Improvements on lower bounds for the blow-up time under local nonlinear Neumann conditions
aa r X i v : . [ m a t h . A P ] M a r Improvements on lower bounds for the blow-up time underlocal nonlinear Neumann conditions
Xin Yang ∗ a and Zhengfang Zhou † ba Department of Mathematical Sciences, University of Cincinati, Cincinnati, OH45220, USA b Department of Mathematics, Michigan State University, East Lansing, MI 48824,USA
Abstract
This paper studies the heat equation u t = ∆ u in a bounded domain Ω ⊂ R n ( n ≥
2) withpositive initial data and a local nonlinear Neumann boundary condition: the normal derivative ∂u/∂n = u q on partial boundary Γ ⊆ ∂ Ω for some q >
1, while ∂u/∂n = 0 on the other part.We investigate the lower bound of the blow-up time T ∗ of u in several aspects. First, T ∗ is provedto be at least of order ( q − − as q → + . Since the existing upper bound is of order ( q − − ,this result is sharp. Secondly, if Ω is convex and | Γ | denotes the surface area of Γ , then T ∗ isshown to be at least of order | Γ | − n − for n ≥ | Γ | − (cid:14) ln (cid:0) | Γ | − (cid:1) for n = 2 as | Γ | → | Γ | − α for any α < n − . Finally, we generalize the results for convexdomains to the domains with only local convexity near Γ . In this paper, unless otherwise stated, Ω represents a bounded open subset in R n ( n ≥
2) with C boundary ∂ Ω. Γ and Γ denote two disjoint relatively open subsets of ∂ Ω. ∂ Γ = ∂ Γ , e Γ is acommon C boundary of Γ and Γ . Moreover, Γ = ∅ and ∂ Ω = Γ ∪ e Γ ∪ Γ . We study the followingproblem: u t ( x, t ) = ∆ u ( x, t ) in Ω × (0 , T ] , ∂u ( x,t ) ∂n ( x ) = u q ( x, t ) on Γ × (0 , T ] , ∂u ( x,t ) ∂n ( x ) = 0 on Γ × (0 , T ] ,u ( x,
0) = u ( x ) in Ω , (1.1) ∗ Email: [email protected] † Email: [email protected] q > , u ∈ C (Ω) , u ( x ) ≥ , u ( x ) . (1.2)The normal derivative in (1.1) is understood in the following way: for any ( x, t ) ∈ ∂ Ω × (0 , T ], ∂u ( x, t ) ∂n ( x ) , lim h → + ( Du )( x h , t ) · −→ n ( x ) , (1.3)where Du denotes the spatial derivative of u , −→ n ( x ) denotes the exterior unit normal vector at x and x h , x − h −→ n ( x ) for x ∈ ∂ Ω. Since ∂ Ω is C , x h belongs to Ω when h is positive and sufficiently small.Throughout this paper, we write M = max x ∈ Ω u ( x ) (1.4)and denote M ( t ) to be the supremum of the solution u to (1.1) on Ω × [0 , t ]: M ( t ) = sup ( x,τ ) ∈ Ω × [0 ,t ] u ( x, τ ) . (1.5) | Γ | represents the surface area of Γ , that is | Γ | = Z Γ dS ( x ) , where dS ( x ) means the surface integral with respect to the variable x . Φ refers to the fundamentalsolution to the heat equation:Φ( x, t ) = 1(4 πt ) n/ exp (cid:16) − | x | t (cid:17) , ∀ ( x, t ) ∈ R n × (0 , ∞ ) . (1.6)In addition, C = C ( a, b . . . ) and C i = C i ( a, b . . . ) represent positive constants which only depend onthe parameters a, b . . . . One should note that C and C i may stand for different constants from line toline. However, C ∗ = C ∗ ( a, b . . . ) and C ∗ i = C ∗ i ( a, b . . . ) will represent the constants which are fixed.When Γ = ∂ Ω, the problem (1.1) and more general parabolic equations with Neumann bound-ary conditions have been studied quite a lot. In addition, the Cauchy problems and the Dirichletboundary value problems related to the nonlinear blow-up phenomenon of the parabolic type werealso investigated. We refer the readers to the surveys [5, 14] and the books [6, 9, 23]. The topicsinclude the local and global existence and uniqueness of the solutions [1–4, 11, 17, 25, 27]; nonexistenceof global solutions and the upper bound estimates for the blow-up time [10–13, 15, 17, 19, 24, 25, 27];lower bound estimates for the blow-up time [16, 19–22, 27, 28]; blow-up sets, blow-up rate and theasymptotic behaviour of the solutions near the blow-up time [7, 8, 10, 11, 17, 18, 24, 26].For the research on the bounds of the blow-up time, the upper bound is usually related to thenonexistence of the global solutions and various methods have been developed. Meanwhile, the lowerbound was not studied as much in the past but was paid more attention in recent years. However, thelower bound can be argued to be more useful in practice, since it provides an estimate of the safe time.As an instance, for the problem (1.1) which was proposed in [27] to describe the re-entry process tothe atmosphere of the Columbia Space Shuttle, the lower bound of the blow-up time would provide asafe time of the landing of the shuttle. In contrast to the upper bound case, not many methods havebeen explored to deal with the lower bound. In addition, when Γ is a proper subset of ∂ Ω, to theauthors’ knowledge, only two papers [27] and [28] investigated the relation between the lower bound2f the blow-up time and the surface area | Γ | . The purpose of this work is to further improve the lowerbound estimate of the blow-up time in terms of | Γ | , especially when | Γ | →
0. Before presenting themain results of this paper, let us review what has been known.The recent paper [27] studied (1.1) systematically. According to it (see Theorem 1.3 in [27]), (1.1)has a unique classical solution u which is positive. Moreover, if T ∗ denotes the maximal existencetime of u , then 0 < T ∗ < ∞ and lim t ր T ∗ M ( t ) = ∞ . In other words, the maximal existence time T ∗ is just the blow-up time of u . For simplicity, we will also call T ∗ to be “the blow-up time”. [27] alsoprovided both upper and lower bounds of T ∗ (see Theorem 1.4 and 1.5 in [27]). For the upper bound,if min x ∈ Ω u ( x ) >
0, then T ∗ ≤ q − | Γ | Z Ω u − q ( x ) dx. (1.7)For the lower bound, T ∗ ≥ C − n +2 (cid:20) ln (cid:16) | Γ | − (cid:17) − ( n + 2)( q −
1) ln M − ln( q − − ln C (cid:21) n +2 , (1.8)where C is a constant which only depends on n , Ω and q . In some realistic problems, small | Γ | is ofinterest. For example in [27], the motivated model for the study of (1.1) is the Columbia space shuttleand Γ stands for the broken part on the left wing of the shuttle during launching, so the surface area | Γ | is expected to be small. As | Γ | → + , the upper bound (1.7) is of order | Γ | − while the lowerbound (1.8) is only of order (cid:2) ln (cid:0) | Γ | − (cid:1)(cid:3) / ( n +2) , so there is a big gap between them. The numericalsimulation in [27] is in the same order as the upper bound, so it is desirable to improve the lowerbound to at least a polynomial order | Γ | − α for some α > | Γ | − α for any α < n − . More precisely, for any α ∈ (cid:2) , n − (cid:1) , there exists C = C ( n, Ω , α ) such that T ∗ ≥ C ( q − M q − | Γ | α (cid:18) min (cid:26) , qM q − | Γ | α (cid:27)(cid:19) n − α − ( n − α . (1.9)In addition to the relation between T ∗ and | Γ | , (1.9) also provides sharp dependence of T ∗ on q and M . As discussed in [28], by sending q → + or M → + , the order of the lower bound in (1.9) is( q − − or M − ( q − , both of which are optimal.Based on the idea in [28], this paper will provide a unified method to enhance the lower boundof T ∗ in several aspects (especially the asymptotic behaviour of T ∗ as | Γ | → + ) according to thegeometric assumptions on Ω. Noticing that the lower bound in (1.8) is negative unless | Γ | or M is sufficiently small or q issufficiently close to 1, so it is desirable to derive a lower bound which is always positive. The firstresult below fulfills this expectation. Moreover, it obtains better asymptotic behavior of the lowerbound when | Γ | → + or q → + . Theorem 1.1.
