Beyond the classical Strong Maximum Principle: sign-changing forcing term and ﬂat solutions

We show that the classical Strong Maximum Principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions the normal derivative on the boundary may also vanish on the boundary (deﬁnition of ﬂat solution). This leads to examples in which the unique continuation property fails. As a ﬁrst application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indeﬁnite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign is also given.


Introduction
In a pioneering article, on 1910, Zaremba [49] established the well-known strong maximum principle saying, in a simple formulation, that if Ω is a smooth bounded domain in N and a function u verifies then f The extension to a more general second-order elliptic operator was due to Hopf, on 1927, in his famous article [38].Moreover, some years later, on 1952, Hopf [39] and Oleinik [42], independently, proved that under the above conditions, the normal derivative of u satisfies the following sign condition: (see the survey [2] for many other historical data).
A "quantitative strong maximum" principle was obtained since 1987 ( [41], [34], [50], [8], [5],...).In order to be more precise, we will work in the class of very weak supersolutions: the theory of very weak solutions to equation (10) was introduced in an unpublished article by Haïm Brezis on 1971, later reproduced in 1996 by Brezis et al. [9] (see a regularity extension in the study by Díaz and Rakotoson [35]).This theory applies to the more general class of data ( ) for which it is possible to give a meaning to the notion of solution of the corresponding problem.We assume Ω such that d , Then, aforementioned "quantitative strong maximum principle" says that if f satisfies (P f ), and thus then we obtain the following estimate, called in [5] as the uniform Hopf inequality (UHI): for some > C 0 only dependent on Ω.Note that, if for instance on ∂Ω.The main goal of this article is to show that the sign assumption (P f ) can be removed so that, under suitable conditions, any supersolution u satisfying equation (1) for suitable sign-changing functions ( ) f x is again strictly positive on Ω.Moreover, under suitable conditions, this strictly positive supersolution for some sign-changing datum ( ) f x does not satisfy condition (2).In some cases, this kind of sign-changing datum ( ) f x may still satisfy condition (3) (see Remark 2.2), but the conclusion (4) may fail (see the notion of flat solution given below).
As far as we know, curiously enough, such a type of extension of the classical strong maximum principle was not presented in the previous literature on the subject (see, e.g., [10,12,44,45,48], among many other articles and books; the survey [2] contains more than 230 references on the subject until 2022, but it seems that none of them deals with the case in which ( ) f x changes sign).There are some articles in the literature that could lead to some related conclusions but their statements are not presented in the same form as in this article (see, e.g., Remark 3.11).
We will pay a special attention to the case in which ( ) < f x 0 in some neighborhood of ∂Ω, but many other cases can also be considered (see Remark 2.10).In order to present our results, we will use the decomposition We assume in that article that the region where ( ) f x is negative has at least a part that touches ∂Ω; however, different cases can also be considered by similar arguments (see Remark 3.9).We assume that there exists an open subset Ω satisfies the interior sphere condition, sup , Ω 0.
x Ω The kind of "new assumptions" on ( ) f x giving positive solutions are of the following type: (H 1 ) A suitable balance expressing that the negative zone of ( ) f x takes place near the boundary ∂Ω: condition (5) holds and there exists a compact set ⊂ + K Ω where ≠ f 0 on K and with * c K < * C , K some positive constants (depending on K ), which will be defined later (see Lemma 3.1 below).(This will allow us to conclude that ≥ + u C on ∂K for a suitable > + C 0, see condition (52).) (H 2 ) A suitable decay of ( ) f x near the boundary ∂Ω: there exists > α 1 such that a.e.Ω, with K the compact mentioned in (H 1 ) and Here, φ 1 denotes the first eigenfunction of the Laplacian operator in Ω, which is given as follows: . We recall that by well-known results φ δ 1 , ( ) Note that the first condition in (H 2 ) has a geometrical meaning (see Remark 3.5).
