Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case

We establish Hardy - Poincare and Carleman estimates for non-smooth degenerate/singular parabolic operators in divergence form with Neumann boundary conditions. The degeneracy and the singularity occur both in the interior of the spatial domain. We apply these inequalities to deduce well-posedness and null controllability for the associated evolution problem.

1. Introduction. This paper deals with a class of degenerate and singular parabolic operators with interior degeneracy and singularity of the form u, associated to Neumann boundary conditions and with (t, x) ∈ Q T := (0, T ) × (0, 1), T > 0 being a fixed number. Here λ ∈ R satisfies suitable assumptions and the functions a and b, that can be non-smooth, degenerate at the same interior point x 0 ∈ (0, 1). The fact that both a and b degenerate at the same point is actually the most complicated situation. Indeed, if a and b degenerated at different points, we could separate the problem in a purely singular one and in a purely degenerate 2010 Mathematics Subject Classification. Primary: 35Q93; Secondary: 93B05, 34H05, 35A23. Key words and phrases. Carleman estimates, singular/degenerate equations, non smooth coefficients, Hardy-Poincaré inequality, Caccioppoli inequality, observability inequality, null controllability.
The first author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). She is supported by the GNAMPA project 2017 Comportamento asintotico e controllo di equazioni di evoluzione non lineari and by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017. The second author is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and supported by the 2017 INdAM-GNAMPA Project Equazioni Differenziali Non Lineari. He is also supported by the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT 009) and by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017. * Corresponding author: Dimitri Mugnai.

GENNI FRAGNELLI AND DIMITRI MUGNAI
one and treat them separately. As in [4] and [7], we shall admit different types of degeneracy for a and b. In all cases, we mean that • there exists x 0 ∈ (0, 1) such that a(x 0 ) = b(x 0 ) = 0 and a, b > 0 on [0, 1]\{x 0 }, 1]. The different nature of the degeneracy depends on the Sobolev space to which a, b belong and on the values of K 1 and K 2 , according to the following cases: (WWD): weakly-weakly degenerate case: a, b ∈ W 1,1 (0, 1) and K 1 , K 2 ∈ (0, 1); (SSD): strongly-strongly degenerate case: a, b ∈ W 1,∞ (0, 1) and K 1 , K 2 ∈ [1, 2); (WSD): weakly-strongly degenerate case: a ∈ W 1,1 (0, 1), b ∈ W 1,∞ (0, 1) and The restriction K i < 2 is essentially related to existence and controllability problems, see [2], [6], [11]. This paper is in some sense a continuation of the previous works [4] and [7], where the authors study well-posedness and null controllability for the following problem via suitable Hardy -Poincaré inequalities and Carleman estimates: Here x) for all t ∈ (0, T ), χ ω is the characteristic function of an open set ω ⊂ (0, 1) and h ∈ L 2 (0, T ; X). Actually, while in [4] the author considers the operator in non divergence form with Dirichlet or Neumann boundary conditions, in [7], the authors take into account the operator in divergence form only with Dirichlet boundary conditions. Hence, this paper is devoted to complete the study of well-posedness and null controllability for (1) in the case of Neumann boundary conditions and Au := (au x ) x + λ u b . Namely, we consider the evolution system where u 0 ∈ L 2 (0, 1) and the control h ∈ L 2 (Q T ) acts on a non empty interval ω ⊂ (0, 1). The main goal of the present paper is to establish Hardy -Poincaré inequalities and global Carleman estimates for operators of the form given in (2) in order to obtain well-posedness and null controllability. We recall that (2) is said globally null controllable if for every u 0 ∈ L 2 (0, 1) there exists h ∈ L 2 (Q T ) such that the solution u of (2) satisfies u(T, x) = 0 for every x ∈ [0, 1] and h 2 L 2 (Q T ) ≤ C u 0 2 L 2 (0,1) for some universal positive constant C. For a brief history on well-posedness, null controllability and Carleman estimates for (2), also in the case of non divergence problem, we refer to [4] and [7].
The present paper is organized in the following way: in Section 2, we establish some Hardy-Poincaré inequalities and, using them, we study the well-posedness. In Section 3, we prove Carleman estimates and we use them, together with a Caccioppoli type inequality, to prove the following observability inequality: there exists a positive constant C T such that every solution v of the adjoint problem As an immediate consequence, one can prove, using a standard technique (e.g., see [9, Section 7.4]), null controllability for the linear degenerate/singular problem (2). We underline that in order to prove (3) the Hardy -Poincaré inequalities proved in Propositions 1 and 2 are essential. They are different from the ones proved in [6], which hold only under Dirichlet boundary conditions. Indeed, as we can see, in the inequalities (5) and (6) there are additional terms with respect to the inequalities in [6] and these additional terms complicate the proof of the observability inequality specially in the case λ > 0. We underline also that (3) is proved only in particular cases. Indeed to obtain this estimate, inequality (6) is crucial, but it is proved only if we are in the (WWD), (WSD) or in the (SWD) case. Observe that, also in [7], null controllability was not proved for (2) with Dirichlet boundary conditions in the (SSD) case for the same reason: the observability inequality was established only in the other three cases. However, using a different technique, in [8] it is proved that null controllability still holds for (2) in the case of Dirichlet boundary conditions also in the (SSD) case, at least when λ < 0. Finally, notice that the results contained in the present paper generalize the ones obtained in [1] in the case of a divergence operator when λ = 0 (that is, in the purely degenerate case).
A final comment on the notation: by C we shall denote universal positive constants, which are allowed to vary from line to line.
2. Well-posedness. In this section we will consider (2) and, for the well-posedness of the problem, we start introducing the following weighted Hilbert spaces, which are suitable to study all situations, namely the (WWD), (SSD), (WSD) and (SWD) cases: H 1 a (0, 1) := u ∈ W 1,1 (0, 1) : and endowed with the inner products respectively. Notice that, if u ∈ H 1 a (0, 1), then au ′ ∈ L 2 (0, 1). The choice of the notation H 1 a (0, 1) is due to the desire to underline the dependence on a, in coherence with the structure of the operator. Remark 1. Of course, it is possible to make different assumptions on the functional setting, and so on the related domain of the operators. For instance, it would be natural to distinguish the case K 1 ∈ (0, 1) from the case K 1 ∈ [1, 2), using H 1 (0, 1) in the former case and H 1 loc ((0, 1)\{x 0 }) in the latter, as done in [2], [5], [6]. However, this choice implies that solutions may be discontinuous at x 0 . We prefer to avoid such unphysical situations a priori, making the set of solutions smaller.
The following properties will be very helpful: and non decreasing on the right of x = x 0 .