Assume (1.2). Let T ∗ be the maximal existence time for (1.1). Then there exists a onstant C = C ( n, Ω) such that T ∗ ≥ Cq − (cid:16) M ) − q − | Γ | − n − (cid:17) , (1.10) where M is given by (1.4). Let us compare (1.10) with (1.8) in more detail on the asymptotic behavior. • As | Γ | → + , the order of (1.10) is ln (cid:0) | Γ | − (cid:1) while the order of (1.8) is only (cid:2) ln (cid:0) | Γ | − (cid:1)(cid:3) n +2 . • As q → + , the order of (1.10) is ( q − − , which is optimal since the order of the upper bound(1.7) is also ( q − − . However, the order of (1.8) is only (cid:0) ln q − (cid:1) n +2 .When the domain Ω is convex, for any α < n − , [28] derives the lower bound (1.9) which is oforder | Γ | − α as | Γ | → + . The next result in this paper improves the order to be | Γ | − / ( n − for n ≥ (cid:0) | Γ | ln | Γ | (cid:1) − for n = 2 as | Γ | → + . Theorem 1.2.
Assume (1.2). Let T ∗ be the maximal existence time for (1.1) and M be defined asin (1.4). Assume Ω is convex. Then there exist constants Y = Y ( n, Ω) and C = C ( n, Ω) such thatthe following statements hold. • Case 1: n ≥ . Denote Y = M q − | Γ | n − . If Y ≤ Y /q , then T ∗ ≥ C ( q − Y . (1.11) • Case 2: n = 2 . Denote Y = M q − | Γ | ln (cid:16) | Γ | + 1 (cid:17) . If Y ≤ Y /q , then T ∗ ≥ C ( q − Y . (1.12)In some practical situations, the convexity of domain Ω is not expected. However, the localconvexity near Γ is usually reasonable. Taking the model in [27] as an example again, since Γ ison the left wing of the shuttle, the region near Γ is indeed convex although the whole shuttle is not.Thus it is desirable to generalize Theorem 1.2 to the domains with only local convexity near Γ . Thethird result realizes this goal. Before the statement of the third result, let us explain the meaning ofthe local convexity near Γ . Definition 1.3 (Local convexity near partial boundary) . Let Ω be a bounded open subset in R n and Γ ⊆ ∂ Ω . We say Ω is locally convex near Γ if there exists d > such that Conv (cid:0) [Γ] d (cid:1) ⊆ Ω , where [Γ] d , { x ∈ ∂ Ω : dist ( x, Γ) < d } (1.13) denotes the boundary part whose distance to Γ is within d and Conv ([Γ] d ) means the convex hull of [Γ] d . Based on this definition, the local convexity near Γ in this paper means Conv (cid:0) [Γ ] d (cid:1) ⊆ Ω forsome d >
0. 4 heorem 1.4.
Assume (1.2). Let T ∗ be the maximal existence time for (1.1) and M be defined asin (1.4). Assume Conv (cid:0) [Γ ] d (cid:1) ⊆ Ω for some d > . Then there exist constants Y = Y ( n, Ω , d ) and C = C ( n, Ω , d ) such that the following statements hold. • Case 1: n ≥ . Denote Y = M q − | Γ | n − . If Y ≤ Y /q , then T ∗ ≥ C ( q − Y | ln Y | . (1.14) • Case 2: n = 2 . Denote Y = M q − | Γ | ln (cid:16) | Γ | + 1 (cid:17) . If Y ≤ Y /q , then T ∗ ≥ C ( q − Y | ln Y | . (1.15)To compare Theorem 1.4 with Theorem 1.2, the estimates in Theorem 1.4 are almost identical tothose in Theorem 1.2 except an extra term | ln Y | in the denominator. If we look at the proofs, thisextra term is due to the lack of the global convexity of Ω. The outlines of the proofs for Theorem1.2 and Theorem 1.4 are very similar, but the computations in the latter one will be much morecomplicated due to the lack of the global convexity again. Although this paper deals with domains with three different geometrical assumptions, the methodsshare many similarities and follow the same outline. Let M ( t ) be the same as in (1.5). The basic ideais to chop the range of M ( t ) into small pieces [ M k − , M k ] ( k ≥
1) and derive a lower bound t k ∗ for t k , the time that M ( t ) increases from M k − to M k . Suppose such lower bound t k ∗ can be found for L steps ( L may be finite or infinite), then P Lk =1 t k ∗ becomes a lower bound for T ∗ . The analysis willbe based on the representation formula (3.1).The common part of the proofs for Theorems 1.1, 1.2 and 1.4 is the second paragraph in the proofof Theorem 1.1 in Section 3. After the equation (3.10), the proofs will be slightly different due to thegeometric properties of the domains. For convenience, we write down the equation (3.10) as below. M k ≤ (cid:2) I + I ) (cid:3) M k − + 4 I M qk , where I , I and I are defined as in (3.7). For the estimates on I + I and I , we will argue indifferent ways under the following three cases.(1) For a general domain Ω, Lemma 3.1 implies I + I ≤ C √ t k for some constant C = C ( n, Ω) andwe will use (2.7) to bound I .(2) For any convex domain Ω, the identities (2.1) and (2.2) yield I + I = 0 and we will applyLemma 2.7 and Lemma 2.10 to bound I .(3) For any domain Ω that is locally convex near Γ , that is Conv (cid:0) [Γ ] d (cid:1) ⊆ Ω for some d >
0, theidentity (2.1) and Corollary 2.2 lead to I + I ≤ C t k exp (cid:16) − d t k (cid:17) C = C ( n, Ω , d ). On the other hand, we will exploit Lemma 2.7 and Lemma2.10 again to bound I .Several remarks will be made in sequel. • First, since small t k is of interest, the bound for I + I in Case (3) is exponential decay as t k →
0. Due to this fast decay, the result in Case (3) is very close to that in Case (2). Inaddition, either the result in Case (2) or Case (3) is far better than that in Case (1) where theestimate on I + I only decays like √ t k . • Secondly, (2.7) implies that I ≤ C − ( n − α | Γ | α t − ( n − α k for some constant C = C ( n, Ω) and for any α ∈ [0 , n − ). Lemma 2.7 and Lemma 2.10 push thepower of | Γ | a little bit further. More precisely, Lemma 2.7 implies I ≤ C | Γ | n − when n ≥ I ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) when n = 2. • Thirdly, for general domains, our method will not gain better lower bound for T ∗ (regardingthe order of | Γ | − ) by increasing the power of | Γ | in the estimate of I , so we just choose α = n − in (2.7) instead of exploiting Lemma 2.7 and Lemma 2.10. • Finally, for convex domains or the domains with local convexity near Γ , the power of | Γ | inthe bound of I makes a difference in the final lower bound estimate of T ∗ (regarding the orderof | Γ | − ), so we apply Lemma 2.7 and Lemma 2.10 instead of (2.7). The organization of this paper is as follows. Section 2 presents some preliminary results which willbe used later. Section 3 verifies Theorem 1.1 for general domain Ω. Section 4 provides the proof forTheorem 1.2 when the domain Ω is convex. Section 5 justifies Theorem 1.4 for the domain Ω that islocally convex near Γ . In [28], it mentioned an elementary identity (see Lemma 2.2 in [28]) about the heat kernel, namelyfor any x ∈ ∂ Ω and t > Z Ω Φ( x − y, t ) dy − Z t Z ∂ Ω ∂ Φ( x − y, t − τ ) ∂n ( y ) dS ( y ) dτ = 12 , ∀ x ∈ ∂ Ω , t > , (2.1)6here ∂ Φ( x − y, t − τ ) ∂n ( y ) , D y (cid:2) Φ( x − y, t − τ ) (cid:3) · −→ n ( y )is the normal derivative. In addition, if Ω is convex, then Z Ω Φ( x − y, t ) dy + Z t Z ∂ Ω (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ = 12 , ∀ x ∈ ∂ Ω , t > . (2.2)This subsection will derive an intermediate result, Corollary 2.2, when the convexity is only as-sumed near Γ rather than in the whole domain. Before presenting Corollary 2.2, we first show anauxiliary lemma. Lemma 2.1.