Under such type of "new assumptions" on f , we will prove (A) The positivity of u, property (P u ), still holds.In addition, if, for instance, ( ) on ∂Ω, (B) under additional conditions on ( ) f x , the positive solution of the linear problem (i.e., now with the equality symbol =, instead ≥) does not satisfy condition (2) but on ∂Ω.Property (B) corresponds to the notion of flat solution already considered by different authors in the framework of some nonlinear problems (see, e.g., [17,30,40]).The existence of flat solutions shows that assumption (P f ) is necessary to conclude condition (4).Note also that a flat solution u on a problem (10) on the domain Ω can be extended by zero to obtain the unique solution ∼ u of a similar problem associated to an extended domain ͠ ⊋ Ω Ω with the right-hand side given as follows: In this way, we can construct solutions with compact support for data with compact support becoming negative near the boundary of its support.This proves that the version of the strong maximum principle obtained in [12] (ensuring that the solution ≥ u 0 of a linear problem (10) corresponding to a datum ≥ f 0 cannot vanish on some positively measured subset of Ω except if ≡ u 0 on Ω) has optimal conditions on ( ) f x .It is a curious fact that the above considerations are motivated, in some sense, after the long experience in the study in different semilinear equations with a non-Lipschitz perturbation in the last 50 years.Indeed, in the study of semilinear problems, with ≥ g 0 and β a continuous function, for instance, such that ( ) = β 0 0, it is well known the existence of a flat solution under suitable conditions on β and g.For the case of β nondecreasing, subdifferential of the convex function j, = ∂ β j such that Beyond the classical strong maximum principle  3 ( )

∫
< +∞ s j s d 0 and ≠ g 0, we send the reader to Theorem 1.16 of [17] (the so called "nondiffusion of the support property": see also the generalization presented in [7] and [1] for the case of β a multivalued maximal monotone graph).For the autonomous case ≡ g 0 and β non-monotone see, e.g., [27,31,32] and its references.This means that if we take with u the flat solution of equation (11), then u is also a flat solution of the corresponding linear problem (10).Note that, necessarily, such ( ) f x becomes negative near the boundary ∂Ω.The above extension of the strong maximum principle admits many generalizations, which will be indicated in form of a series of remarks (Schrödinger equation [Remark 3.7], operators with a first-order term [Remark 3.9], linear nonlocal operators [Remark 3.8], nonlinear elliptic operators and obstacle problem [Remark 3.10], etc.).
The organization of this article is as follows: we start by proving, in Section 2, a version, as simple as possible, for the one-dimensional case in which assumptions (H 1 ) and (H 2 ) can be easily formulated in an optimal way.The N -dimensional case without symmetry conditions is presented in Section 3.An application to some sublinear indefinite semilinear equations (see, e.g., [37] and [33]) will be given in Section 4. Finally, in Section 5, we will consider the linear parabolic problem We will show that the above arguments for stationary equations, jointly to some related results [23], allow us to prove that, under suitably changing sign conditions on ( ) u x 0 and/or on ( )

The symmetric one-dimensional linear problem
For the sake of the exposition, here we consider supersolutions ( ) u x of the symmetric one-dimensional linear problem on the domain We assume the symmetry condition and we will work in the framework of the space ( ) and we consider very weak supersolutions, i.e., functions such that for any Cδ x for any ∈ x Ω, for some > C 0, then the expressions in equation ( 15) make sense.The notion of a very weak solution is similar, except the symbol ≥ is replaced with =.In some parts of our exposition, we will refer to symmetric solutions (and not merely supersolutions).By well-known results (see, e.g., [9,35]), we have that Let us see how the type of assumptions (H 1 ) and (H 2 ), mentioned in the Introduction, can be easily formulated, and even in an optimal way.
