A refined Green's function estimate of the time measurable parabolic operators with conic domains
aa r X i v : . [ m a t h . A P ] A ug A REFINED GREEN’S FUNCTION ESTIMATE OFTHE TIME MEASURABLE PARABOLIC OPERATORSWITH CONIC DOMAINS
KYEONG-HUN KIM, KIJUNG LEE, AND JINSOL SEO
Abstract.
We present a refined Green’s function estimate of thetime measurable parabolic operators on conic domains that in-volves mixed weights consisting of appropriate powers of the dis-tance to the vertex and of the distance to the boundary. Introduction
In recent years we have been interested in the stochastic heat diffu-sion occurring in wedge shaped subdomains of R , which are probablysimplest non-smooth Lipschitz domains. In the literature there exist al-most fully developed regularity results for the stochastic heat diffusionon C domains, but when it comes to Lipschitz domains the results arequite unsatisfactory and very little is known. To fill in the gap betweenthe theory for C domains and the theory for Lipschitz domains, thewedge domains are what we decided to pay attention first.Along the way, we set the theme that the angle around the vertexaffects regularity of the temperature when the boundary temperatureis controlled. We believe that our previous work [4] captured suchrelation in a certain way. Based on this work, in [3] we proceeded toconstruct a theory on the stochastic diffusion in polygonal domains.The main tool of our results was an estimate on Green’s function forthe heat operator with the wedge domains obtained in [5]. Lookingback, what we feel sorry about is that the estimate only involves theweight of powers of the distance to the vertex. “only” means that Mathematics Subject Classification.
Key words and phrases.
Green’s function estimate, conic domains, wedge do-mains, stochastic parabolic equation.The first and third authors were supported by the National Research Founda-tion of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1A5A1028324).The second author was supported by the National Research Foundationof Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2019R1F1A1058988). it could be better or much better if the estimate also involves weightof the distance to the boundary. Having weight depending only onthe distance to the vertex in the estimate did not yield satisfactoryboundary regularity of the solution and caused quite a bit of troublewhen we constructed a global regularity theory for polygonal domains.Aiming more natural and hopefully complete theory for polygonaldomains, we imagined a refined Green’s function estimate that involvesboth the distance to the vertex and the distance to the boundary. Thispaper is about this improvement task.The main contents of this paper are as follows. In Section 2, weintroduce a Green’s function estimate of the time measurable parabolicoperator L = ∂∂t − P di,j =1 a ij ( t ) D ij defined on a conic domain D ⊂ R d with a vertex at the origin. We prove an estimate of the type G ( t, s, x, y ) ≤ N e − σ | x − y | t − s ( t − s ) d/ (cid:18) | x |√ t − s ∧ (cid:19) β (cid:18) | y |√ t − s ∧ (cid:19) β × (cid:18) ρ ( x ) √ t − s ∧ (cid:19) (cid:18) ρ ( y ) √ t − s ∧ (cid:19) , β , β ≥ , (1.1)where ρ ( x ) := dist( x, ∂ D ). The ranges of β and β are determined by D and L and described in Remark 2.2. Note that estimate (1.1) involvesboth the distance to the vertex and the distance to the boundary, andgives a subtle decay rate as x, y approach the boundary or the origin.In Sections 3 and 4, we obtain some critical upper bounds of β , β forthe operator L .In this paper we use the following notations:- We use := to denote a definition.- α ∧ β = min { α, β } , α ∨ β = max { α, β } - N ( · · · ) means a constant depending only on what are indicated.- D ij u = ∂ u∂x j ∂x i and - B R ( x ) = { y ∈ R d | | y − x | < R } - B D R ( x ) = B R ( x ) ∩ D - Q R ( t, x ) = ( t − R , t ] × B R ( x )- Q D R ( t, x ) = ( t − R , t ] × ( B R ( x ) ∩ D ).Also, we will frequently use the following sets of functions (see [6]).- V ( Q R ( t , x )) : the set of functions u defined at least on Q R ( t , x )and satisfyingsup t ∈ ( t − R ,t ] k u ( t, · ) k L ( B R ( x )) + k∇ u k L ( Q R ( t ,x )) < ∞ . REEN’S FUNTION WITH WEDGE BOUNDARY 3 - V loc ( Q R ( t , x )) : the set of functions u defined at least on Q R ( t , x ) and satisfying u ∈ V ( Q r ( t , x )) , ∀ r ∈ (0 , R ) . - V ( Q D R ( t , x )) : the set of functions u defined at least on Q D R ( t , x )and satisfyingsup t ∈ ( t − R ,t ] k u ( t, · ) k L ( B D R ( x )) + k∇ u k L ( Q D R ( t ,x )) < ∞ . - V loc ( Q D R ( t , x )) : the set of functions u defined at least on Q D R ( t , x ) and satisfying u ∈ V ( Q D r ( t , x )) , ∀ r ∈ (0 , R ) . Main result
We define our conic domain in R d by D = n x ∈ R d \ { } (cid:12)(cid:12)(cid:12) x | x | ∈ M o , where M is a connected open subset in the sphere S d − = { ξ ∈ R d || ξ | = 1 } which has C boundary. Here, C boundary means that for anyfixed point p ∈ S d − \ D and the stereographic projection of S d − \ { p } onto the tangent hyperplane at − p , the antipode of p , the image of D has C boundary in the hyperplane. κ − κ d = 2 d = 3 Figure 1.