Let Ω and Γ be the same as in (1.1). Then for any d > , there exists C = C ( n, Ω , d ) such that for any x ∈ Γ and t > , Z t Z ∂ Ω \ [Γ ] d (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ C t exp (cid:16) − d t (cid:17) . (2.3) Proof.
In this proof, C denotes a constant which depends only on n , Ω and d . By a change of variablein τ and the definition of Φ, Z t Z ∂ Ω \ [Γ ] d (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ = Z t Z ∂ Ω \ [Γ ] d (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ C Z t Z ∂ Ω \ [Γ ] d | ( x − y ) · −→ n ( y ) | τ n +1 exp (cid:16) − | x − y | τ (cid:17) dS ( y ) dτ. (2.4)Since ∂ Ω is assumed to be C , then | ( x − y ) · −→ n ( y ) | ≤ C | x − y | . In addition, | x − y | ≥ d for any x ∈ Γ and y ∈ ∂ Ω \ [Γ ] d . As a result, | ( x − y ) · −→ n ( y ) | τ n +1 exp (cid:16) − | x − y | τ (cid:17) ≤ C | x − y | − n (cid:18) | x − y | τ (cid:19) n exp (cid:16) − | x − y | τ (cid:17) ≤ C | x − y | − n exp (cid:16) − | x − y | τ (cid:17) ≤ Cd − n exp (cid:16) − d τ (cid:17) . (2.5)Plugging (2.5) into (2.4), Z t Z ∂ Ω \ [Γ ] d (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ Cd − n Z t Z ∂ Ω \ [Γ ] d exp (cid:16) − d τ (cid:17) dS ( y ) dτ ≤ Cd − n | ∂ Ω | Z t exp (cid:16) − d τ (cid:17) dτ ≤ Cd − n | ∂ Ω | t exp (cid:16) − d t (cid:17) .
7y exploiting Lemma 2.1, the following (2.6) is a variant of the identity (2.2), and it will play thesame role in the proof of Theorem 1.4 as (2.2) will do in the proof of Theorem 1.2.
Corollary 2.2.
Let Ω and Γ be the same as in (1.1). Assume there exists d > such thatConv (cid:0) [Γ ] d (cid:1) ⊆ Ω . Then there exists C = C ( n, Ω , d ) such that for any x ∈ Γ and t > , Z Ω Φ( x − y, t ) dy + Z t Z ∂ Ω (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤
12 +
C t exp (cid:16) − d t (cid:17) . (2.6) Proof.
Since x ∈ Γ and Conv (cid:0) [Γ ] d (cid:1) ⊆ Ω, we have ∂ Φ( x − y, t − τ ) ∂n ( y ) = C ( x − y ) · −→ n ( y )( t − τ ) n/ exp (cid:16) − | x − y | t − τ ) (cid:17) ≤ , ∀ y ∈ [Γ ] d . As a result, Z t Z ∂ Ω ∂ Φ( x − y, t − τ ) ∂n ( y ) dS ( y ) dτ + Z t Z ∂ Ω (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ = Z t Z ∂ Ω \ [Γ ] d ∂ Φ( x − y, t − τ ) ∂n ( y ) + (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ C t exp (cid:16) − d t (cid:17) , where the last inequality is due to Lemma 2.1. Therefore Z Ω Φ( x − y, t ) dy + Z t Z ∂ Ω (cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ Z Ω Φ( x − y, t ) dy − Z t Z ∂ Ω ∂ Φ( x − y, t − τ ) ∂n ( y ) dS ( y ) dτ + C t exp (cid:16) − d t (cid:17) = 12 + C t exp (cid:16) − d t (cid:17) , where the last equality is because of (2.1). The estimate for the boundary-time integral of the heat kernel is a basic tool in the derivation of thelower bound in (1.9). More precisely (see Lemma 2.3 in [28]), there exists C = C ( n, Ω) such that forany Γ ⊆ ∂ Ω, α ∈ (cid:2) , n − (cid:1) , x ∈ ∂ Ω and t > Z t Z Γ Φ( x − y, t − τ ) dS ( y ) dτ ≤ C − ( n − α | Γ | α t − ( n − α . (2.7)According to the method in [28], the power α in (2.7) determines the power on | Γ | − of the lowerbound for T ∗ in (1.9). However, the range of the power α in (2.7) missed n − since the coefficientwill blow up as α ր n − . So it is natural to ask whether α can be taken as n − by other methods.In this subsection, the above expectation will be justified for n ≥ n = 2 (withan extra log term and bounded time t ) in Lemma 2.10.We first introduce a simple fact which can be regarded as a rearrangement result.8 emma 2.3. Let n ≥ and f : (0 , ∞ ) → [0 , ∞ ) be a decreasing function. Then for any boundedsubset U of R n and for any x ∈ R n , Z U f ( | x − y | ) dy ≤ Z B R (0) f ( | z | ) dz (2.8) where R satisfies | B R (0) | = | U | (namely the volume of B R (0) equals the volume of U ).Proof. Define U = U − { x } . Then by a change of variable z = y − x , Z U f ( | x − y | ) dy = Z U f ( | z | ) dz = Z U ∩ B R (0) f ( | z | ) dz + Z U \ B R (0) f ( | z | ) dz , J + J , (2.9)Since f is decreasing, J ≤ f ( R ) | U \ B R (0) | . Due to the definition of R , | B R (0) | = | U | = | U | . So we have | B R (0) \ U | = | U \ B R (0) | . As a result, J ≤ f ( R ) | B R (0) \ U | ≤ Z B R (0) \ U f ( | z | ) dz, (2.10)where the last inequality is again due to the decay of f . Combining (2.9) and (2.10), we finish theproof. Definition 2.4.