Theorem 2.1.We assume that ( ) f x becomes negative near the boundary in the following sense: there exists (A) Assume the "balance condition" and the "decay condition" Then, any symmetric supersolution u satisfies In addition, assumed condition (19), if u is a solution, then > u 0 if and only if the decay condition (20) holds.(B) Assume now equations (19), (20), and Let u be the unique solution u of problem (16).
if and only if the following condition holds: Remark 2.2.It is easy to see that assumptions (18) and (19) imply that ( ) f x satisfies the positivity of the weighted integral (condition (3)).Indeed, As a matter of fact, one of the main goals of the present article is to prove a conjecture raised by the first author and communicated to Jean Michel Morel in 1985 (when preparing the joint article [34]), concerning the possibility for sign-changing data ( ) f x to keep the positivity of the weighted integral (3) and guaranteeing the positivity of the supersolution.
Example 2.3.Before to give the proof, let us see the different kinds of behavior of solutions for a simple example with a changing sign right-hand side term.Consider for different values of ≥ a 1 with Figure 1 shows different cases for the values of = a 1, = a 1.8, = a 2, and = a 2.2.In the first case of Figure 1 = a 1, the forcing ( ) f x is strictly positive and the classical strong maximum principle applies.In the case = a 1.8 we see that the solution is strictly positive and its normal derivative at the boundary is again strictly negative.For = a 2 the conditions of part B of Theorem 2.1 hold and we see that the solution is flat.Finally, for = a 2.2, the assumptions of part A of Theorem 2.1 fail and the solution becomes negative near the boundary.(25), and the value of its normal derivative at the boundary, when the forcing is given by (26), for different values of a.
Proof of Theorem 2.1.By well-known results (see, e.g., [8,9,35]), we may assume that u is symmetric and has some regularity properties, i.e., u is such that To establish part (A), let us start by proving that Multiplying the equation by ( , , 0 and using condition (18), we have Integrating by parts, and using the boundary condition, we obtain , we obtain On the other hand, from condition ( 14) and the structure assumptions (18), we have that in fact Then, for any [ ] ∈ r r 0, 0 , we have which clearly implies that the maximum of u is taken at = r 0. Thus, from condition (18), Substituting in expression (29), since which proves that condition (19) implies property (28), i.e., ( ) > u r 0 0 .[In fact, it is easy to see that if u is a solution (and not merely a supersolution), then assumption ( 19) is also a necessary condition to have ( ) > u r 0 0 .]Note also that, from condition (18), we always have Then, we obtain which implies, from inequality (30), that Beyond the classical strong maximum principle  7 To complete the proof of (A), we see that from the structure conditions (18), for any Then, integrating again, for any  ( ) ∈ r r R , , we have Assumption (20) implies that and thus, for any Then, making ↗ r R, in equation ( 34), we obtain which leads to the positivity conclusion (21).Obviously, if, for instance, Let us prove (B).We observe that assumed (19) and (20), then if u is a solution of (16) the inequality (33) becomes an equality.Making = r R and using condition (31), since This proves the necessary and sufficient condition to have flat solutions.□ The existence of nonnegative solutions with compact support in a larger domain , is a simple consequence of the existence of flat solutions on the small domain Note that this proves a failure of the unique continuation property under the below conditions.
be the extension of a given function Assume that f satisfies conditions (19), (20), and (24).Let u be the unique solution of problem (16), and let ∼ u be the extension of u defined as follows: , and it is the unique weak solution of the problem Proof.By Theorem 2.1, we know that ( ) u x is a flat solution of the problem (16) associated to ( ) on Ω, and then, the extension ∼ u is a weak solution of the extended problem (37).By the uniqueness of solutions for such problem, ( ) ∼ u x is the unique function satisfying problem (37).□ Remark 2.5.No flat solution may satisfy the Hopf conclusion (2) nor the decay estimate (4).This proves that the nonnegative condition assumed on ( ) f x in the corresponding results is necessary.Analogously, the Corollary 2.4 proves that the version of the strong maximum principle obtained in [12] (ensuring that the solution ≥ u 0 of a linear problem (10) corresponding to a datum ≥ f 0 cannot vanish on some positively measured subset of Ω except if ≡ u 0 on Ω) fails for positive solutions corresponding to changing sign data ( ) f x .