Cases of d = 2 and d = 3 KYEONG-HUN KIM, KIJUNG LEE, AND JINSOL SEO
For example, when d = 2, for each fixed angle κ ∈ (0 , π ) we canconsider D = D κ = n ( r cos θ, r sin θ ) ∈ R | r ∈ (0 , ∞ ) , − κ < θ < κ o . (2.1)In this paper we consider the Green’s function of the operator L = ∂∂t − X i,j a ij ( t ) D ij (2.2)with the domain D . We assume that the diffusion coefficients a ij , i, j = 1 , . . . , d , are real valued measurable functions of t , a ij = a ji , i, j =1 , . . . , d , and satisfy the uniform parabolicity condition, i.e. there existsa constant ν ∈ (0 ,
1] such that for any t ∈ R and ξ = ( ξ , . . . , ξ d ) ∈ R d , ν | ξ | ≤ X i,j a ij ( t ) ξ i ξ j ≤ ν − | ξ | . (2.3)We denote the Green’s function by G ( t, s, x, y ). By the definition ofGreen’s function G is nonnegative and, for any fixed s ∈ R and y ∈ D ,the function v = G ( · , s, · , y ) satisfies L v = 0 in ( s, ∞ ) ×D ; v = 0 on ( s, ∞ ) × ∂ D ; v ( t, · ) = 0 for t < s. Also, in this paper we use the notations ρ ( x ) = | x | , ρ ( x ) = dist( x, ∂ D )and R t,x := | x |√ t ∧ ρ ( x ) √ t ∧ , J t,x := ρ ( x ) √ t ∧ . Remark . Since aa +1 ≤ a ∧ ≤ · aa +1 for any a ≥
0, we can alsodefine R t,x and J t,x by R t,x := ρ ( x ) ρ ( x ) + √ t , J t,x := ρ ( x ) ρ ( x ) + √ t . From the probabilitstic point of view related to a Brownian motionkilled at the boundary of ∂ D , G is essentially a transition probabilityand bounded by a constant multiple of Gaussian density function:0 ≤ G ( t, s, x, y ) ≤ N t − s ) d/ e − σ | x − y | t − s , t > s, x, y ∈ D , (2.4)where the constants N , σ > d and ν in the assumption (2.3).Having further information of the domain, the right hand side of(2.4) can be refined. Especially, for our conic domains D , one canpursue the following type of estimate G ( t, s, x, y ) ≤ N t − s ) d/ R λ + t − s,x R λ − t − s,y e − σ | x − y | t − s , t > s, x, y ∈ D REEN’S FUNTION WITH WEDGE BOUNDARY 5 for some positive constants λ + , λ − . Since R t,x is less than equal to 1,this estimate is sharper as we find bigger λ + , λ − satisfying the estimate. Remark . As in [6, Section 2], the critical upper bound λ + c > λ + can be characterized by the supremum of all λ such that for someconstant K = K ( L , M , λ ) it holds that | u ( t, x ) | ≤ K (cid:18) | x | R (cid:19) λ sup Q D R ( t , | u | , ∀ ( t, x ) ∈ Q D R/ ( t ,
0) (2.5)for any t > R >
0, and u belonging to V loc ( Q D R ( t , L u = 0 in Q D R ( t ,
0) ; u ( t, x ) = 0 for x ∈ ∂ D . The value of λ + c does not change if one replaces in (2.5) by anynumber in (1 / ,
1) (see [6, Lemma 2.2]).Moreover, the critical upper bound λ − c > λ − is characterized bythe supremum of λ with above property for the operatorˆ L = ∂∂t − X i,j a ij ( − t ) D ij . (2.6)Both λ + c and λ − c will definitely depend on M = D ∩ S d − . Especiallywhen D = D κ in (2.1), λ + c and λ − c will depend on the opening angle κ .If in addition L is the heat opeartor, L = ∂∂t − ∆ x , then λ + c = λ − c = πκ . See Section 2 of [6] and Section 3 of this paper for details.The following lemma is, we think, the most updated estimate of G among the ones involving R t,x only. Lemma 2.3.
