Math. J. Okayama Univ. 56 (2014), 157–169
B OUBAKER-K ALED S DALLAH Abstract. In this paper we consider the heat equation with Dirichlet boundary conditions in a conical domain. We look for a sufficient con- dition on the lateral surface of the cone in order to have the optimal regularity of the solution in an anisotropic Sobolev space when the right hand side of the equation is in a Lebesgue space.
1. I NTODUCTION
Let φ be a locally Lipschitz function φ : (0,T] → R+  where T > 0, such that
and Ω ⊂ R³ be the open subset of conical type :
Consider in Ω the parabolic problem
where D(s,r) denotes the disc of radius r centred at (s,0,0), and L 2 (Ω) is the usual Lebesgue space on Ω. We look for the solution u in the anisotropic Sobolev space
here, H 1 (Ω) stands for the Sobolev space defined by H 1 (Ω) = { u ∈ L 2 (Ω) : ∂tu ∈ L 2 (Ω), ∂zu ∈ L 2 (Ω),∂yu ∈ L 2 (Ω)} . The space H 1,2(Ω) is equipped with the natural norm, that is The belonging of the solution u  to H 1,2(Ω) depends on the function ϕ . Our aim is to find a sufficient condition on φ , as weak as possible in order to obtain a solution u ∈ H 1 ,2 (Ω). Our main result is Theorem 1. Assume that ϕ(0) = 0 and
or ϕ′(t) ≥ 0 a.e. in a neighborhood of  0. Then, for all f ∈ L 2 (Ω), Problem (1) admits a (unique) solution u ∈ H 1,2(Ω). For example, if ϕ(t) = αtα , Problem (1) admits a unique solution for any positive constants α and α .  The same result holds true if ϕ(t) = tα ln t , with α >キObserve that the main difficulty is due to the condition ϕ(0) = 0 (instead of ϕ(0) > 0) which gives a conical point at (0,0,0) and then, it does not allow to transform the cone Ω into a usual cylindrical domain. This type of problems has been studied by Sadallah [14] in one space dimension for the parabolic operator ∂t u + (−1)m∂zm u with m ≥ 1. Alkhutov [1] has treated, in some weighted Sobolev L P -spaces, the heat equation in bounded and unbounded domains of paraboloid type. He has considered in [2] the case of the heat equation in a ball; this case corresponds here to ϕ(t) = √t(2R − t) in the neighborhood of 0 (for a ball of radius R centred at ( R, 0, 0)). It is clear that this function satisfies the hypothesis of Theorem 1. In [15] Sadallah has obtained the optimal regularity of the solution for any disc Ω = √ (t, x) ∈ R² : 0 ≤ √ t² + x² < R λ In the case of one space variable, the heat equation in non Hilbretian spaces has been considered in some works, for instance in Labbas et al. [10] and [11]. Berroug et al. [3] have treated the same equation with a singular domain in Hölder spaces, whereas Kheloufi et al. [7] have studied the case when the domain is cylindrical, not with respect to the time variable, but with respect to one of space variables. Some authors have considered the singularities which appear in the solu- tions when the domain is non cylindrical as Kozlov and Maz'ya in [8] and [9]. We can find in Nazarov [13] some results about the solution of Neumann problem in a conical domains. In Degtyarev [5] a more general parabolic equation has been studied in domains with a conical point. The method used here is to approach the domain Ω for T small enough by a sequence of subsomains (Ω n ) n ∈ N in which we can solve Problem (1). More precisely, when T is any positive real number, we divide Ω into two parts : Ωτי, with T ′ small enough,
and
So, we obtain two solutions u 1 ∈ H 1 ; 2 (Ω T ′ ) in Ω T ′ and u 2 ∈ H 1 ; 2 (Ω T ′ T ) in Ωτ′τ. Finally, we prove that the function u defined by
is the solution of Problem (1) and has the optimal regularity, that is u ∈ H 1,2(Ω). It is not difficult to prove the uniqueness of the solution. Indeed, if we consider the inner product (∂tu − ∂z u − ∂z u, u) L 2  with ∂tu − ∂z u − ∂z u = ∂z u = 0 and take into account the boundary conditions, we obtain
where |∇ u |² = (∂ x u )² +(∂ y u )² . Then |∇ u |² = 0. This leads to ∂ x u = ∂ y u = 0 and therefore, ∂žu = ∂žu = 0. From the equation ∂t u − ∂žu − ∂žu = 0 we deduce that ∂tu = 0. Accordingly, the solution u is constant, but it is null on one part of the boundary. So, u = 0. This proves the uniqueness of the solution of Problem (1). This is why we will be interested in the sequel only by the question of the existence of the solution when f ∈ L 2 (Ω). Our work is motivated by the interest of researchers for many mathe- matical questions related to non-regular domains. In fact, some important applied problems reduce to the study of boundary-value problems for par- tial differential equations (Laplace equation, heat equation, Fokker-Planck equation, Chapman-Kolmogorov equation…) in domains with non-regular points on the boundary. The plan of the paper is : In Section 2 we give a change of variables which transforms a subdomain of Ω into a cylindrical domain, so we confine ourselves to the neighborhood of the origin (0,0,0). Section 3 is devoted to the proof of an estimate in order to get a subsequence of functions converging to the solution of Problem (1) in a subdomain of Ω. Finally, in Section 4, we complete the proof of Theorom 1.