Let Ω be a bounded, open subset of R n with C boundary. Let Γ be a subset of ∂ Ω .We say Γ is given by a graph if (upon relabelling and reorienting the coordinates axes) there exists abounded subset U ⊆ R n − and a C function φ : R n − → R such that Γ = { (˜ y, φ (˜ y )) : ˜ y ∈ U } . In the following, for any x ∈ R n , we will decompose it to be x = (˜ x, x n ), where ˜ x denotes the first n − x . Lemma 2.5.
Let Ω be a bounded, open subset of R n ( n ≥ with C boundary. Let Γ be a subset of ∂ Ω that is given by a graph as in Definition 2.4. Then there exists a constant C = C ( n, ||∇ φ || L ∞ ( U ) ) ,where φ and U are the same as in Definition 2.4, such that for any x ∈ R n , Z Γ | x − y | n − dS ( y ) ≤ C | Γ | / ( n − . Proof.
By Definition 2.4, without loss of generality, we can assume there exists a C function φ : R n − → R and a bounded subset U of R n − such thatΓ = { (˜ y, φ (˜ y )) : ˜ y ∈ U } . (2.11)9hus, Z Γ | x − y | n − dS ( y ) = Z U p |∇ φ (˜ y ) | | (˜ x, x n ) − (˜ y, φ (˜ y )) | n − d ˜ y ≤ Z U p |∇ φ (˜ y ) | | ˜ x − ˜ y | n − d ˜ y ≤ C Z U | ˜ x − ˜ y | n − d ˜ y. Define f ( r ) = 1 r n − , ∀ r > . Then it follows from Lemma 2.3 that Z U | ˜ x − ˜ y | n − d ˜ y = Z U f ( | ˜ x − ˜ y | ) d ˜ y ≤ Z B R (0) f ( | ˜ z | ) d ˜ z = CR = C | U | / ( n − . Again by the parametrization (2.11), it is readily seen that | U | ≤ | Γ | . Hence, Z Γ | x − y | n − dS ( y ) ≤ C | U | / ( n − ≤ C | Γ | / ( n − . Corollary 2.6.
Let Ω be a bounded open subset of R n ( n ≥ with C boundary. Let Γ be any subsetof ∂ Ω . Then there exists a constant C = C ( n, Ω) such that for any x ∈ R n , Z Γ | x − y | n − dS ( y ) ≤ C | Γ | / ( n − . Proof.
Since ∂ Ω is C , for any point x ∈ ∂ Ω, the boundary part of Ω near x is given by a graph asin Definition 2.4). Therefore we can split ∂ Ω into finite pieces: ∂ Ω = K [ i =1 A i , (2.12)where each A i (1 ≤ i ≤ K ) is given by the graph of some C function φ i on some bounded set U i ⊆ R n − . The number of total pieces K and ||∇ φ i || L ∞ ( U i ) only depend on Ω.For any 1 ≤ i ≤ K , Γ ∩ A i is also a boundary part given by a graph. Therefore by Lemma 2.5,there exists a constant C = C ( n, Ω) such that for any 1 ≤ i ≤ K , Z Γ ∩ A i | x − y | n − dS ( y ) ≤ C | Γ ∩ A i | / ( n − . Z Γ | x − y | n − dS ( y ) ≤ K X i =1 Z Γ ∩ A i | x − y | n − dS ( y ) ≤ C K X i =1 | Γ ∩ A i | / ( n − ≤ CK | Γ | / ( n − = C | Γ | / ( n − . Lemma 2.5 and Corollary 2.6 will be applied to show the desired Lemma 2.7 which pushes thepower α in (2.7) to n − when n ≥ Lemma 2.7.
Let Ω be a bounded open subset of R n ( n ≥ with C boundary. Let Γ be any subset of ∂ Ω . Then there exists C = C ( n, Ω) such that for any x ∈ R n and t ≥ , Z t Z Γ Φ( x − y, t − τ ) dS ( y ) dτ ≤ C | Γ | / ( n − . (2.13) Proof.
In this proof, unless otherwise stated, C represents constants which only depend on n and Ω.First, by the explicit formula (1.6) of Φ and a change of variable in τ , we have Z t Z Γ Φ( x − y, t − τ ) dS ( y ) dτ = C Z Γ Z t τ − n/ e −| x − y | / (4 τ ) dτ dS ( y ) . Then by the change of variable s = | x − y | / (4 τ ) for τ , Z Γ Z t τ − n/ e −| x − y | / (4 τ ) dτ dS ( y ) ≤ C Z Γ | x − y | n − Z ∞| x − y | / (4 t ) s n − e − s ds dS ( y ) . (2.14)Since n ≥ s n − e − s is integrable on (0 , ∞ ). As a result, Z Γ | x − y | n − Z ∞| x − y | / (4 t ) s n − e − s ds dS ( y ) ≤ Z Γ | x − y | n − Z ∞ s n − e − s ds dS ( y )= C Z Γ | x − y | n − dS ( y ) . Now applying Corollary 2.6, Z Γ | x − y | n − dS ( y ) ≤ C | Γ | / ( n − . The following Lemma 2.8, Corollary 2.9 and Lemma 2.10 are parallel results as Lemma 2.5, Corol-lary 2.6 and Lemma 2.7, but they deal with dimension n = 2 rather than n ≥ emma 2.8. Let Ω be a bounded, open subset of R with C boundary. Let Γ be any subset of ∂ Ω that is given by a graph as in Definition 2.4. Then there exists a constant C = C (Ω , ||∇ φ || L ∞ ( U ) ) ,where φ and U are the same as those in Definition 2.4, such that for any x ∈ Ω , Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) , where d Ω denotes the diameter of Ω .Proof. By Definition 2.4, without loss of generality, we can assume there exists a C function φ : R → R and a bounded set U ⊆ R such that Γ = { (˜ y, φ (˜ y )) : ˜ y ∈ U } . (2.15)In addition, we define f ( r ) = ( ln (cid:0) d Ω r (cid:1) , < r ≤ d Ω , , r > d Ω . (2.16)Since x = (˜ x, x n ) ∈ Ω, then for any (˜ y, φ (˜ y )) ∈ Γ, | ˜ x − ˜ y | ≤ | (˜ x, x n ) − (˜ y, φ (˜ y )) | ≤ d Ω . As a result, Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) = Z U ln (cid:16) d Ω | (˜ x, x n ) − (˜ y, φ (˜ y )) | (cid:17)p |∇ φ (˜ y ) | d ˜ y ≤ C Z U ln (cid:16) d Ω | ˜ x − ˜ y | (cid:17) d ˜ y = C Z U f ( | ˜ x − ˜ y | ) d ˜ y. (2.17)Now it follows from Lemma 2.3 that Z U f ( | ˜ x − ˜ y | ) d ˜ y ≤ Z B R (0) f ( | ˜ z | ) d ˜ z = 2 Z R f ( r ) dr, (2.18)where | B R (0)) | = | U | , namely 2 R = | U | . For any ˜ y , ˜ y ∈ U , we have | ˜ y − ˜ y | ≤ | (˜ y , φ (˜ y )) − (˜ y , φ (˜ y )) | ≤ d Ω , which implies diam( U ) ≤ d Ω . Moreover, since U ⊆ R , then | U | ≤ diam( U ). Thus, R = | U | / ≤ d Ω / Z R f ( r ) dr = Z R ln (cid:16) d Ω r (cid:17) dr = R h ln (cid:16) d Ω R (cid:17) + 1 i . (2.19)12gain by the parametrization (2.15), it is readily seen that | U | ≤ | Γ | . Therefore, R ≤ min n | Γ | , d Ω o . Define g ( r ) = r h ln (cid:16) d Ω r (cid:17) + 1 i , ∀ r > . Then g is increasing when r ∈ (0 , d Ω ] and (2.19) implies R R f ( r ) dr = g ( R ). Next, we will estimate g ( R ) in the following two situations. • | Γ | ≤ d Ω . g ( R ) ≤ g ( | Γ | ) = | Γ | h ln (cid:16) d Ω | Γ | (cid:17) + 1 i ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) (2.20)for some constant C only depending on Ω. • | Γ | > d Ω . g ( R ) ≤ g ( d Ω ) = d Ω . Define h ( r ) = r ln (cid:16) r + 1 (cid:17) , ∀ r > . (2.21)Then h ′′ ( r ) = − r (1 + r ) < , ∀ r > . This implies h ′ ( r ) > r >
0, since lim r →∞ h ′ ( r ) = 0. Hence, h is an increasing functionand | Γ | ln (cid:16) | Γ | + 1 (cid:17) = h ( | Γ | ) ≥ h ( d Ω ) = d Ω ln (cid:16) d Ω + 1 (cid:17) . Thus, g ( R ) ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) , (2.22)where C = 1 / ln (cid:0) d Ω + 1 (cid:1) is a constant depending only on Ω.Combining (2.17), (2.18), (2.20) and (2.22), the conclusion follows. Corollary 2.9.
