f L 2 (Ω), (1.4) for example see [4, Theorem IX.26]. By Morrey s Theorem, if N = 1, 2, 3, then H 2 (Ω) C 0,α ( Ω), so that (1.4) implies u C 0 (Ω) K f L 2 (Ω), where, of course, u C 0 (Ω) = max Ω u = L (Ω). Our purpose is to combine the spirit of all the results cited so far showing that, although a (UAMP) cannot hold, a quasi (UAMP) does, in the sense of Definition 1.3 below. In order to make our setting precise, we start describing the abstract framework we are working within. By Ω we denote a bounded domain of R n endowed

4 4 Genni Fragnelli and Dimitri Mugnai by a positive and finite measure µ, and we write L p (Ω) := L p (Ω, µ), p [1, ]. Given f L 2 (Ω) and k > 0 we define f := 1 fdµ, f := f f, µ(ω) and Ω L := f L 2 (Ω) : f = 0}, C := C 0 ( Ω) L, F k := f L 2 (Ω) : f L 2 (Ω) k f L 1 (Ω)}. It is clear that any f L 2 (Ω) belongs to a suitable F k and to F l for any l k, and that k F k L 1 (Ω) strictly. We now consider a linear operator L : Dom(L) C 0 ( Ω) L 2 (Ω) satisfying the following properties: Ker(L) = constant functions}, Im(L) = L, (1.5) the problem Lu = f has a unique solution ũ C (1.6) and K = K(L) > 0 such that ũ C 0 (Ω) K f L 2 (Ω). We remark that these requirements are the natural extensions to our setting of the assumptions made in [6]. Therefore, having in mind Definition 1.1, we give the following Definition 1.3. Given λ R \ 0}, we say that the operator L + λi satisfies a k uniform maximum principle (k (UMP) for short) if for every f F k equation (1.2) has a unique solution with λu 0 for any f 0. We say that a strong k (UMP) holds if λu(x) > 0 for any x Ω whenever f 0 and f(x) > 0 in a subset of Ω with positive measure. Remark 1.4. As in the case of Definition 1.1, the case λ < 0 corresponds to a classical maximum principle, while the case λ > 0 states the validity of an antimaximum principle, which is almost uniform due to the fact that f F k and not to the whole of L 2 (Ω). In view of the cited results stating that for the Laplace operator with Neumann conditions a (UAMP) can hold only in dimension 1, we want to prove that a k (UMP) does hold also in some higher dimensions. In this context, we believe that our result, stated in Theorem 1.5, can shed new light in the general understanding of the matter. More precisely, we prove the following complete characterization of the set of real λ s for which L + λi has a k (UMP). Theorem 1.5. Assume that (1.5) and (1.6) are satisfied and fix k > 0. Then there exist λ = λ (k) and λ + = λ + (k) such that λ < 0 < λ + + and L + λi has a k (UMP) if and only if λ [λ, 0) (0, λ + ]. Moreover a strong k (UMP) holds if λ (λ, 0) (0, λ + ).

5 A k uniform maximum principle 5 We remark that in this way we can extend the result about an antimaximum principle for the Laplace operator to higher dimensions, say N = 2, 3, also in a quasi uniform way, being impossible to extend it in a uniform way by the cited results. Indeed, although in [23] it is shown that the condition p > n is sharp (being f L p (Ω)) and that one cannot have δ(f) to be bounded away from 0 uniformly for all positive f, we can prove that in F k there is the desired uniformity. In some sense, it seems that the validity of (UAMP) is strongly related to the fact that L 1 L 2! On the other hand, we must also underline the fact that if solutions exist in the right Sobolev space, independently of the Lebesgue spaces containing f, standard maximum principle can be proved also for inhomogeneous inequalities, possibly set on Riemannian manifolds (see the recent [2], [19], [20]), and also when everything is settled in anisotropic Sobolev and Lebesgue spaces vith variable exponent ([11]). 2. Proof of Theorem 1.5 In this section we want to extend the technique and the spirit of [6] to our functional setting. The first lemma we prove gives a condition that ensures the existence of the resolvent of L. We recall that the resolvent of L is the operator R λ : L 2 (Ω) C 0 ( Ω) which is the inverse of L + λi, whenever it exists. Moreover, we denote by R 0 the operator R 0 : L C defined by ũ = R 0 f Lũ = f, which is well defined by assumption (1.6). Lemma 2.1. There exist Λ 1 > 0 such that for all λ [ Λ 1, Λ 1 ] \ 0} the resolvent R λ : L 2 (Ω) C 0 (Ω) of L is well defined. Moreover, there exists C > 0 such that if f L and λ [ Λ 1, Λ 1 ] \ 0} then where C := K 1 Λ 1 R 0 C C R λ f C 0 (Ω) C f L 2 (Ω), and K is the constant appearing in (1.6). Here R 0 C C denotes the norm of the restriction of the operator R 0 from C to C, which is well defined, since C L. Proof. Rewrite (1.2) as the system Lũ + λũ = f, λu = f. (2.1) Applying R 0, the first equation in (2.1) can be rewritten as (I + λ R 0 )ũ = R 0 f. (2.2)