2. CHANGE OF VARIABLES
Let ε > 0 be a real which we will choose small enough. The continuity of φ at 0 proves the existence of a real number T ′ < T such that : (2.1) ∀t ∈ (0,T') , | ϕ(t)| ≤ ε . Consider a decreasing sequence (tn)n∈N of real numbers such that 0 < t n < T ′ for all n ∈ N and  lim t n = 0. We define the sequence of subdomains t→0+ (Ωn) n of Ωτι by
The solution u of Problem (1) will be approached by the solutions u n of the problems
where fn = f|Ωn• Then, we perform the change of variables :
here fn(t,x,y) = gn(t,x',y'), un(t,x,y) = vn(t,x',y') and Δvn = ∂ 2,vn + dy vn. It is easy to see that the operator
admits a unique solution ν n ∈ H 1,2 in the cylindrical domain ( t n ,T ′ ) × B(0, 1). This proves that Problem (4) has a unique solution v n ∈ H 1,2 ((t n ,T') × B (0,1)). The previous change of variables shows that Problem (3) has also a unique solution u n ∈ H 1,2 (Ω n ) for all n ∈ N . The goal of the following section is to establish an estimate concerning the sequence ( u n ) which allows us to extract a subsequence converging to the solution of Problem (1). 3. A N ESTIMATE Consider the sequence ( u n ) and look for a constant C > 0 independent of n satisfying the estimate
For this purpose we will estimate ‖ ∂t u n − ∂z u n − ∂z u ‖ z (Ωn) · Lemma 2. There exists a constant C > 0 independent of n such that for almost every t ∈ (0,T')
Proof. Let H 2 and H 1 be the usual Sobolev spaces defined, for instance, in Lions-Magenes [12]. We know that the Laplace operator △ : H 2 ( D ( 0 , 1 )) ∩ H}(D(0,1)) → L 2 (D(0,1)) is an isomprphism. Then, there exists a constant C > 0 such that ||v||H²(D(0,1)) ≤ C ||Δv||L²(D(0,1)) ,∀v ∈ H²(D(0,1)). Let t n < t < T ′ . The change of variables
allows to write, thanks to (6) :
with the notation v(x,y) = un(x',y'). Notice that un ∈ H 2 (D(t,ϕ(t)) for almost every t ∈ ( t n , T ′ ). Hence
On the other hand, we have
161 =   Cϕ(t) ||∆un|| L²(D(t,ϕ(t)) · The same result is true for ‖ ∂y ′ Un‖ L 2 (D(t,ϕ(t)))  · The following lemma (see Theorem 2.1, Lions-Magenes [12]) will justify some calculations in this section. Lemma 3. The space { u ∈ H² ((0,T) × D(0,1)) : u |{0}×D(0,1) ＝ 0,U |(0,T')×∂D(0,1) ＝ 0} is dense in the space { u ∈ H¹,²((0,T) × D(0,1)) : u|{0}×D(0,1) ＝ 0,U|(0,T')×∂D(0,1) ＝ 0} . Now, consider the solution un ∈ H 1,2 (Ω n ) of Problem (3) and let Δ un = ∂žun +∂žun. Expanding the inner product ‖ƒn‖Ż2(Ωn) = (∂ŧun − ∆un, ∂ŧun − △ un ) L 2 (Ω n ) we obtain (3.3) || fn|| ž²(Ωn) = || ∂tun|| ž²(Ωn) + || ∆un|| ž²(Ωn) = 2(∂tun, ∆un) L²(Ωn) We deduce the following result from Lemma 2 and Grisvard-Ioss [6] (see Theorem 2.2). Lemma 4. There exists a constant C > 0 independent of n such that (3.4) ||∂xyun|| L²(s n ) + ||∂žun|| L²(s n ) + ||∂žun|| L²(s n ) ≤ C ||Δun|| L²(s n ) + We have to estimate the term (∂ℓun, ∆ un) L 2 (Ωn) · For this end, we need to prove the result Proposition 5. One has
Proof. To establish this relationship we perform the following parametriza- tion of Γ 1 :
(t,θ)   →   (t,φ(t)cosθ,φ(t)sinθ) = (t,χ,y) and we denote by ν = √ 1+φ r2 (t) ( −φ ′ (t),cos θ,sin θ) = ( νt,ν x,ν y ) the out- ward unit normal vector to the part Γ 1 of ∂Ω n defined by
STUDY OF A PARABOLIC PROBLEM IN A CONICAL DOMAIN The other part of ∂Ω n will be denoted by Γ T ′ , i.e., Γ T ′ = D(T ′ , ϕ(T ′ )).We