Let Ω be a bounded, open subset of R with C boundary. Let Γ be any subset of ∂ Ω .Then there exists a constant C = C (Ω) such that for any x ∈ Ω , Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) , where d Ω denotes the diameter of Ω .Proof. Similar to the proof of Corollary 2.6, we first decompose ∂ Ω as that in (2.12). Then Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) ≤ K X i =1 Z Γ ∩ A i ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) . ∩ A i is given by a graph, we can apply Lemma 2.8 to conclude there exists a constant C = C (Ω) such that for each 1 ≤ i ≤ K , Z Γ ∩ A i ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) ≤ C | Γ ∩ A i | ln (cid:16) | Γ ∩ A i | + 1 (cid:17) . Recalling the function h defined in (2.21) is an increasing function, so | Γ ∩ A i | ln (cid:16) | Γ ∩ A i | + 1 (cid:17) ≤ | Γ | ln (cid:16) | Γ | + 1 (cid:17) . As a result, Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) . Next, Lemma 2.8 and Corollary 2.9 will be applied to show our desired Lemma 2.10 which is animprovement of (2.7) when n = 2. Lemma 2.10.
Let Ω be a bounded, open subset of R with C boundary. Let Γ be any subset of ∂ Ω .Then there exists C = C (Ω) such that for any x ∈ Ω and t ∈ [0 , , Z t Z Γ Φ( x − y, t − τ ) dS ( y ) dτ ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) . (2.23) Proof.
We proceed similarly as that in the proof of Lemma 2.7 until (2.14). Next, the situation isdifferent since s n/ − e − s is not integrable near s = 0 when n = 2. For convenience, we rewrite (2.14)when n = 2 as following: Z Γ Z t τ − e −| x − y | / (4 τ ) dτ dS ( y ) ≤ C Z Γ Z ∞| x − y | / (4 t ) s − e − s ds dS ( y ) . (2.24)Since t ≤ x ∈ Ω, | x − y | / (4 t ) ≥ | x − y | /
4. Thus, Z ∞| x − y | / (4 t ) s − e − s ds ≤ Z ∞| x − y | / s − e − s ds = Z d | x − y | / s − e − s ds + Z ∞ d s − e − s ds ≤ Z d | x − y | / s − ds + 1 d Z ∞ d e − s ds = 2 ln (cid:16) d Ω | x − y | (cid:17) + C. As a result, Z Γ Z ∞| x − y | / (4 t ) s − e − s ds dS ( y ) ≤ Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) + C | Γ | . (2.25)Now applying Corollary 2.9, Z Γ ln (cid:16) d Ω | x − y | (cid:17) dS ( y ) ≤ C | Γ | ln (cid:16) | Γ | + 1 (cid:17) . | Γ | ≤ (cid:16) | ∂ Ω | + 1 (cid:17) | Γ | ln (cid:16) | Γ | + 1 (cid:17) = C | Γ | ln (cid:16) | Γ | + 1 (cid:17) , the lemma is proved. The starting point of the proofs in this paper is the representation formula of the solution u (seeLemma A.1 in [28]): for any T ∈ [0 , T ∗ ) and ( x, t ) ∈ ∂ Ω × [0 , T ∗ − T ), u ( x, T + t ) = 2 Z Ω Φ( x − y, t ) u ( y, T ) dy − Z t Z ∂ Ω ∂ Φ( x − y, t − τ ) ∂n ( y ) u ( y, T + τ ) dS ( y ) dτ +2 Z t Z Γ Φ( x − y, t − τ ) u q ( y, T + τ ) dS ( y ) dτ. (3.1)To estimate the integral of ∂ Φ( x − y,t − τ ) ∂n ( y ) on ∂ Ω × [0 , t ], we apply the lemma below. Lemma 3.1.
There exists C = C ( n, Ω) such that for any x ∈ ∂ Ω and t > , Z t Z ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ C √ t. (3.2) Proof.
By the definition of Φ, (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = C | ( x − y ) · −→ n ( y ) | ( t − τ ) n +1 exp (cid:16) − | x − y | t − τ ) (cid:17) . Since ∂ Ω is assumed to be C , there exists a constant C such that | ( x − y ) · −→ n ( y ) | ≤ C | x − y | for any x, y ∈ ∂ Ω. As a result, (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | x − y | ( t − τ ) n +1 exp (cid:16) − | x − y | t − τ ) (cid:17) . Noticing the term | x − y | t − τ exp (cid:16) − | x − y | t − τ ) (cid:17) is bounded by some constant, so (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( t − τ ) n/ exp (cid:16) − | x − y | t − τ ) (cid:17) . Z t Z ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x − y, t − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dS ( y ) dτ ≤ C Z t Z ∂ Ω t − τ ) n/ exp (cid:16) − | x − y | t − τ ) (cid:17) dS ( y ) dτ = C Z t Z ∂ Ω τ n/ exp (cid:16) − | x − y | τ (cid:17) dS ( y ) dτ. By the change of variable σ = 2 τ , Z t Z ∂ Ω τ n/ exp (cid:16) − | x − y | τ (cid:17) dS ( y ) dτ = C Z t Z ∂ Ω σ n/ exp (cid:16) − | x − y | σ (cid:17) dS ( y ) dσ = C Z t Z ∂ Ω Φ( x − y, σ ) dS ( y ) dσ = C Z t Z ∂ Ω Φ( x − y, t − σ ) dS ( y ) dσ. Finally, invoking (2.7) with Γ = ∂ Ω and α = 0, the proof is finished. Proof of Theorem 1.1.