Let λ + ∈ (0 , λ + c ) , λ − ∈ (0 , λ − c ) , and denote K +0 := K ( L , M , λ + ) and K − := K ( ˆ L , M , λ − ) . Then there exist positiveconstants N = N ( M , ν, λ ± , K ± ) and σ = σ ( ν ) such that G ( t, s, x, y ) ≤ N ( t − s ) d/ R λ + t − s,x R λ − t − s,y e − σ | x − y | t − s (2.7) and |∇ x G ( t, s, x, y ) | ≤ N ( t − s ) ( d +1) / R λ + − t − s,x R λ − t − s,y e − σ | x − y | t − s for any t > s , x, y ∈ D . Proof.
See [6, Theorem 3.10]. We only remark that in [6] the depen-dency of N on K ± is taken for granted and omitted. By inspectingthe proof of [6, Theorem 3.10] one can check that constant N actuallydepends also on K ± . (cid:3) KYEONG-HUN KIM, KIJUNG LEE, AND JINSOL SEO
Remark . In fact, [6] has the estimates of the derivatives of G upto the second order that contain Lemma 2.3 as a part. We refer toTheorem 3.10 of [6]. Yet, the estimates involve R t,x only. Remark . Despite the beauty in estimate (2.7), we note that theright hand side of (2.7) does not go to zero as x or y approaches bound-ary of D , meaning that the estimate is not sharp enough in terms ofthe boundary behavior of the Green’s function.On the other hand, for any domain satisfying, for instance, the uni-form exterior ball condition, the corresponding Green’s function of L is bounded by the constant multiple of1( t − s ) d/ J t − s,x J t − s,y e − σ | x − y | t − s , which is now forcing the degeneracy of the Green’s function at theboundary (see e.g. [2]).Of course, our domains, for instance, like D κ in (2.1) does not satisfythe uniform exterior ball condition if κ > π . However, for any κ , D κ ismostly flat except a samll neighborhood of the vertex and we hoped arefined estimate that involves both R t,x and J t,x together. After all, wesettled down with the following theorem, which is the refined estimatewe mentioned in the introduction and is the main result of this paper. Theorem 2.6.
Let λ + ∈ (0 , λ + c ) , λ − ∈ (0 , λ − c ) , and denote K +0 := K ( L , M , λ + ) and K − := K ( ˆ L , M , λ − ) . Then there exist positiveconstants N = N ( M , ν, λ ± , K ± ) and σ = σ ( ν ) such that G ( t, s, x, y ) ≤ N ( t − s ) d/ R λ + − t − s,x R λ − − t − s,y J t − s,x J t − s,y e − σ | x − y | t − s (2.8) for any t > s , x, y ∈ D .Remark . Obviously estimate (2.8) is sharper than estimate (2.7)since J t,x ≤ R t,x . Moreover, estimate (2.8) gives delicate boundarybehavior of Green’s funciton. Remark . The strategy of our proof of Theorem 2.8 is inspired by[2] and [7] although the details are quite different.In the proof of Theorem 2.6, we will use the following two lemmasfrom [6].
Lemma 2.9 (Proposition 3.2 of [6]) . Let u belong to V ( Q R ( t , x )) andsatisfy L u = 0 in Q R ( t , x ) , then |∇ u ( t, x ) | ≤ NR sup Q R ( t ,x ) | u | , ∀ ( t, x ) ∈ Q R/ ( t , x ) , REEN’S FUNTION WITH WEDGE BOUNDARY 7 where the constant N depends only on ν and d . Lemma 2.10 (Proposition 3.4 of [6]) . There exists a sufficently samll δ such that the following holds for any δ ∈ (0 , δ ) : Let x ∈ D , ρ ( x ) < δ | x | , and R ≤ | x | . Then if u belongs to V ( Q D R ( t , x )) andsatisfies L u = 0 in Q D R ( t , x ) and u ( t, x ) = 0 for x ∈ ∂ D , then |∇ u ( t, x ) | ≤ NR sup Q D R ( t ,x ) | u | , ∀ ( t, x ) ∈ Q D R/ ( t , x ) , where the constant N depends only on M , ν, δ . Proof of Theorem 2.6.1 . First, we fix s ∈ R , y ∈ D . We show that there exist positiveconstants N = N ( M , ν, λ ± , K ± ) and σ = σ ( ν ) such that for any t ∈ ( s, ∞ ) and x ∈ D , G ( t, s, x, y ) ≤ N ( t − s ) d/ J t − s,x R λ + − t − s,x R λ − t − s,y e − σ | x − y | t − s . (2.9)For given t ∈ ( s, ∞ ), we consider the following two cases of x ∈ D . √ t − s √ t − s Figure 2.