6 6 Genni Fragnelli and Dimitri Mugnai Now, if λ R 0 C C < 1 then I + λ R 0 is invertible from C to C and (2.2) is solved by ũ = (I + λ R 0 ) 1 R0 f. In conclusion, R λ f = (I + λ R 0 ) 1 R0 f + f ( ) λ. 1 Now, take Λ 1 0, R. Thus for all λ such that λ Λ 1, one has, 0 C C from the triangle inequality, (2.2) and (1.6), ũ L (Ω) Λ 1 R 0 C C ũ L (Ω) ũ L (Ω) λ R 0 C C ũ L (Ω) The thesis follows. (I + λ R 0 )ũ L (Ω) = R 0 f L (Ω) = ũ L (Ω) K f L 2 (Ω). Once proved that R λ exists, we can start with some preliminary results about maximum and antimaximum principles when the data belong to F k. Lemma 2.2. Take k > 0; then there exists Λ 2 := Λ 2 (k) (0, Λ 1 ] such that for all λ [ Λ 2, Λ 2 ] \ 0} the operator L + λi has a k (UMP). Moreover, a strong k (UMP) holds if λ ( Λ 2, Λ 2 ) \ 0}. Proof. If f F k, f 0, then f = 1 µ(ω) f L 1 (Ω). Thus, using the second equation in (2.1), one has λu = λr λ ( f + f) = λr λ ( f) + f = λr λ ( f) + 1 µ(ω) f L 1 (Ω) 1 µ(ω) f L 1 (Ω) λ R λ f L (Ω). By the previous Lemma it results that if λ [ Λ 1, Λ 1 ] \ 0}, then The thesis follows taking λu 1 µ(ω) f K L 1 (Ω) λ 1 Λ 1 R f L 0 2 (Ω) C C ( ) 1 µ(ω) k λ K 1 Λ 1 R f L 0 1 (Ω). C C Λ 2 = min Λ 1, 1 Λ 1 R 0 C C kkµ(ω) We now prove that the validity of a maximum principle according to Definition 1.3 for one value of λ immediately extends to a whole interval, like in [6], but also under our notion of k uniformity. Lemma 2.3. Assume that L + λ 0 I has a k (UMP) for some λ 0 > 0. Then }.