In this proof, C will denote constants which only depend on n and Ω, thevalues of C may be different in different places. But C ∗ and C ∗ i ( i ≥
1) will represent fixed constantswhich only depend on n and Ω. M ( t ) represents the same function as in (1.5).For any strictly increasing sequence { M k } k ≥ whose initial term is the same as the M defined in(1.4), we denote T k to be the first time that M ( t ) reaches M k . Obviously, T = 0. For any k ≥ t k = T k − T k − (3.3)to be the time spent in the kth step. By the maximum principle and the Hopf lemma, there exists x k ∈ Γ such that u ( x k , T k ) = M k . (3.4)Applying the representation formula (3.1) with T = T k − and ( x, t ) = ( x k , t k ), then u ( x k , T k ) = 2 Z Ω Φ( x k − y, t k ) u ( y, T k − ) dy − Z t k Z ∂ Ω ∂ Φ( x k − y, t k − τ ) ∂n ( y ) u ( y, T k − + τ ) dS ( y ) dτ + 2 Z t k Z Γ Φ( x k − y, t k − τ ) u q ( y, T k − + τ ) dS ( y ) dτ. (3.5)16ombining (3.4) and (3.5), M k ≤ M k − Z Ω Φ( x k − y, t k ) dy +2 M k Z t k Z ∂ Ω (cid:12)(cid:12)(cid:12) ∂ Φ( x k − y, t k − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12) dS ( y ) dτ +2 M qk Z t k Z Γ Φ( x k − y, t k − τ ) dS ( y ) dτ. Replacing the term R Ω Φ( x k − y, t k ) dy by the identity (2.1), then M k ≤ M k − (cid:20)
12 + Z t k Z ∂ Ω ∂ Φ( x k − y, t k − τ ) ∂n ( y ) dS ( y ) dτ (cid:21) + 2 M k Z t k Z ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x k − y, t k − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dS ( y ) dτ + 2 M qk Z t k Z Γ Φ( x k − y, t k − τ ) dS ( y ) dτ. Moving the term on the second line of the right hand side to the left, we obtain(1 − I ) M k ≤ (1 + 2 I ) M k − + 2 I M qk , (3.6)where I = Z t k Z ∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x k − y, t k − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dS ( y ) dτ,I = Z t k Z ∂ Ω ∂ Φ( x k − y, t k − τ ) ∂n ( y ) dS ( y ) dτ, (3.7) I = Z t k Z Γ Φ( x k − y, t k − τ ) dS ( y ) dτ. It is readily seen that | I | ≤ I . In addition, by Lemma 3.1, there exists a constant C ∗ such that I ≤ C ∗ √ t k . (3.8)If t k satisfies t k ≤ C ∗ ) , (3.9)then | I | ≤ I ≤ . As a result, 1 − I ≥ and1 + 2 I − I = 1 + 2( I + I )1 − I ≤ I + I ) . Hence by dividing 1 − I from both sides of (3.6), we obtain M k ≤ (cid:2) I + I ) (cid:3) M k − + 4 I M qk . (3.10)In the following, by choosing a suitable sequence { M k } k ≥ and obtaining a lower bound t k ∗ for17ach t k , the sum of all t k ∗ becomes a lower bound for T ∗ . First, due to the estimate (3.8) again, I + I ≤ I ≤ C ∗ √ t k . (3.11)Next in order to estimate I , we apply (2.7) for Γ = Γ and α = n − , then there exists someconstant C such that I ≤ C | Γ | α t / k . (3.12)Plugging (3.11) and (3.12) into (3.10) yields M k ≤ (1 + C √ t k ) M k − + C | Γ | α t / k M qk . (3.13)Define M k = 2 k M . (3.14)Then 2 k M ≤ (1 + C √ t k ) 2 k − M + C | Γ | α t / k qk M q . Subtracting 2 k − M from both sides, we obtain2 k − M ≤ C √ t k k − M + C | Γ | α t / k qk M q . Dividing by 2 k − M , 1 ≤ C √ t k + C | Γ | α t / k ( q − k M q − . Thus, √ t k + | Γ | α M q − ( q − k t / k − C ≥ . Regarding the left hand side of the above inequality to be a quadratic function in t / k , then t / k hasto be greater than its positive root, that is t / k ≥ (cid:18) − | Γ | α M q − ( q − k + r | Γ | α M q − q − k + 4 C (cid:19) . Consequently, t / k ≥ C (cid:16) | Γ | α M q − ( q − k + q | Γ | α M q − q − k + C (cid:17) ≥ C q | Γ | α M q − q − k + C . Hence, there exists C ∗ such that t k ≥ C ∗ (cid:0) | Γ | α M q − q − k + 1 (cid:1) . (3.15)As a summary of the above paragraph, by choosing M k = 2 k M , then (3.9) implies (3.15). There-fore, t k ≥ min (cid:26) C ∗ ) , C ∗ (cid:0) | Γ | α M q − q − k + 1 (cid:1) (cid:27) . C ∗ = min n C ∗ ) , C ∗ o , then t k ≥ C ∗ | Γ | α M q − q − k + 1 . (3.16)Hence, T ∗ = ∞ X k =1 t k ≥ C ∗ ∞ X k =1 | Γ | α M q − q − k + 1 ≥ C ∗ Z ∞ | Γ | α M q − q − x + 1 dx = C ∗ q −
1) ln(2) ln | Γ | α M q − q − ! . Recalling α = n − , (1.10) follows. Define E q = ( q − q − /q q , ∀ q > . (4.1)By elementary calculus, 13 q < E q < min n q , q − e o < . (4.2)The lemma below is a simple generalization of Lemma 3.2 in [28]. Lemma 4.1.
For any q > and m > , write E q as in (4.1) and define g : ( m, ∞ ) → R by g ( λ ) = λ − mλ q , ∀ λ > m. (4.3) Then the following two claims hold.(1) For any y ∈ (cid:0) , m − q E q (cid:3) , there exists unique λ ∈ (cid:0) m, qq − m (cid:3) such that g ( λ ) = y .(2) For any y > m − q E q , there does not exist λ > m such that g ( λ ) = y .Proof. Since g is strictly increasing on the interval (cid:0) m, qq − m (cid:3) and strictly decreasing on the interval (cid:2) qq − m, ∞ (cid:1) , it reaches the maximum at λ = qq − m . Noticing that g (cid:16) qq − m (cid:17) = m − q E q , then the claims (1) and (2) follow directly.Now we can carry out the main proof in this section. Proof of Theorem 1.2.
We will demonstrate detailed proof for the case n ≥
3, the proof for the case n = 2 is similar and will be briefly mentioned at the end. In the proof below, C will denote the19onstants which only depend on n and Ω, the values of C may be different in different places. But C ∗ will represent a fixed constant which only depends on n and Ω. Let M ( t ) be defined as in (1.5). Step 1.