Two cases of x - Case ρ ( x ) ≥ (cid:0) | x | ∧ √ t − s (cid:1) .In this case, by assumption we have2 ρ ( x ) √ t − s ≥ (cid:18) | x |√ t − s ∧ (cid:19) . Therefore, R t − s,x = | x |√ t − s ∧ ≤ ρ ( x ) √ t − s ∧ (cid:18) ρ ( x ) √ t − s ∧ (cid:19) . (2.10)Then, using Lemma 2.3, we immediately get (2.9). KYEONG-HUN KIM, KIJUNG LEE, AND JINSOL SEO - Case ρ ( x ) < (cid:0) | x | ∧ √ t − s (cid:1) ; the point close to the boundary.For such point x ∈ D , there exists x ∈ ∂ D such that | x − x | = ρ ( x ).For this x ∈ ∂ D , G ( t, s, x , y ) = 0 and there exists θ ∈ (0 ,
1) such that G ( t, s, x, y ) = G ( t, s, x, y ) − G ( t, s, x , y ) ≤ | x − x ||∇ x G ( t, s, ¯ x, y ) | = ρ ( x ) |∇ x G ( t, s, ¯ x, y ) | , (2.11)where ¯ x = (1 − θ ) x + θx ∈ D .To estimate the gradient part, we make use of Lemma 2.3. Now,since | ¯ x | ≥ | x |− θ | x − x | ≥ | x |− ρ ( x ) > | x | , | ¯ x | ≤ | x | + θ | x − x | ≤ | x | + ρ ( x ) < | x | , we note that 12 R t − s,x ≤ R t − s, ¯ x ≤ R t − s,x . In addition, the inequalities | x − y | ≤ | ¯ x − y | + | ¯ x − x | ≤ | ¯ x − y | + | x − x | ≤ | ¯ x − y | + √ t − s give −| ¯ x − y | ≤ − | x − y | + t − s. Hence, |∇ x G ( t, s, ¯ x, y ) | is bounded by N ′ t − s ) ( d +1) / R λ + − t − s,x R λ − t − s,y e − σ ′ | x − y | t − s , where N ′ = N ′ ( M , ν, λ ± , K ± ) > σ ′ = σ ′ ( ν ) >
0. This, (2.11),and ρ ( x ) ≤ √ t − s lead us to (2.9) again. . Now, we consider the operator ˆ L defined in (2.6). Let ˆ G denotethe Green’s function for ˆ L with the same domain D . Note that thediffusion coefficients a ij ( − t ), i, j = 1 , . . . , d , also satisfy the uniformparabolicity condition (2.3) with the same ν . Since for any s ∈ R and y ∈ D , ˆ L ˆ G ( · , s, · , y ) = 0 on ( s, ∞ ) × D and ˆ G ( · , s, · , y ) = 0 on( s, ∞ ) × ∂ D , we can repeat the argument in Step 1 literally line byline. Hence, denoting the critical upper bounds of λ for the operatorˆ L by ˆ λ + c , ˆ λ − c and noting that ˆ λ + c = λ − c , ˆ λ − c = λ + c by Remark 2.2, withthe same constants N, σ in (2.9), we obtain thatˆ G ( t, s, x, y ) ≤ N ( t − s ) d/ J t − s,x R λ − − t − s,x R λ + t − s,y e − σ | x − y | t − s (2.12)for any t > s and x, y ∈ D . Note that the locations of λ + , λ − in (2.12) incomparison with the locations of them in (2.9). This is simply because λ − ∈ (0 , ˆ λ + c ) and λ + ∈ (0 , ˆ λ − c ). REEN’S FUNTION WITH WEDGE BOUNDARY 9 . Next, using the result of Step 2 and the following identity G ( − s, − t, y, x ) = ˆ G ( t, s, x, y ) or G ( t, s, x, y ) = ˆ G ( − s, − t, y, x ) , t > s which is due to a duality argument (see (3.12) of [6] for the detail), weobserve that with the same constants N, σ in (2.9) we have G ( t, s, x, y ) ≤ N ( t − s ) d/ J t − s,y R λ − − t − s,y R λ + t − s,x e − σ | x − y | t − s = N ( t − s ) d/ R λ + t − s,x J t − s,y R λ − − t − s,y e − σ | x − y | t − s (2.13)for any t > s and x, y ∈ D . . Finally to finish the proof of (2.8) we repeat the argument in Step1. For the points x away from the boundary the argument is the same.Indeed, if ρ ( x ) ≥ (cid:0) | x | ∧ √ t − s (cid:1) , then (2.10) and (2.13) certainly give(2.8).Therefore, for the rest of the proof, we may assume ρ ( x ) < (cid:0) | x | ∧ √ t − s (cid:1) . In this case we first show |∇ x G ( t, s, x, y ) | ≤ N t − s ) ( d +1) / J t − s,y R λ + − t − s,x R λ − − t − s,y e − σ | x − y | t − s . (2.14)For this, we fix ( s, y ) and set u ( t, x ) = G ( t, s, x, y ) . Take δ ∈ (0 , δ ∧ / δ is from Lemma 2.10 which dependsonly on M . We consider the following two cases.- Case ρ ( x ) ≥ δ | x | . Put R = δ ( | x | ∧ √ t − s ) which is less than ρ ( x ) so that ¯ B R ( x ) ⊂ D . Since u belongs to V ( Q R ( t, x )) and satisfies L u = 0 in Q R ( t, x ), by Lemma 2.9, we get |∇ x u ( t, x ) | ≤ NR sup Q R ( t,x ) | u | . We note that for ( r, z ) ∈ Q R ( t, x ),0 ≤ t − r ≤ t − s ,