Remark 2.6.It is possible to give an alternative proof of Theorem 2.1 by using the Green function associated to the Dirichlet problem when the datum ( ) − f x is symmetric.Indeed, as before we can argue only for very weak solutions of the problem (25).Let us assume = a 1.Then, the Green function , is given by (see, e.g., [47], p. 54) and since (( Recall that, in this special case, ( ) . Then, some straightforward computations lead to the explicit formula x x 0 1 (40) From this formula, by assuming ( 18)-( 20), we can deduce again the expressions ( 36) and (34), and the proof follows.Once again, we see that > u 0 if and only if ( 18)-( 20) hold.
Remark 2.7.Clearly, condition ( 14) is weaker than assumption (23).On the other hand, we point out that the decay assumption (20) indicates that the indefinite integral ( ) should decay to zero, as ↗ r R, with a rate less than a linear decay (and in fact on the whole interval ( ) r R , 0 ).Note that this does not require a pointwise decay of the type ( ) → f r Beyond the classical strong maximum principle  9 Then, the decay condition (20) holds if we assume C small and ( ) ∈ α 0, 1 .Note that it fails for ( ) ∈ α 1, 2 (similar computations can be carried out for = α 1).
Remark 2.8.Concerning the optimality of the decay condition, as indicated in Theorem 2.1, in the class of solutions of problem ( 16), the condition is optimal.It is not difficult to build explicit examples of functions ( ) f r satisfying assumption (14) and such that ( ) > f r 0 on a very large zone on ( ) ∈ r r 0, 0 and only negative in a very small region , for which the unique solution u of problem ( 16) is negative near the boundary = r R.This is the case, for instance, if we consider for some positive constants F and C f .It is clear that 1 , and thus, there exists a unique solution u of the corresponding problem (16).Moreover, by using formula (36) (which becomes an for the case of solutions and not merely supersolutions), it is easy to check that, even if = − ε R r 0 is small and ∕ F C f is large enough, then there exists Remark 2.9.Note that, under the structure condition ( 18) on ( ) f x , we have and thus, condition ( 24) is equivalent to Remark 2.10.The case in which > f 0 on a large part of the domain but with < f 0 in an interior subset of Ω can be also considered by this type of techniques.In that case the positive solutions satisfy that ( ) 0, but they may generate interior points ∈ x Ω 0 , where ( ) and also the formation of an internal "dead core."Note that, again, this proves a failure of the unique continuation property under such conditions.Here is an example with a similar structure to Example 1 but now reversing the positive and negative subsets of ( ) f x .In the first case of Figure 2, = a 4, the forcing ( ) f x is now positive close to the boundary and negative on ( ) −1, 1 : the solution is positive and a local minimum is created at = x 0. The dependence on a of the local minimum ( ) u 0 is also included in Figure 2.For = a 3.4142, we have ( ) = u 0 0. Finally, in Figure 2, it is also represented the solution u corresponding to a forcing term ( ) ( We see the formation of an internal "dead core" (the interval ( ) −0.5, 0.5 ) where = u 0.