7 A k uniform maximum principle 7 1. R λ is defined for all λ (0, 2λ 0 ) and the function λ R λ f is analytic for all f L 2 (Ω); 2. the operator L + λi satisfies k (UMP) for all λ (0, λ 0 ]; 3. for any k > 0, L + λi has a strong k (UMP) for all λ (0, λ 0 ). Proof. First take f C 0 ( Ω); then it obviously results f C 0 (Ω) f f C 0 (Ω). Moreover, by (2.1), R λ0 1 = 1 λ 0. Thus, by assumption, we easily get Therefore f C 0 (Ω) λ 0 R λ0 f f C0 (Ω) λ 0. R λ0 = 1 λ 0, where R λ0 is the norm of R λ0 as an operator from C 0 ( Ω) to C 0 ( Ω). Now, if λ R, we can rewrite Lu + λu = f as and applying R λ0, we get Lu + λ 0 u = f + (λ 0 λ)u, u (λ 0 λ)r λ0 u = R λ0 f. (2.3) If λ 0 λ R λ0 < 1, then I (λ 0 λ)r λ0 is invertible from C 0 ( Ω) to C 0 ( Ω) and (2.3) is solvable. Note that, since R λ0 = 1 λ 0, λ 0 λ R λ0 < 1 λ 0 λ < 1 0 < λ < 2λ 0. λ 0 This implies that if λ (0, 2λ 0 ), then (2.3) is solvable. Now, if f F k then R λ0 f C 0 ( Ω) and R λ0 f 0 by assumption; since u = [I (λ 0 λ)r λ0 ] 1 R λ0 f by (2.3), applying the consideration above and Neumann s formula for operators, we get ( ) R λ f = [I (λ 0 λ)r λ0 ] 1 R λ0 f = (λ 0 λ) n Rλ n 0 R λ0 f. (2.4) Thus the function λ R λ f is analytic and nonnegative if λ (0, λ 0 ] and if f 0. In conclusion, claims 1 and 2 are proved. Now, take k > 0, f F k, f 0 with f > 0 in a subset of Ω having positive measure, and let λ (0, λ 0 ]. First note that by (2.4) the following monotonicity property is immediate: n=0 if 0 < λ 2 < λ 1 λ 0 then R λ2 f R λ1 f. (2.5) Then, fix x Ω and consider the function g : (0, λ 0 ] C 0 ( Ω) defined as g(λ)(x) = R λ f(x). Then, by (2.4) and (2.5), it is clear that g λ is nonnegative and nonincreasing. By the first statement of this Lemma, we can assume that Λ 1 < 2λ 0. By Lemma 2.2 there exists Λ 2 = Λ 2 (k) (0, Λ 1 ] such that for all λ (0, Λ 2 ) the

8 8 Genni Fragnelli and Dimitri Mugnai operator L+λI has a strong k (UMP), or equivalently, g(λ) > 0 for all λ (0, Λ 2 ), so that g is not identically trivial. On the other hand, by the second claim g is analytic in (0, λ 0 ], so by analytic continuation we immediately get that g(λ) > 0 for all λ (0, λ 0 ). Now we are able to prove the our main theorem. Proof of Theorem 1.5. Fix k > 0 and take first λ > 0. From Lemmas 2.2 and 2.3 we know that the set Λ + (k) := λ > 0 : L + λi has a k (UMP)} is a nonempty interval. Setting λ + = λ + (k) := sup Λ + (k), by Lemma 2.3 we know that a strong k (UMP) holds if λ (0, λ + ). Now, if λ + R, we show that λ + Λ +. Indeed take an increasing sequence λ n such that lim n + λ n = λ + and let n 0 be such that λ + < 2λ n0. From Lemma 2.3 we know that R λ+ is well defined and by (2.4) applied with λ 0 = λ + we can conclude that u n = R λn f u = R λ+ f uniformly in Ω. If f 0, again by (2.4) we get u n 0, and so u 0. If λ < 0 we can replace L with L and the thesis follows immediately in the same way. 3. Applications In this section we give three differential problems where Theorem 1.5 can be applied. The first two examples are almost straightforward, after the considerations made in Section 1, and consist in extending to higher dimensions the uniform maximum principle proved in [6] for elliptic operators, of course under our version of k (UMP). The third application requires some additional calculations, but we think is an interesting one: in fact, in the last example we consider some classes of linear time periodic parabolic problem, which have raised growing interest in the last year, also in their nonlinear versions, mainly for the large number of biological applications (see [1], [12], [13], [14], [18], [21],... ), and for which a general approach for the validity of a maximum principle seemed to miss so far. On the other hand, we prove the validity of a k (UMP) only in dimension 1, which seems to follow coherently the previous results (see Remark 3.2) Laplace operator Let us consider the classical Neumann problem u + λu = f in Ω, u ν = 0 on Ω, (3.1) where Ω is a smooth bounded domain of R N, N 1, 2, 3}, and f L 2 (Ω). Then it is well known that problem (3.1) has a solution if and only if f = 0. On the Ω