34 ( t − s ) ≤ r − s ≤ t − s, | z | ≤ | x | + R ≤ | x | , | z | ≥ | x | − R ≥ | x | and | z − y | ≥ | x − y | − R ≥ | x − y | − √ t − s, − | z − y | ≤ − | x − y | + ( t − s ) , − | z − y | r − s ≤ − | x − y | t − s + 43 . Hence, using (2.13) we get | u ( r, z ) | ≤ N ( r − s ) d/ R λ + r − s,z J r − s,y R λ − − r − s,y e − σ | z − y | r − s ≤ N ( t − s ) d/ R λ + t − s,x J t − s,y R λ − − t − s,y e − σ ′ | x − y | t − s . Consequently, we have |∇ x u ( t, x ) | ≤ NR sup Q R ( t,x ) | u |≤ N | x | ∧ √ t − s t − s ) d/ R t − s,x R λ + − t − s,x J t − s,y R λ − − t − s,y e − σ ′ | x − y | t − s = N ( t − s ) ( d +1) / J t − s,y R λ + − t − s,x R λ − − t − s,y e − σ ′ | x − y | t − s , and thus (2.14) is proved.- Case ρ ( x ) ≤ δ | x | . In this case, we put R = ( | x | ∧ √ t − s ). Since u belongs to V ( Q D R ( t, x )) and satisfies L u = 0 in Q D R ( t, x ), and u ( t, x ) = 0for x ∈ ∂ D , we can apply Lemma 2.10, and have |∇ x u ( t, x ) | ≤ NR sup Q D R ( t,x ) | u | . Similarly as before, we again obtain (2.14).Finally, by (2.11), the computations below (2.11), and (2.14), weobtain (2.8). This ends the proof. (cid:3) On the critical upper bounds λ ± c In this section we discuss some detailed informations of the criticalupper bounds λ + c and λ − c , whose characterizations are given in Remark2.2.We first introduce some known results on λ ± c . The following state-ments are the 3rd, the 8th, and the 7th in Theorem 2.4 of [6]: REEN’S FUNTION WITH WEDGE BOUNDARY 11 • If L = L := ∂∂t − ∆ x , then λ ± c ( L , D ) = − d −
22 + r Λ + ( d − , (3.1)where Λ is the first eigenvalue of Laplace-Beltrami operatorwith the Dirichlet condition on domain M = D ∩ S d − , where S d − is the sphere with radius 1 in R d . • Suppose that ( a ij ) d × d is a constant matrix. Then λ ± c ( L , D ) = λ ± c ( L , e D ) = − d −
22 + re Λ + ( d − , (3.2)where e Λ is the first eigenvalue of the Dirichlet boundary valueproblem to Beltrami-Laplacian in the domain f M = e D ∩ S d − while cone e D is the image of D under the change of variables x → y that reduces ( a ij ) d × d to the canonical form ( δ ij ) d × d withthe Kronecker delta δ ij , i, j = 1 , . . . , d . • For the general operator L = ∂∂t − P di,j =1 a ij ( t ) D ij in (2.2), wehave λ ± c ≥ − d ν r Λ + ( d − , (3.3)where ν is the uniform parabolicity constant in (2.3). Remark . One big difference between (3.2) and (3.3) is that “ d ”appears in (3.3) in place of “ d − d = 2, D = D κ in(2.1), and L = L = ∂∂t − ( D x x + D x x ). Then we can easily find Λ in(3.1), which is the same as ˜Λ in (3.2). To find Λ, we just need to findthe smallest eigenvalue λ > φ = φ ( θ ) satisfying − φ ′′ = λφ, − κ < θ < κ , ; φ (cid:16) κ (cid:17) = φ (cid:16) − κ (cid:17) = 0 , which yields φ ( θ ) = cos( √ λθ ) and cos (cid:16) √ λ κ/ (cid:17) = 0. Hence, theeigenvalues satisfy √ λ κ/ π/ kπ , k = 0 , , , . . . , and thus Λ = π /κ .In this example, if for instance κ = π , then (3.3) yields, as we cantake ν = 1, a trivial information λ ± c ≥
0, whereas (3.2) gives λ ± c = 1.In this section we improve (3.3). In particular, we will replace d in(3.3) by d −
2. We assume that the coefficients a ij ( t ), i, j = 1 , · · · , d, satisfy a ij ( t ) = a ji ( t ), and there exist constants ν , ν > any t ∈ R and ξ ∈ R d , ν | ξ | ≤ X i,j a ij ( t ) ξ i ξ j ≤ ν | ξ | . (3.4)The condition (2.3) is a special case of this condition: ν = ν, ν = ν − . Theorem 3.2.