Remark 2.11.In fact, the property is a local property, and it may occur only on a part of ∂Ω.A mixed situation, exhibiting this fact, is given in the following example (see also [14]).Consider the problem with Note that ( ) It is easy to see that the unique solution of equation ( 42) is On the other hand, since the roots of ( ) = u x Remark 2.12.Note that a necessary condition in order to have the positivity of the solutions of problem ( 42) is that (it suffices to the equation by φ 1 and integrate twice by parts).The above example shows that this necessary condition is not sufficient to have the positivity of the solution.Indeed, we can study the values of a for such function f we have for > a 2 and, otherwise, the positivity of u requires ≥ a 3 (i.e., for ( ] ∈ a 2, 3 we have that ( ) then the solution of and thus, ( ) > u x 0 for any ( ) ( ) 1, 0 0, 1 and ( ) = u 0 0. In addition, if we take for some > b 0, then the solution has a dead core in ( ) The N-dimensional case and a general bounded domain Ω Now we consider the N -dimensional case and a general bounded regular domain Ω.Our strategy will be quite close to the main idea of the proof of Theorem 2.1 for the one-dimensional case.We will assume ( ) f x such that there exists an open subset ⊂ + Ω Ω verifying the hypothesis (5).In a first step, we will prove something weaker that in the first step of the proof of Theorem 2.1, we will not prove the positivity of u on + Ω (in Theorem 2.1, the interval [ ] r 0, 0 ) but on a regular compact K contained in + Ω : we will show that there exists a positive constant + C such that any supersolution u of equation ( 1) satisfies see the expression (52) below.
For the proof, we will need to work with some auxiliary problems of the type where ∈ ∂ y K, χ A denotes the characteristic function of A and > ϱ 0 is small enough such that Due to the compactness of K and the classical strong maximum principle (Hopf-Oleinik boundary lemma, since ∂Ω satisfies the interior sphere condition), it is well known that for any ∈ ∂ y K, there exist two positive constants ( ) where Using the compactness of K , it is possible to obtain the following result (that will be proved later) giving some uniform estimates: Lemma 3.1.There exists two positive constants < x for any y K Ω .
K y K (48) As in the one-dimensional case, there will be a second step in the proof of the main result of this section, where we prove that, under the balance and decay conditions mentioned in the Introduction, the unique solution v of the problem on the ring ⧹K Ω is a positive subsolution, and thus, For simplicity in the exposition (since there are other different options), this subsolution ( ) v x will be con- structed in terms of a suitable power of φ 1 , the normalized first eigenfunction of the Laplacian operator on Ω.We recall that u is a very weak supersolution, which means that ( ) for any Cδ x for any ∈ x Ω, for some > C 0, then the expressions in condition (50) make sense.The notion of very weak solution is similar, but replacing the symbol ≥ by = , we have the following theorem.In addition, if then the unique weak solution ( ) of the linear problem (10) is a flat solution.
Proof.First step.As in the proof of the first part of Lemma 3.2 of [8] (in which the authors offer an alternative proof to the main result of [41]), we will use the mean value theorem (in our case on + Ω ) but in a different way.As mentioned before, let K be a regular compact contained in + Ω .Since K is compact and Using condition (50), i.e., integrating twice by parts (since and ≥ ς 0 y ), and using the uniform estimates (condition ( 48)), we have Then, thanks to the balance condition (H 1 ), we obtain that Moreover, since we obtain (45) when ( ) In the general case, it is enough to approximate f and u Δ by a sequence of regular functions such that ( ) , satisfying the assumptions of the statement for each ∈ n .We know that the corresponding sequences of functions f n and u n converge in ( ) L δ Ω : respectively (see [9,35]).Then, we arrive to the conclusion since the inequality ( 45) is stable for the strong convergence in ( ) L Ω 1 .Second step.Let us prove that, under (H 1 ) and (H 2 ), the function is a subsolution to problem (49), for some positive constant k and for some > α 1, where φ 1 denotes the first eigenfunction of the Laplacian operator in Ω (see (9)).We have Thus, by assumption (H 2 ), over the ring − K Ω , there exists > ε 0 such that Then, we have On the other hand, the inequality Then, from the definition of M given in condition (8), we have that − = M εkα 0 and thus (the trace, on ∂K , of the very weak solution u is well defined by the regularity results of [35]).Then, since we can apply the comparison principle on the ring ⧹K Ω (see, e.g., [9]), and we obtain that which ends the proof of (A).