9 A k uniform maximum principle 9 other hand, setting L =, it is clear that Ker(L) = constant functions} and that λ 1 = 0. Moreover, it is evident that the problem Lu = f in Ω, u ν = 0 on Ω, has a unique solution u H 2 (Ω) H0 1 (Ω) C 0 (Ω) with u = 0, that is u C, Ω according to the notations introduced in Section 1. Moreover, as already remarked above, there exists K > 0 such that u H 2 (Ω) K f L 2 (Ω). Summing up, all the abstract requirements (1.5) and (1.6) for L are fulfilled, where the underlying measure µ is simple Lebesgue s measure in Ω. Applying Theorem 1.5 we immediately get that if N 1, 2, 3}, for any k > 0 there exist λ (k) and λ + (k) with λ (k) < 0 < λ + (k) such that L + λi has a k (UMP) if and only if λ [λ (k), 0) (0, λ + (k)], and that a strong k (UMP) holds if λ (λ (k), 0) (0, λ + (k)). In [6] it was already proved that this result was valid for N = 1, but the authors underlined the fact that they could not prove it for N = 2, so that they were naturally turned to consider polyharmonic operators in low dimensions. We consider the same operators in the following section Polyharmonic operator Let us now consider a classical elliptic Neumann problem in presence of an m polyharmonic operator, m N, in a smooth bounded domain Ω of R N, N 1,..., 4m 1}, m u + λu = f in Ω, u ν = u ν... = (3.2) m 1 u ν = 0 on Ω, so that Ker(L) = constant functions}, with L = m. As already remarked in [6], the assumption (1.5) for L is satisfied, and by elliptic regularity the weak solution u H m (Ω) of (3.2) actually belongs to H 2m (Ω) and satisfies the estimate ũ H 2m K f L 2 (Ω) for a suitable constant K. Since N 4m 1, H 2m (Ω) C 0 ( Ω), so that also assumption (1.6) holds. Theorem 1.5 can be immediately applied, extending the analogous result showed in [6] for N 2m 1 to higher dimensions Periodic parabolic problems Let us consider the following parabolic problem: u t αu xx + λu = f in Ω (0, ), u x = 0 on Ω (0, ), u(0) = u(t ). (3.3) Here Ω is a bounded interval of R, α > 0, T > 0, λ R and f L 2 (Q T ), where we have put Q T = Ω (0, T ) for shortness. Using the notation of Section 1, we

10 10 Genni Fragnelli and Dimitri Mugnai set Lu := u t αu xx with D(L) = u H 1 (0, T ; H 2 (Ω)) : u x = 0 on Ω for a.e. t (0, T ) }. As usual, we define weak solutions of (3.3) as functions u L 2 (0, T ; H 1 (Ω)) such that d uv dx + α u x v x dx + λ uv dx = fv dx (3.4) dt Ω Ω Ω Ω for a.e. t in (0, T ) and for all v H 1 (Ω). Moreover u has to satisfy u(0) = u(t ). First, let us note that by parabolic regularity, any solution of (3.3) actually belongs to C([0, T ]; H 1 (Ω)) (this is an obvious consequence of [4, Theorem X.11] applied to time periodic solutions of Neumann problems). This fact lets us prove that Ker(L) = constant functions}: indeed, if u is a solution of u t αu xx = 0 in Ω (0, ), u x = 0 on Ω (0, ), (3.5) u(0) = u(t ), integrating over (0, T ) the related equality given by (3.4) with λ = f = 0 and v = u, gives T d 0 = u 2 dx dt+α u x 2 dx dt = [u 2 (T ) u 2 (0)] dx+ u x 2 dx dt, 0 dt Ω Q T Ω Q T from which we get that u is a constant by the periodicity condition. Moreover, we now prove that f Im(L) if and only if } f L = f L 2 (Q T ) : f dx dt = 0. Q T Indeed, if f Im(L) and u is the related solution, integrating over (0, T ) the definition of weak solution with v = 1, gives T d u dx dt = f dx dt, 0 dt Ω Q T and by periodicity this implies Q T f = 0, so that Im(L) L. Viceversa, let f L L 2 (Q T ) (L 2 (0, T ); H 1 (Ω)) ; then, by [25, Theorem 32.D] there exists a weak solution u L 2 (0, T ; H 1 (Ω)) of u t αu xx = f in Ω (0, ), u x = 0 on Ω (0, ), (3.6) u(0) = u(t ). Any other solution of (3.6) is found adding a constant to u, since their difference solves (3.5); thus there exists a unique solution } ũ C := v C 0 (Q T ) : v dx dt = 0. Q T