Let ν , ν be the uniform parabolicity constants in (3.4) . If λ < − d −
22 + r ν ν r Λ + ( d − , then there exists a positive constant K = K ( ν , ν , M , λ ) such that | u ( t, x ) | ≤ K (cid:18) | x | R (cid:19) λ sup Q D R ( t , | u | , ∀ ( t, x ) ∈ Q D R/ ( t , for any t > , R > , and u belonging to V loc ( Q D R ( t , and satisfying L u = 0 in Q D R ( t ,
0) ; u ( t, x ) = 0 for x ∈ ∂ D . In particular, we have λ ± c ≥ − d −
22 + r ν ν r Λ + ( d − . (3.5)Note that if ν ≤ ν ≤ ν ≤ ν − , the right hand side of (3.5) is quitebigger that that of (3.5). Indeed, − d −
22 + r ν ν r Λ + ( d − ! − − d ν r Λ + ( d − ! = 1 + (cid:16)r ν ν − ν (cid:17)r Λ + ( d − ≥ . To prove the above theorem, we start with the following lemma whichis a slight modificaiton of Lemma A.1 of [6].
Lemma 3.3.
Let µ < ν ν (cid:16) Λ + ( d − (cid:17) and < ǫ < ǫ ≤ . Thenthere exists a constant N depending only on µ, ǫ , ǫ , ν , ν , M such that Z Q D ǫ R ( t , | x | µ |∇ u | dxdt + Z Q D ǫ R ( t , | x | µ − | u | dxdt ≤ N R µ − Z Q D ǫ R ( t , | u | dxdt for any R > and any function u belonging to V loc ( Q D R ( t , andsatisfying L u = 0 in Q D R ( t , , u = 0 on R × ∂ D . REEN’S FUNTION WITH WEDGE BOUNDARY 13
Proof.
The proof of this lemma is almost the same as that of LemmaA.1 of [6]. The only difference is that we use conditon (3.4) instead ofcondition (2.3). (cid:3)
Proof of Theorem 3.2.1 . Refering to Remark 2.2, we note that it is enough to show thatfor any µ ∈ R satisfying µ < ν ν (cid:16) Λ + ( d − (cid:17) , there exists a constant N depending only on M , µ, ν , ν such that | u ( t, x ) | ≤ N (cid:18) | x | R (cid:19) − d − + µ sup Q D R ( t , | u | , ∀ ( t, x ) ∈ Q D R/ ( t ,
0) (3.6)for any t > R >
0, and u belonging to V loc ( Q D R ( t , L u = 0 in Q D R ( t ,
0) ; u ( t, x ) = 0 for x ∈ ∂ D . Also, we note that we may assume t = 0, R = 1. . Take any function u satisfying the conditions in Step 1 with t = 0 , R = 1 and take any ( t, x ) ∈ Q D / (0 , r = | x | (cid:16) < (cid:17) , D r = ( t − r / , t ] × ( B r (0) \ B r (0)) . Then as in the proof of statement 7 of Theorem 2.4 in [6], we have | u ( t, x ) | ≤ N r − d − Z D r | u ( τ, y ) | dydτ ≤ N r − d +2 µ Z D r | y | − µ − | u ( τ, y ) | dydτ. (3.7)The last inequality in (3.7) holds since for the points y in D r , | y | arecomparable with r .Now, we define a time-changed function of u : v ( s, y ) := u ( t + r s, y ) . This function is well defined at least on Q D (0 ,
0) due to t + r s ∈ ( − , s ∈ ( − , v belongs to V loc ( Q D (0 , L v = 0 in Q D (0 ,
