Exit Time, Green Function and Semilinear Elliptic Equations

Let D be a bounded Lipschitz domain in R n with n ≥ 2 and τ D be the ﬁrst exit time from D by Brownian motion on R n . In the ﬁrst part of this paper, we are concerned with sharp estimates on the expected exit time E x [ τ D ] . We show that if D satisﬁes a uniform interior cone condition with angle θ ∈ ( cos − 1 ( 1 / p n ) , π ) , then c 1 ϕ 1 ( x ) ≤ E x [ τ D ] ≤ c 2 ϕ 1 ( x ) on D . Here ϕ 1 is the ﬁrst positive eigenfunction for the Dirichlet Laplacian on D . The above result is sharp as we show that if D is a truncated circular cone with angle θ < cos − 1 ( 1 / p n ) , then the upper bound for E x [ τ D ] fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation ∆ u = u p in D , p ∈ R , that vanish on an open subset Γ ⊂ ∂ D decay at the same rate as ϕ 1 on Γ .


Introduction
Let n ≥ 2 and X a Brownian motion on n . Suppose D is a bounded Lipschitz domain in n . Denote by τ D := inf{t : X t / ∈ D} the first exit time from D by X , and X D the subprocess obtained from X by letting it be killed upon exiting D. It is well-known that X D has a jointly continuous Green function G D (x, y) on D × D except along the diagonal: for every Borel function f ≥ 0 on D. The right hand side of the above display will be denoted as G D f (x). The infinitesimal generator of X D is the Laplacian 1 2 ∆ on D with zero Dirichlet boundary condition. This Dirichlet Laplacian on D has discrete spectrum 0 > −λ 1 > −λ 2 ≥ −λ 3 ≥ · · · . Let ϕ 1 be the positive eigenfunction corresponding to the eigenvalue −λ 1 normalized to have D ϕ 1 (x) 2 d x = 1. The random exit time τ D , the Green function G D and the first eigenfunction ϕ 1 are fundamental objects in probability theory and analysis. In many occasions, one needs to estimate x [τ D ] = G D 1(x). When D is a bounded C 1,1 domain, using the known two-sided Green function estimate on G D (see Lemma 2.1 below), one can easily deduce (1.1) Here, and throughout the paper, δ D (x) denotes the Euclidean distance between x and D c , and for two positive functions f , g, the notation f ≍ g means that there are positive constants c 1 and c 2 so that c 1 g(x) ≤ f (x) ≤ c 2 g(x) in the common domain of definition for f and g. Since ϕ 1 (x) = λ 1 G D ϕ 1 , using the two-sided Green function estimate on G D again, we have ϕ 1 Thus on a bounded C 1,1 domain D, While it is clear that in general (1.1) no longer holds on bounded Lipschitz domains, it is reasonable to ask if (1.2) remains true for a bounded Lipschitz domain D. We show in this paper that, in fact, (1.2) holds on any bounded Lipschitz domain D with Lipschitz constant strictly less than 1/ n − 1, and fails on some Lipschitz domain with Lipschitz constant strictly larger than 1/ n − 1. In fact, our result is somewhat stronger than that. To state it, let us recall the following notions. For θ ∈ (0, π), let (θ ) be the truncated circular cone in n with angle θ , defined by (θ ) := x ∈ n : |x| < 1 and x · e 1 > |x| cos θ , (1.3) where e 1 := (1, 0, · · · , 0) ∈ n . We say that a bounded Lipschitz domain D satisfies the interior cone condition with common angle θ , if there is some a > 0 such that for every point x ∈ ∂ D, there is a cone ⊂ D with vertex at x that is conjugate to a (θ ); that is, is the cone with vertex at x that is obtained from (θ ) through parallel translation and rotation.
The result below states that (1.2) holds for bounded Lipschitz domains in n satisfying the interior cone condition with common angle strictly larger than cos −1 (1/ n). This includes as special case bounded Lipschitz domains in n whose Lipschitz constant is strictly less than 1/ n − 1.
Theorem 1.1. Let D be a bounded Lipschitz domain in n with n ≥ 2 satisfying the interior cone condition with common angle θ ∈ cos −1 (1/ n), π . Then there is a constant c ≥ 1 such that for every x ∈ D.
(ii) A variant but equivalent form of Theorem 1.1 has also been obtained independently by M.
Bieniek and K. Burdzy [4] using a different method. In [4], it is shown under the same condition of Theorem 1.1 that for a fixed compact set A ⊂ D with non-empty interior, u(  .2)).
(iv) The above result is in sharp contrast to the situation when X is replaced by a rotationally symmetric α-stable process Y on n with 0 < α < 2. For any bounded Lipschitz domain D ⊂ n and for the process Y , [17,Theorem 8] shows that for some compact subset K of D. By a similar argument as that for Lemma 3.2, one can show that ϕ 1 ≍ ϕ 0 on D. Thus for any rotationally symmetric α-stable process Y on n with 0 < α < 2, (1.4) holds on every bounded Lipschitz domain D.
We now consider the positive solutions of the following semilinear equation on a bounded Lipschitz domain D ⊂ n : where p ∈ , and φ is a non-negative continuous function on ∂ D that vanishes on an open subset Γ ⊂ ∂ D. If φ > 0 then existence of positive solutions is standard and we briefly review the vast literature at the end of this section. If φ vanishes on a portion of the boundary we show that there is a p 0 ∈ such that above Dirichlet problem has a positive solution if p ≥ p 0 and does not if p < p 0 . We investigate whether positive solutions of (1.5) go to zero at the same rate as ϕ 1 on Γ.
The primary motivation for such a study comes from the BHP for positive harmonic functions.
for every x, y ∈ D ∩ B(z, r).
(1.6) From Remark 1.2(i) and the above result it is clear that all harmonic functions that vanish continuously on a part of the boundary go to zero at the same rate as ϕ 1 on that part. One quickly observes that positive solutions of (1.5) are subharmonic functions on D, and in general, subharmonic functions need not go to zero on Γ ⊂ ∂ D at the same rate as ϕ 1 . To state our results precisely, we need some definitions.