163 BOUBAKER-KHALED SADALLAH
This ends the proof of the proposition. Remark 6.  Observe that the integral ſD(T',ϕ(T')) |∇ un|² (T', x, y)dxdy which appears in Proposition 5 is non negative.This is a good sign for our estimate because we can deduce immediately
hand, if ϕ is an increasing function in the interval (t n,T ′ ), then
Consequently,
(5) under the hypothesis  lim φ(t)φ'(t) = 0. For this purpose, we need the following proposition Proposition 7. One has
164 STUDY OF A PARABOLIC PROBLEM IN A CONICAL DOMAIN
Proof. For t n < t < T ′ , denote by D t the disc D(t, ϕ(t)) and consider the inner product (∆un, x∂xun + y∂un)L²(Dt). We have
cos θθyun. Consequently
165 and
Theorem 8. Assume that limφ(t)φ'(t) = 0 or φ'(t) ≥ 0 in a neighborhood of  0.Then, there exists a constant C > 0 independent of n satisfying the estimate
Proof. The case when ϕ′(t) ≥ 0 in a neighborhood of 0 has been treated in Remark 6. Then, assume that limφ(t)φ ′ (t) = 0. We have
but Lemma 2 yields
166
From the previous theorem we deduce that ‖ u ‖ H 1,2(Ω n ) ≤ C ‖ f ‖ L 2(Ω) for all n ∈ N because ∥ f n ∥ L 2 (Ω n ) ≤ ∥ f ∥ L 2 (Ω) ⋅ For any function v defined in Ω n , denote by v  the function
It is then possible to extract a subsequence ( ũnj ) j from ( ũn ) ∈ L 2 (Ω T ' ) which converges weakly in H 1 , 2 ( Ω T ′ ). Denote by u 1 the limit of ( ũn j ) , Therefore, passing to the limit for nj → +∞, we get
This proves Theorem 1 in the subdomain Ω T ′ . Let T  be any positive real number, f ∈ L 2 (Ω), and T ′ < T small enough in order to Problem 11 admits a solution u 1 ∈ H 1 , 2 (Ω τ ′ ) We have to solve Problem (1) in Ω. We know (see Section 2) that the Cauchy-Dirichlet problem
has a unique solution ν ∈ H 1 ,2 (Ω τיτ ). We need the following trace result (see Lions-Magenes [12]) Lemma 9. For v ∈ H 1 ;2 (Ω T ′ T ), one has V | D ( T ′ ,ϕ(T ′ )) ∈ H 1 (D(T ′ ,ϕ(T ′ ))). Moreover, for each ψ ∈ H l (D(T', ϕ(T'))), there exists w ∈ H 1,2 (Ω T r r ) such that w|D(T',ϕ(T')) = ψ  and w|∂Ω,,,,,(T'))  = 0 . Since u 1 ∈ H 1 ; 2 (Ω τ ′ ), from this lemma and the homogeneous Problem (12), we obtain Proposition 10. The problem
admits a unique solution u² ∈ H 1 ,2 (Ω T r T ). Now, define the function u in Ω by
where u1 and u2 are the solutions of Problem (11) and Problem (13) respec- tively.  Observe that inasmuch as the boundary condition u2|D(T',ϕ(T')) ＝ U1|D(T',φ(T')) is satisfied, it follows that ∂žu ∈ L 2 (Ω) and ∂žu ∈ L 2 (Ω), so, we have also ∂ xy ∈ L 2 (Ω) . Moreover, in view of the equations given in Problem (11) and Problem (13) we deduce that ∂t u ∈ L 2 (Ω) and ∂t u − ∂z u − ∂z u = f . Finally, the function u ∈ H 1 , 2 (Ω) is a solution of Problem (1).  This completes the proof of Theorem 1. Remark 11. This study can be extended to the case when the function φ depends also on an angle θ ∈ (0,2π). For instance, the domain Ω may be defined by
where ϕ is a Lipschitz function ϕ : [0,T] × [0,2π] → R+  satisfying the condition ϕ(0,θ) = 0 for 0 ≤ θ ≤ 2π and ϕ(t,θ) = ϕ(t,2π) Then, a new condition on φ depending on θ will be appear.  In particular, if φ(t,θ) = 0 for 0 < t < T  and 0 ≤ θ ≤ θ 0  with 0 < θ 0 < 2π  the  solution  may  contain  some singularities, and no conditions on ϕ could avoid these singularities.This problem will be treated later.