0) ; v = 0 on R × ∂ D , where ˜ L = ∂∂s − P i,j r a ij ( s ) D ij . We note that r ν | ξ | ≤ X i,j r a ij ( s ) ξ i ξ j ≤ r ν | ξ | is the uniform parabolicity condition for ˜ L and the ratio r ν r ν is thesame as ν ν and hence we can apply Lemma 3.3 for ˜ L and v . Havingthis in mind, we continue with (3.7) as below. Since ( t + r s, y ) ∈ D r ⇒ ( s, y ) ∈ ( − / , × B r (0)and ( − / , × B r (0) ⊂ Q D (0 , N r − d +2+2 µ Z Q D (0 , | y | − µ − | v ( s, y ) | dyds. (3.8)Then we apply Lemma 3.3 with ǫ = , ǫ = and see Z Q D (0 , | y | − µ − | v ( s, y ) | dyds ≤ N Z Q D (0 , | v ( s, y ) | dyds ≤ N sup Q D (0 , | v | ≤ N sup Q D (0 , | u | , (3.9)where the last quantity in (3.9) follows the observation t + r s ∈ ( − (cid:0) (cid:1) ,
0] for any s ∈ ( − (cid:0) (cid:1) , N in this Step 2 depend only on M , µ , and d .Hence, (3.7), (3.8) and (3.9) yield (3.6), and the claim in Step 1 isproved. (cid:3) Remark . For instance, let d = 3 and for any fixed κ ∈ (0 , π ) take D = D κ = n ( r sin θ cos φ, r sin θ sin φ, r cos θ ) ∈ R | r ∈ (0 , ∞ ) , ≤ θ < κ , < φ ≤ π o . Then the first eigenvalue Λ of Laplace-Beltrami operator with theDirichlet condition on domain D κ ∩ S satisfies12 | log(cos( κ/ | ≤ Λ ≤ j κ (3.10)where j ≈ . J (see [1]).Hence, using (3.10) and Theorem 3.2 we can obtain rough lower boundsof λ ± c . 4. Evaluation of λ ± c when d = 2Finding the exact values of λ ± c are very difficult in general. In Section3 we presented a decent estimation of them from below. In this sectionwe attempt to evaluate λ ± c when d = 2 and the diffusion coefficients a ij , i, j = 1 ,
2, in our operator L are constants. REEN’S FUNTION WITH WEDGE BOUNDARY 15 As a = a , we can set A := ( a ij ) × := (cid:18) a bb c (cid:19) . By (2.3) matrix A is positive-definite and the eigenvalues are greaterthan equal to ν and in particular there is a symmetric matrix B suchthat A = B .For any fixed κ ∈ (0 , π ) and α ∈ [0 , π ) we denote D κ,α := n x = ( r cos θ, r sin θ ) ∈ R | r ∈ (0 , ∞ ) , − κ α < θ < κ α o , calling κ the central angle of the domain D κ,α .We consider the operator L = ∂∂t − (cid:0) aD x x + b ( D x x + D x x ) + cD x x (cid:1) with the conic (angular) domain D κ,α .Below arctan is a map from R → ( − π/ , π/ Proposition 4.1.
For L and D κ,α defined above, we have λ ± c ( L , D κ,α ) = π e κ , where e κ = π − arctan (cid:16) ¯ c cot( κ/
2) + ¯ b p det( A ) (cid:17) − arctan (cid:16) ¯ c cot( κ/ − ¯ b p det( A ) (cid:17) (4.1) with constants ¯ a, ¯ b from the relation (cid:18) ¯ a ¯ b ¯ b ¯ c (cid:19) = (cid:18) cos α sin α − sin α cos α (cid:19) (cid:18) a bb c (cid:19) (cid:18) cos α − sin α sin α cos α (cid:19) . (4.2) In particular,(i) if κ = π , then ˜ κ = π ;(ii) if α = 0 and b = 0 , then ˜ κ is determined by the relation tan (cid:16) e κ (cid:17) = r ac tan (cid:16) κ (cid:17) (4.3) for κ ∈ (0 , π ) \ { π } . Proof. 1 . We first consider the operator L := ∂∂t − ∆ x with domain D κ,α . In this case we note ˜ κ = κ and, as in Remark 3.1,we again have λ + c = λ − c = √ Λ = πκ .