The proof of the second conclusion will follow again from the Green's formula.We recall that now , which gives a meaning to the Green's formula since From condition (51), integrating in equation (10), we obtain and since we already know that ( ) The conclusion can be obtained in some different ways.We will give here some direct arguments.By applying the UHI to problem (46), we obtain ϱ is a continuous function on the compact ∂K.Thus, there exists > μ 0 such that ( ) ≥ μ y μ for any ∈ ∂ y K, and then, we obtain Then, by the comparison principle and well-known regularity results, we obtain Ω, y for some > C 0 and for any ∈ ∂ y K. Thus, As in the one-dimensional problem, the existence of nonnegative solutions with compact support in a larger domain ͠ ⊋ Ω Ω is a simple consequence of the existence of flat solutions on the small domain.
Assume that f satisfies conditions (H 1 ), (H 2 ), and (51).Let u be the unique solution of problem (10), and let ∼ u be the extension of u defined as follows: Then, ∼ u is the unique weak solution of the problem (54) Remark 3.4.The comments on the optimality of the nonnegative character of ( ) f x in previous statements of the strong maximum principle made in the previous section apply also for N-dimensional linear problems.Nevertheless, the extension to other second-order elliptic operators is a delicate point.The case of Lipschitz coefficients can be treated (see Remarks 3.9 and 3.10), but, even if ( ) f x is there are several counterexamples in the literature showing that the strong maximum principle may fail for merely bounded coefficients (see, e.g., [2,5] and its references).It would be interesting to see if the application of the Green's function allows us to obtain sharp conditions on the datum ( ) f x as in the one-dimensional case (Remark 2.6).
Remark 3.5.The first condition in (H 2 ) has a geometrical meaning.It requires that if , this condition requires that ∈ ⊂ + K 0 Ω.This holds in the case of Theorem 2.1.Note also that the value of > α 1, which makes possible this condition, must increase if ( ) ⧹ + d Ω Ω , 0 decreases.Remark 3.6.Arguing as in [1], it seems possible to obtain a "local" condition on ( ) f x on some neighborhood of some points ∈ ∂ x Ω 0 in order to construct suitable local supersolutions implying that ( ) (see also [14]).
Remark 3.7.It is possible to adapt the above proof to the case of the Schrödinger operator with an absorption potential ( ) The easy case concerns bounded absorption potentials V Indeed, the second part of the proof consists in finding a subsolution on the region − K Ω , and using again ( ) ( ) = w x kφ x α 1 V , we see that V It is not difficult to see that we can take ≥ α α V (the exponent when there is no absorption term).Then, ( ) V , i.e., the subsolution in the case with absorption is smaller than the one for the case without absorption.The case of an unbounded potential requires additional approximation arguments.For some related results, see [18][19][20]24,25,28,29,43].
Remark 3.8.The case of the Schrödinger equation for a nonlocal diffusion (the fractional Laplace operator) and an unbounded absorption potential can also be considered with the techniques of the article [26] (exposition made in [22]).
Remark 3.9.The study of the presence of linear transport terms in the equation can also be carried out (see some similar techniques in [6,24,25]).When the negative part of ( ) f x takes place on an interior subregion of Ω (as indicated in Remark 2.10) and ( ) − f x satisfies some suitable conditions, it is possible to prove the positivity of the solution of by means of probabilistic methods (see [16]), which also allow us to consider the case of a general elliptic operator with Lipschitz coefficients.

0
Beyond the classical strong maximum principle  5

Figure 1 :
Figure 1: Representation of the exact solution of (25), and the value of its normal derivative at the boundary, when the forcing is given by(26), for different values of a.

0 1 Figure 2 :
Figure 2: Representation of the solution of problem (25) when the forcing is like (26) but with the oposite sign on ( ) f x .

ϱ
solution of the auxiliary problem(46).Let us assume, for the moment, that ( )∈ + u C Ω 0 .Then, since − ≥ u Δ 0 in + Ω ,we conclude that there exists a positive constant  c such that, for any ∈ ∂ y K, Beyond the classical strong maximum principle  13