The first part is exactly the same as the second paragraph in the proof of Theorem 1.1,namely we adopt the same notations and the same estimates from (3.3) through (3.10). In particular,we make the assumption (3.9).
Step 2.
In this step, we will find a constant t ∗ > { M k } ≤ k ≤ L such that t k ≥ t ∗ for 1 ≤ k ≤ L . Then in Step 3, a lower bound for Lt ∗ will be derived.Due to the convexity of Ω, the normal derivative ∂ Φ( x k − y, t k − τ ) ∂n ( y ) in (3.7) is always nonpositive.As a result, I + I = Z t k Z ∂ Ω ∂ Φ( x k − y, t k − τ ) ∂n ( y ) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ Φ( x k − y, t k − τ ) ∂n ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dS ( y ) dτ = 0 . (4.4)To estimate I , we apply Lemma 2.7 to conclude I ≤ C | Γ | / ( n − (4.5)for some constant C = C ( n, Ω). Hence plugging (4.4) and (4.5) into (3.10), we get M k ≤ M k − + C ∗ | Γ | / ( n − M qk (4.6)for some constant C ∗ = C ∗ ( n, Ω). As a summary, the argument so far claims that if (3.9) holds, then M k will satisfy (4.6).Based on the above observation, if we choose δ = 2 C ∗ | Γ | / ( n − (4.7)and define M k to be the solution (if it exists) to M k − M k − M qk = δ , (4.8)then (3.9) can not hold since otherwise (4.6) will be violated. Consequently t k > t ∗ , where t ∗ = 116( C ∗ ) . (4.9)Due to Lemma 4.1, the existence of a solution M k to (4.8) is equivalent to the inequality M q − k − δ ≤ E q . In addition, as long as such a solution exists, M k can be chosen to satisfy M k − < M k ≤ qq − M k − . Thus, the strategy of constructing { M k } is summarized as below. First, define M and δ as in (1.4)and (4.7). Next suppose M k − has been constructed for some k ≥
1, then whether defining M k depends on how large M k − is. ⋄ If M q − k − δ ≤ E q , then we define M k ∈ (cid:0) M k − , qq − M k − (cid:3) to be the solution to (4.8). ⋄ If M q − k − δ > E q , then there does not exist M k > M k − which solves (4.8). So we do not define20 k and stop the construction.According to this construction, if { M k } ≤ k ≤ L have been defined, then T k − T k − ≥ t ∗ for any1 ≤ k ≤ L . Therefore, T k ≥ kt ∗ for any 1 ≤ k ≤ L . Since T ∗ is finite, L ≤ T ∗ /t ∗ < ∞ , whichmeans the cardinality of { M k } has to be finite (actually this fact can also be justified by analysingthe construction directly, see Lemma 4.2). So we can assume the constructed sequence is { M k } ≤ k ≤ L for some finite L . Step 3.
By Lemma 4.2,
L > q − (cid:16) M q − δ − q (cid:17) . To obtain an effective lower bound, M q − δ should be greater than 3 q . If requiring1 M q − δ ≥ q, (4.10)then L > q − M q − δ = 140 C ∗ ( q − M q − | Γ | / ( n − . Denote Y = M q − | Γ | / ( n − . Then T ∗ ≥ Lt ∗ > C ( q − Y for some constant C . Finally, noticing that (4.10) is equivalent to Y ≤ C ∗ q , the proof for the case n ≥ Y = 1 / (12 C ∗ ).When n = 2, the process is almost identical as the above except two differences. First, Lemma2.10 will be applied instead of Lemma 2.7, so the term | Γ | / ( n − in the above proof needs to bereplaced by | Γ | ln (cid:0) | Γ | + 1 (cid:1) . Secondly, due to the restriction t ≤ t k should satisfyboth t k ≤ t ∗ will be t ∗ = min n C ∗ ) , o (4.11)instead of (4.9). Fortunately, this additional requirement will not bring major changes to the proof.Actually, without loss of generality, we can choose C ∗ to be larger than 1 /
4, which makes C ∗ ) ≤ Lemma 4.2.
Given q > , M > and δ > , denote E q as (4.1) and construct a (finite) sequence { M k } k ≥ inductively as follows. Suppose M k − has been constructed for some k ≥ , then based onLemma 4.1, whether defining M k depends on how large M k − is. If M q − k − δ ≤ E q , then we define M k ∈ (cid:0) M k − , qq − M k − (cid:3) to be the solution to M k − M k − M qk = δ . (4.12) ⋄ If M q − k − δ > E q , then there does not exist M k > M k − which solves (4.12). So we do not define M k and stop the construction.We claim this construction stops in finite steps and if the last term is denoted as M L , then L > q − (cid:16) M q − δ − q (cid:17) . (4.13) Proof.
First, we will show the construction has to stop in finite steps. In fact, it follows from (4.12)that the sequence { M k } is strictly increasing and M k = M k − + M qk δ ≥ (cid:0) M q − δ (cid:1) M k − . As a result, M k ≥ (cid:0) M q − δ (cid:1) k M . Thus M q − k will exceed E q /δ when k is sufficiently large, which forces the construction to stop.Next suppose the constructed sequence is { M k } ≤ k ≤ L . The lower bound (4.13) for L will bejustified below. Case 1. M q − δ > E q . In this case, it follows from (4.2) that1 M q − δ < E q < q. Thus (4.13) holds automatically since the right hand side of (4.13) is negative.
Case 2. M q − δ ≤ E q . In this case, it is evident from the construction that L ≥
1. Moreover, M q − L − δ ≤ E q and M q − L δ > E q . According to the recursive relation (4.12), M k − = M k (cid:0) − M q − k δ (cid:1) . Raising both sides to the power q − δ , M q − k − δ = M q − k (cid:0) − M q − k δ (cid:1) q − δ . Let x k = M q − k δ . Then x k − = x k (1 − x k ) q − , ∀ ≤ k ≤ L. (4.14)Moreover, x = M q − δ , x L − ≤ E q and x L > E q . M L ≤ qq − M L − , so x L = (cid:18) M L M L − (cid:19) q − x L − ≤ (cid:16) qq − (cid:17) q − E q = 1 q . Since the right hand side of (4.14) is nonlinear in x k , it seems impossible to express x k as anexplicit formula in terms of x k − . This motivates us to consider the “reversed” relation of (4.14),namely a new sequence { y k } ≤ k ≤ L defined in the following way: y , min { / , E q } and y k , y k − (1 − y k − ) q − , ∀ ≤ k ≤ L. (4.15)To analyse the sequence { y k } , we define h : (0 , → R by h ( t ) = t (1 − t ) q − so that y k = h ( y k − ) for 1 ≤ k ≤ L . It is easy to see that h is strictly increasing on (0 , /q ] andstrictly decreasing on [1 /q, < y < x L ≤ /q , so y = h ( y ) < h ( x L ) = x L − . Keep doing this, we get y k < x L − k for any 0 ≤ k ≤ L . In particular, y L < x = M q − δ .Since { y k } is a decreasing positive sequence and y ≤ /
2, then y k ≤ / ≤ k ≤ L . As aresult, it follows from (4.15) and the mean value theorem that for any 1 ≤ k ≤ L , y k ≥ y k − (cid:2) − q − y k − (cid:3) . (4.16)Recalling (4.2) again, y k − ≤ y ≤ E q < q − e , so 1 − q − y k − > − e > . Hence, taking the reciprocal in (4.16) yields1 y k ≤ y k − (cid:2) − q − y k − (cid:3) = 1 y k − + 2( q − − q − y k − < y k − + 10( q − . (4.17)Summing up (4.17) for k from 1 to L , then1 y L < y + 10( q − L. (4.18)Since y L < M q − δ and y = min n , E q o > q ,
23t follows from (4.18) that 1 M q − δ < q + 10( q − L. Thus,
L > q − (cid:16) M q − δ − q (cid:17) . Proof of Theorem 1.4.