(4)
Then, there exists a constant C > 0 such that for any function w, locally absolutely the following inequality holds: Using the previous result, one can prove new Hardy Poincaré inequalities in the space H 1 a (0, 1) or H 1 a,b (0, 1): Proposition 2. The following holds: holds for every u ∈ H 1 a,b (0, 1).
The space we will use is obviously the space where (6) holds; thus in view of the previous Propositions, such a space is When Proposition 2 holds, then the standard norm · 2 H is equivalent to for c = C HP + 1.
When λ < 0, an equivalent norm in H is From now on, we make the following assumptions on a, b and λ.
Observe that the assumption λ = 0 is not restrictive, since the case λ = 0 is considered in [1].
Thanks to the previous propositions, as in [7, Proposition 2.18], one can prove the next inequality.
Proof. By using Proposition 2, one has We recall the following definition:  Finally, we introduce the Hilbert space , so that u ∈ H 1 a,b (0, 1) and inequality (6) holds.
Proof. If λ < 0 the proof is similar to the one of [7, Theorem 2.22], so we omit it. If λ ∈ (0, 1/C HP ), by Proposition 3 for all u ∈ D(A) we have 3. Carleman estimates for singular/degenerate problems and its application to observability inequality in the Neumann case. In this section we prove an estimate of Carleman type for the adjoint problem of (2), that is for where T > 0 is given and h ∈ L 2 (Q T ). As it is well known, to prove Carleman estimates the final datum is irrelevant, only the equation and the boundary conditions are important. For this reason we consider only the problem   3.1. Carleman estimates. In order to deal with Carleman estimates for (13), we introduce the functionã, which is an extension of a to the interval [−1, 2]: Since a, b have no regularity property beyond Sobolev regularity, for technical reasons (see [4], [6] and [7]), we make the following assumptions: In addition, when K 1 > 3 2 the function in (14) is bounded below away from 0 and there exists a constant Σ > 0 such that