Indeed, the eigenvalue/eigenfunction problem − φ ′′ ( θ ) = λφ ( θ ) , θ ∈ (cid:16) − κ α, − κ α (cid:17) ; φ (cid:16) − κ α (cid:17) = φ (cid:16) κ α (cid:17) = 0leads us to have φ ( θ ) = cos (cid:0) √ λ ( θ − α ) (cid:1) and cos (cid:0) √ λ κ/ (cid:1) = 0. Hence,the first eigenvlaue Λ again satisfies √ Λ κ/ π/
2. Thus we have λ ± c ( L , D κ,α ) = √ Λ = π ˜ κ . . General case. Having (3.2) and the accompanied explanationin mind, we take a symmetric matrix B satisfying A = B . Thechange of variables x = By transforms the operator aD x x + bD x x + bD x x + cD x x into ∆ y = D y y + D y y in y -coordinates, that is,putting v ( t, y ) = u ( t, By ), we obtain (cid:0) aD u + bD u + bD u + cD u )( t, By ) = ∆ y v ( t, y ) , ( t, y ) ∈ R × e D , where e D is the image of D κ,α under a linear transformation defined by e D := B − D κ,α := (cid:8) B − x : x ∈ D κ,α (cid:9) . We note that e D is also a conic (angular) domain with a certaincentral angle e κ . In fact, we can use (3.2) and Step 1 to have λ ± c ( L , D κ,α ) = λ ± c ( L , e D ) = π e κ . Let us verify the formula for e κ . We first note e κ π = | e D ∩ B (0) | ℓ | B (0) | ℓ and hence e κ = 2 · | e D ∩ B (0) | ℓ , where | E | ℓ denotes the Lebesgue measure of E ⊂ R . By the relation y = B − x , we then have REEN’S FUNTION WITH WEDGE BOUNDARY 17 | e D ∩ B (0) | ℓ = Z { y ∈ e D : | y |≤ } dy = 1 | det( B ) | Z { x ∈D : | B − x |≤ } dx = 1 p det( A ) Z κ/ α − κ/ α Z | B − v θ | − r dr dθ = 12 p det( A ) Z κ/ α − κ/ α | B − v θ | dθ = 12 p det( A ) Z κ/ α − κ/ α v Tθ A − v θ dθ, where v θ := (cid:18) cos θ sin θ (cid:19) . Now, a direct calculation based on translation,symmetry, and change of variable gives | e D ∩ B (0) | ℓ = 12 p det( A ) Z κ/ v Tθ A − v θ dθ + Z − κ/ v Tθ A − v θ dθ ! = p det( A )2 Z κ/ (cid:18) c cot θ − b cot θ + ¯ a + 1¯ c cot θ + 2¯ b cot θ + ¯ a (cid:19) · θ dθ = p det( A )2 Z ∞ cot( κ/ c t − b t + ¯ a + 1¯ c t + 2¯ b t + ¯ a dt = 12 π − arctan (cid:16) ¯ c cot( κ/ − ¯ b p det( A ) (cid:17) − arctan (cid:16) ¯ c cot( κ/
2) + ¯ b p det( A ) (cid:17)! , where A = (cid:18) ¯ a ¯ b ¯ b ¯ c (cid:19) with ¯ a , ¯ b , and ¯ c defined in (4.2). Hence, we obtain (4.1) for ˜ κ and theproof is done. (cid:3) Remark . Let us consider the simple but essential case of b = 0 and α = 0, i.e., L with A = (cid:18) a c (cid:19) and domain D κ . Then, from (4.3), weobserve that the ratio r := ac of the diffusion constants, rather than theexact values of a and c , along with κ decides ˜ κ and hence the values λ ± c . We also note that for κ ∈ (0 , π )˜ κ → π − as r → ∞ ; ˜ κ → + as r → + and for κ ∈ ( π, π )˜ κ → π + as r → ∞ ; ˜ κ → π − as r → + . In particular, if κ ∈ (0 , π ), or domain D κ is convex, and the diffusionconstant to x direction is relatively much lager than the the diffusionconstant to x direction, then λ ± c are much bigger than 1 and henceGreen’s function estimate (2.8) gives better decay near the vertex since R t,x ≤ References [1] C. Betz, G.A. C´amera, and H. Gzyl,
Bounds for the first eigenvalue of aspherical cap , Appl. Math. Optim. , no. 1, 193–202 (1983).[2] Sungwon Cho, Two-sided global estimates of the Green’s function of parabolicequations , Potential Anal. , no. 4, 387–398 (2006).[3] Petru A. Cioica, Kyeong-Hun Kim, and Kijung Lee, On the regularity of thestochastic heat equation on polygonal domains in R , J. Differential Equations , 6447–6479 (2019).[4] Petru A. Cioica, Kyeong-Hun Kim, Kijung Lee, and Felix Lindner, An L p -estimate for the stochastic heat equation on an angular domain in R , Stoch.Partial Differ. Equ. Anal. Comput. , no. 1, 45–72 (2018).[5] Vladimir A. Kozlov, Asymptotics of the Green function and Poisson kernelsof a mixed parabolic problem in a cone. II. , Z. Anal. Anwendungen , no. 1,27–42, (in Russian) (1991).[6] Vladimir A. Kozlov, Alexander Nazarov, The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in awedge , Math. Nachr. , no. 10, 1142–1165 (2014).[7] Riahi, L.
Comparison of Green functions and harmonic measures for parabolicoperators . Potential Anal. , no. 4, 381–402 (2005). Kyeong-Hun Kim, Department of Mathematics, Korea University, 1Anam-Dong, Sungbuk-gu, Seoul, 136–701, Republic of Korea
E-mail address : [email protected] Kijung Lee, Department of Mathematics, Ajou University, Suwon,443–749, Republic of Korea
E-mail address : [email protected] Jinsol Seo, Department of Mathematics, Korea University, 1 Anam-Dong, Sungbuk-gu, Seoul, 136–701, Republic of Korea
E-mail address ::