We will demonstrate detailed proof for the case n ≥
3, the proof for the case n = 2 is similar and will be briefly mentioned at the end. In the proof below, C and C i ( i ≥
1) willdenote the constants which only depend on n , Ω and d , the values of C and C i may be different indifferent places. But C ∗ and C ∗ i ( i ≥
1) will represent fixed constants which only depend on n , Ω and d . Let M ( t ) be defined as in (1.5). Step 1.
The first part is exactly the same as the second paragraph in the proof of Theorem 1.1,namely we adopt the same notations and the same estimates from (3.3) through (3.10). In particular,we make the assumption (3.9).
Step 2.
In this step, we will find a constant t ∗ > { M k } ≤ k ≤ L such that t k ≥ t ∗ for 1 ≤ k ≤ L . Then in Step 3, a lower bound for Lt ∗ will be derived.Due to the local convexity near Γ , it follows from (2.1) and Corollary 2.2 that I + I ≤ C t k exp (cid:16) − d t k (cid:17) . Because of the assumption (3.9), the above inequality implies I + I ≤ C exp (cid:16) − d t k (cid:17) . (5.1)On the other hand, we estimate I in the same way as (4.5). Now plugging (5.1) and (4.5) into (3.10),then M k ≤ h C ∗ exp (cid:16) − d t k (cid:17)i M k − + C ∗ | Γ | / ( n − M qk , (5.2)for two constants C ∗ and C ∗ . Next if t k is so small thatexp (cid:16) − d t k (cid:17) ≤ C ∗ M k − M k − M k − , (5.3)which is equivalent to M k − h C ∗ exp (cid:16) − d t k (cid:17)i M k − ≥
12 ( M k − M k − ) . Then it follows from (5.2) that M k − M k − M qk ≤ C ∗ | Γ | / ( n − . (5.4)As a summary, the argument so far claims if both (3.9) and (5.3) hold, then M k will satisfy (5.4).24ased on this observation, if we choose δ = 4 C ∗ | Γ | / ( n − (5.5)and define M k to be the solution (if it exists) to M k − M k − M qk = δ , (5.6)then either (3.9) or (5.3) can not hold since otherwise (5.4) will be violated. The invalidity of (3.9)means t k > C ∗ ) . (5.7)On the other hand, due to (5.6), the failure of (5.3) impliesexp (cid:16) − d t k (cid:17) > C ∗ M k − M k − M k − = M qk δ C ∗ M k − ≥ M q − δ C ∗ . (5.8)If M q − δ ≤ C ∗ , (5.9)then the right hand side of (5.8) is smaller than 1. Therefore, (5.8) is equivalent to t k > d (cid:20) ln (cid:16) C ∗ M q − δ (cid:17)(cid:21) − . (5.10)In summary, if (5.9) holds, then it follows from (5.7) and (5.10) that t k ≥ t ∗ , where t ∗ = min (cid:26) C ∗ ) , d (cid:20) ln (cid:16) C ∗ M q − δ (cid:17)(cid:21) − (cid:27) . (5.11)Due to Lemma 4.1, the existence of a solution M k to (5.6) is equivalent to the inequality M q − k − δ ≤ E q . In addition, as long as such a solution exists, M k can be chosen to satisfy M k − < M k ≤ qq − M k − . Thus, the strategy of constructing { M k } is summarized as following. First, define M and δ as in(1.4) and (5.5). Next suppose M k − has been constructed for some k ≥
1, then whether defining M k depends on how large M k − is. ⋄ If M q − k − δ ≤ E q , then we define M k ∈ (cid:0) M k − , qq − M k − (cid:3) to be the solution to (5.6). ⋄ If M q − k − δ > E q , then there does not exist M k > M k − which solves (5.6). So we do not define M k and stop the construction.According to this construction, if { M k } ≤ k ≤ L have been defined, then T k − T k − ≥ t ∗ for any1 ≤ k ≤ L . Therefore, T k ≥ kt ∗ for any 1 ≤ k ≤ L . Since T ∗ is finite, L ≤ T ∗ /t ∗ < ∞ , whichmeans the cardinality of { M k } has to be finite (actually this fact can also be justified by analysingthe construction directly, see Lemma 4.2). So we can assume the constructed sequence is { M k } ≤ k ≤ L for some finite L . 25 tep 3. By Lemma 4.2,
L > q − (cid:16) M q − δ − q (cid:17) . If M q − δ ≤ q , (5.12)then L ≥ q − M q − δ . Combining the assumptions (5.9) and (5.12), if M q − δ ≤ min n C ∗ , q o , (5.13)then T ∗ ≥ Lt ∗ ≥ q − M q − δ min (cid:26) C ∗ ) , d (cid:20) ln (cid:16) C ∗ M q − δ (cid:17)(cid:21) − (cid:27) . (5.14)Denote Y = M q − | Γ | / ( n − . Recalling δ = 4 C ∗ | Γ | / ( n − , we can rewrite (5.13) and (5.14) as Y ≤ min n C ∗ C ∗ , C ∗ q o (5.13 ′ )and T ∗ ≥ C ∗ ( q − Y min (cid:26) C ∗ ) , d (cid:20) ln (cid:16) C ∗ C ∗ Y (cid:17)(cid:21) − (cid:27) . (5.14 ′ )In order to simplify (5.14 ′ ), if Y ≤ min n C ∗ C ∗ , C ∗ C ∗ exp (cid:2) − d ( C ∗ ) (cid:3)o , (5.15)then 2 d ( C ∗ ) ≤ ln (cid:16) C ∗ C ∗ Y (cid:17) ≤ (cid:16) Y (cid:17) = 2 | ln Y | . Taking reciprocal of the above inequality and multiplying by d /
8, we obtain d | ln Y | ≤ d (cid:20) ln (cid:16) C ∗ C ∗ Y (cid:17)(cid:21) − ≤ C ∗ ) . Therefore, (5.14 ′ ) yields T ∗ ≥ C ∗ ( q − Y d | ln Y | = C ( q − Y | ln Y | . Combining the assumptions (5.13 ′ ) and (5.15) together, it suffices to require Y ≤ Y /q , where Y = min n C ∗ , C ∗ C ∗ , C ∗ C ∗ exp (cid:2) − d ( C ∗ ) (cid:3)o
26s a constant which only depends on n , Ω and d . Hence, we finish the proof when n ≥ n = 2, we can argue in the same way as the last paragraph of the proof for Theorem 1.2 tojustify the conclusion. Acknowledgements
The authors appreciate Willie Wong’s suggestions which simplify several proofs in this paper and makethe ideas clearer. The authors also thank the referee for the careful reading and helpful suggestions.
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