Hypothesis 3. Assume (WWD) or (WSD). Suppose that there exist
and two strictly positive constants g 0 , h 0 such that g(x) ≥ g 0 and As in [5] or in [6, Chapter 4], we introduce the functions ϕ(t, x) := Θ(t)ψ(x) and and, for A < B, Here y − x 0 a(y) dy, r, c 1 > 0 (c 1 will be taken sufficiently large for the observability inequality), c > 0 is such that max Proof. First, assume that x 0 , B 1 , 2 − B 2 ∈ ω. Then, we can fix two subintervals whereλ i = (λ i +β i )/2, i = 1, 2. Then, define w := ξv, where v is any fixed solution of (13). Hence w satisfies Now, define a smooth function τ : [−1, 2] → [0, 1] such that and where v satisfies (13). Thus W satisfies the problem   Defining z := τ W , we have that z satisfies the nondegenerate problems for all s ≥ s 0 . Now, as in [6], we can prove that exist two positive constants k i , i = 1, 2, such that Thus, by definitions of τ , W and z, by using (17), we have Using the fact that τ x is supported in , the boundedness ofã ′ (far away from x 0 if a ∈ W 1,1 (0, 1)) and the fact that Hence, for all s ≥ s 0 . Analogously, by using (18), we can choose s 0 so large that, for all s ≥ s 0 and for a positive constant C: h 2 e 2sΦ2(t,x) dxdt .

GENNI FRAGNELLI AND DIMITRI MUGNAI
Nothing changes in the proof if ω = ω 1 ∪ ω 2 and each of these intervals lies on different sides of x 0 , as the assumption implies.
3.2. Application to observability inequality. In this section we consider problem (2) and we assume that the control set ω is an interval which contains the degeneracy point or the union of two intervals each of them lying on one side of the degeneracy point, i.e.
Remark 5. Observe that, if (21) holds, we can find two subintervals

Hypothesis 4.
1. One among (WWD), (WSD) or (SWD) with K 1 + K 2 ≤ 2 , then there exists a constant θ ∈ (0, K 1 ] such that (14) holds, namely (14) is bounded below away from 0 and there exists a constant Σ > 0 such that 5. Hypothesis 3 holds with B 1 = β 1 and B 2 = 2 − λ 2 . Remark 6. Though it seems that Hypothesis 4 is simply a re-listing of previous ones with the additional two last conditions, this is not the case, since in Hypothesis 1.1 we assumed λ > 0, while in Hypothesis 4.1 we don't. On the other hand, the case λ < 0 in Hypothesis 1 is considered also in the (SSD) case, while in Hypothesis 4, it is not. This is due to the fact that we are dealing with sufficient conditions for controllability, not for existence, i.e. if we have a solution, then it satisfies the observability inequality below. Now, we associate to (2) the homogeneous adjoint problem where T > 0 is given. By the Carleman estimate given in Theorem 3.1, we will deduce the analogous observability and controllability results of [ Hence, given u 0 ∈ L 2 (0, 1), there exists h ∈ L 2 (Q T ) such that the solution u of (2) satisfies u(T, x) = 0 for every x ∈ [0, 1]. Moreover for some positive constant C.

Proof of Theorem 3.2.
As a consequence of the Carleman estimate given in Theorem 3.1, here we will prove the observability inequality (24), the null controllability property following in a standard way. The proof of (24) starts as the one given in [1]. However, due to the presence of 1 0 u 2 dxdt in (6) and of u(0) and u (1) in (5), it is more difficult and in some sense different from the one of [1] (see in particular the proof of Lemma 3.4 below).
First of all, we consider the adjoint problem with more regular final-time datum where D(A 2 ) = u ∈ D(A) Au ∈ D(A) . As usual, we define the following class of functions: W := v is a solution of (25) , which is strictly contained in C 1 [0, T ] ; H 2 a,b (0, 1) ⊂ V ⊂ U ⊂ C([0, T ]; L 2 (0, 1)) ∩ L 2 (0, T ; H). For any solution v of (25), as a corollary of Theorem 3.1, we get the following estimate: Here Θ and ϕ are as in (15), with c 1 sufficiently large.
Lemma 3.3 is the key tool to prove the following result, which is essential for proving Theorem 3.2: Hence, If λ < 0, we have obviously that the function is non decreasing for all t ∈ [0, T ]. In particular, If λ ∈ (0, 1/C HP ), by (26) and Proposition 3, we have i.e. d dt Thus the function t → e 2λC HP t Now, we will concentrate on the last integral and we will prove that for a positive constant C. Indeed, using the Young inequality, we find Consider the integral 1 0 a 1/3 |x − x 0 | 2/3 v 2 (t, x)dx.