11 A k uniform maximum principle 11 Moreover, by parabolic regularity, we get that ũ C 0 ([0, T ]; H 1 (Ω)). In addition, the following estimate holds: ũ L (0,T ;H 1 (Ω)) C } ũ(0) H 1 (Ω) + f L 2 (Q T ) (3.7) for a universal constant C = C(α, T, Ω). We remark that this is again an adaptation of classical estimates to solutions of periodic problems, for example, see [22, Theorem 8.13]. At this point we assume to deal with data (α, T, Ω) such that the related constant C(α, T, Ω) appearing in (3.7) is strictly less than 1, i.e. C(α, T, Ω) < 1. (3.8) We remark that condition (3.8) is satisfied if, for example, α or Ω are sufficiently large. Thus from (3.7) we easily get ũ L (0,T ;H 1 (Ω)) C 1 C f L 2 (Q T ). (3.9) By Poincaré Wirtinger inequality (see [4, Chapter VIII]) we know that so that (3.9) implies ũ(t) L (Ω) Ω ũ(t) H 1 (Ω) t, ũ L (Q T ) C Ω 1 C f L 2 (Q T ); thus (1.6) is satisfied with K = C Ω 1 C. Once done that, by applying the abstract Theorem 1.5 to problem (3.3), we have proved: Theorem 3.1. Assume (3.8), fix k > 0 and set Lu := u t αu xx. Then there exist λ = λ (k) and λ + = λ + (k) such that λ < 0 < λ + + and L + λi has a k (UMP) if and only if λ [λ, 0) (0, λ + ]. Moreover a strong k (UMP) holds if λ (λ, 0) (0, λ + ). Remark 3.2. To our best knowledge, there are not many results for (UMP) or (UAMP) for parabolic problems like (3.3). For example, we quote [10], where the authors prove a result which reminds an antimaximum principle but for certain Cauchy problems with homogeneous Dirichlet boundary conditions. Their result, however, is different in nature from ours, since they show what we could call a kind of eventual antimaximum principle, in the sense that they prove that solutions of Cauchy Dirichlet problems are positive for large times, also when the data is negative. We are aware of the recent paper [16], where the authors consider a periodic parabolic problem under both homogeneous Dirichlet or Neumann conditions, and they show an (AMP) also in presence of a weight. On the other hand, if N = 1, they assume that the right hand side of the parabolic equation belongs to L p with p > 3, and in addition their result is not uniform. On the contrary, with our

12 12 Genni Fragnelli and Dimitri Mugnai approach we can handle the case f L 2 and we can prove a k (UMP), so that certain uniformity for the validity of a maximum or antimaximum principle with data in L 2 is guaranteed, although with some restrictions on the coefficient α and on the interval Ω. Of course, a result analogous to Theorem 3.1 can be proved if in dimension 1 is replaced in higher dimensions by a polyharmonic operator with the natural Neumann boundary conditions, so that the associated spatial Sobolev space is embedded in the space of continuous functions; thus one can consider the following problem: u t α m u + λu = f in Ω (0, ), u ν = u ν... = m 1 u ν = 0 on Ω (0, ), u(0) = u(t ), where all the assumptions made above for m = 1 are obviously generalized according to the new setting, and in particular N 2m 1, since C 0 ([0, T ]; H m (Ω)) C 0 (Q T ). The details are left to the reader. References [1] W. Allegretto, P. Nistri, Existence and optimal control for periodic parabolic equations with nonlocal term IMA J. Math. Control Inform 16 (1999), [2] P. Antonini, D. Mugnai, P. Pucci, Quasilinear elliptic inequalities on complete Riemannian manifolds J. Math. Pures Appl. 87 (2007), no. 6, [3] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vazquez, An L 1 theory of existence and uniqueness of solutions of nonlinear elliptic equations Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22 (1995), no. 2, [4] H. Brezis, Analyse fonctionnelle: Thorie et applications Masson, Paris, [5] A. Cabada, S. Lois, Maximum principles for fourth and sixth order periodic boundary value problems Nonlinear Anal. 29 (1997), no. 10, [6] J. Campos, J. Mawhin, R. Ortega, Maximum principles around an eigenvalue with constant eigenfunctions Commun. Contemp. Math. 10 (2008), no. 6, [7] P. Clément, L. A. Peletier, An anti-maximum principle for second-order elliptic operators J. Differential Equations 34 (1979), no. 2, [8] P. Clément, G. Sweers Uniform anti-maximum principle for polyharmonic boundary value problems Proc. Amer. Math. Soc. 129 (2000), no. 2, [9] P. Clément, G. Sweers Uniform anti-maximum principles J. Differential Equations 164 (2000), no. 1, [10] J.I. Díaz, J. Fleckinger Pellé, Positivity for large time of solutions of the heat equation: the parabolic antimaximum principle Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, [11] R. Fortini, D. Mugnai, P. Pucci, Maximum Principles for anisotropic elliptic inequalities Nonlinear Analysis TMA 70 (2009),

13 A k uniform maximum principle 13 [12] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations J. Math. Anal. Appl., in press. [13] G. Fragnelli, A. Andreini, Non negative periodic solution of a system of (p(x); q(x)) Laplacian parabolic equations with delayed nonlocal terms Differential Equations and Dynamical Systems 15 (2007), [14] G. Fragnelli, P. Nistri, D. Papini, Positive periodic solutions and optimal control for a distributed biological model of two interacting species, 35 pages, submitted for publication. [15] T. Godoy, J.P. Gossez, S. Paczka, A minimax formula for principal eigenvalues and application to an antimaximum principle Calc. Var. Partial Differential Equations 21 (2004), no. 1, [16] T. Godoy, E. Lami Dozo, S. Paczka, On the antimaximum principle for parabolic periodic problems with weight Rend. Semin. Mat. Univ. Politec. Torino 60 (2002), no. 1, [17] H.C. Grunau, G. Sweers, Optimal conditions for anti-maximum principles Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 3-4, [18] P. Hess, Periodic-parabolic boundary value problems and positivity Pitman Research Notes 247, John Wiley, New York, [19] D. Mugnai, P. Pucci, Maximum Principles for inhomogeneous elliptic inequalities on complete Riemannian manifolds Adv. Nonlinear Stud. 9 (2009), [20] P. Pucci, J. Serrin, The maximum principle Progress in Nonlinear Differential Equations and Their Applications 73. Basel: Birkhäuser, [21] T. I. Seidman, Periodic solutions of a non-linear parabolic equation J. Differential Equations 19 (1975), no. 2, [22] S. Salsa, Partial differential equations in action. From modelling to theory Berlin: Springer, [23] G. Sweers, L n is sharp for the anti-maximum principle J. Differential Equations 134 (1997), no. 1, [24] Y.Wang, J. Yin, Z.Wu, Periodic solutions of evolution p-laplacian equations with nonlinear sources J. Math. Anal. Appl. 219 (1998), no. 1, [25] E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B. Nonlinear monotone operators Springer, Acknowledgment The research of G.F. is supported by the M.I.U.R. National Project Metodi di viscosità, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari. The research of D.M. is supported by the M.I.U.R. National Project Metodi variazionali ed equazioni differenziali nonlineari.

14 14 Genni Fragnelli and Dimitri Mugnai Genni Fragnelli Dipartimento di Ingegneria dell Informazione Università degli Studi di Siena Via Roma Siena Italy Dimitri Mugnai Dipartimento di Matematica e Informatica Università degli Studi di Perugia Via Vanvitelli Perugia Italy mugnai@dmi.unipg.it