Definition 1.4.
We say that u ∈ C(D) is a mild solution to (1.5) if where h ∈ C(D) is a harmonic function in D satisfying h = φ on ∂ D.
Here C(D) denotes the space of continuous functions on D. It is easy to see that a function u ∈ C(D) is a mild solution of (1.5) if and only if it is a weak solution of (1.5) (cf. [6]). We consider the following classes of functions.     Assume, in addition, that D satisfies for every x ∈ D. (1.7) Then there exists p 0 ∈ (−∞, 0) such that We conjecture that for p ∈ (p 0 , 1), = p H p + . Note that Theorem 1.1 gives a sufficient condition on a Lipschitz D to satisfy condition (1.7). We can say more when D is a bounded C 1,1 -domain. Recall that a bounded domain D ⊂ n is said to be C 1,1 -smooth if for every point z ∈ ∂ D, there is r > 0 such that D ∩ B(z, r) is the region in B(z, r), under some z-dependent coordinate system, that lies above the graph of a function whose first derivatives are Lipschitz continuous.
where a i j has continuous derivatives on D (i.e. it is C 1 (D)), and A(x) = (a i j (x)) is a symmetric matrix-valued function that is uniformly bounded and elliptic. Then by Theorem 3.3 of Grüter and Widman [16], the Green function G D (x, y) of in D satisfies the following estimate where c > 0. On the other hand, we know from Lemma 4.6.1 and Theorem 4.6.11 of Davies [8] that It is well known that Harnack and boundary Harnack principles hold for and the Green function Hence by a similar argument as that in Bogdan [5], we conclude that the estimate (2.1)-(2.2) hold for the Green function G D of in D . Finally, by imitating the proof of Theorem 1.6 we can obtain the result for as well.
is the first coordinate of x = (x 1 , . . . , x n ). Now u ∈ p + clearly and due to the one dimensional nature of this example one can establish u ∈ p H . This suggests that Theorem 1.6(ii) could be replaced by: However, we were not able to generalize the above example to general bounded C 1,1 -domains D.
(iii) We have stated all our results for solutions of the equation (1.5). However if we assume that f (u) ≍ u p then the proofs of our main results can be suitably modified to yield the same quantitative behavior for solutions of the equation (1.9) (iv) When D is a bounded C 1,1 domain in n , and p ≥ 1, the result p + = is established in [6] (in fact the result is proved to be true for any bounded regular domain D), while for −1 < p < 1, There is a wealth of literature on the semilinear elliptic equations. Under certain regularity conditions on D ⊂ n and φ, where n ≥ 3, the existence of solutions to (1.9), bounded below by a positive harmonic function, was established in [6] when f satisfies the condition that −u ≤ f (u) ≤ u for |u| < ǫ for some ǫ > 0, and in [1] the case when 0 ≤ f (u) ≤ u −α for some α ∈ (0, 1) was resolved.
The equation ∆u = u p in D with u = φ on ∂ D has also been widely studied. For 1 ≤ p ≤ 2, it has been studied probabilistically using the exit measure of super-Brownian motion (a measure valued branching process), by Dynkin, Le Gall, Kuznetsov, and others [11; 12; 18]. Properties of solutions when f (u) = u p , p ≥ 1, with both finite and singular boundary conditions have also been studied by a number of authors using analytic techniques [2; 13; 15; 19].
Our proofs of Theorem 1.5 and Theorem 1.6 employ implicit probabilistic representation of solutions of (1.5) and Schauder's fixed point theorem. We emphasize that our main new contributions in these two theorems are for subcases (ii)-(iii), that is for p < 1. Some part of the results that address the case p ≥ 1 (Theorem 1.5(i) and Theorem 1.6(i)) may be known (cf. [11; 12; 18]). However, the proofs we provide for these results appear to be more elementary than those available in the literature.
In the sequel, we use C ∞ (D) to denote the space of continuous functions in D that vanish on ∂ D. For two real number a and b, a ∧ b := min{a, b} and a ∨ b := max{a, b}. We will use B(x, r) to denote the open ball in n centered at x with radius r.
The rest of the paper is organized as follows. In the next section we present some estimates on the Green function which are required for the proof of Theorem 1.1, Theorem 1.5 and Theorem 1.6. In Section 3, we prove Theorem 1.1 and show that the condition on the common angle is sharp (Theorem 3.3). Finally in Section 4 we prove Theorem 1.5 and in Section 5 we prove Theorem 1.6.

Green function estimates
Recall that a bounded domain D ⊂ n is said to be Lipschitz if there are positive constants r 0 and r so that for every z ∈ ∂ D, there is an orthonormal coordinate system C S z and a Lipschitz function y n ) ∈ C S z : | y| < r and y n > F z ( y 1 , · · · , y n−1 ) .
The constants (r 0 , λ) are called the Lipschitz characteristics of D. If each F z is a C 1 function whose first order partial derivatives are Lipschitz continuous, then the Lipschitz domain D is said to be a C 1,1 domain.
We begin with an estimate for Green function in C 1,1 domains.
For the rest of this subsection we will assume that D is a bounded Lipschitz domain in n with n ≥ 2.
Recall that we defined ϕ( On the other hand, by [7, Lemma 6.7], for every c 1 , there is a constant c 2 > 0 such that From these, inequality (2.4) can be proved in the same way as the proofs for [5, Proposition 6 and Theorem 2].

Exit time and boundary decay rate
In this section we prove Theorem 1.1, and the sharpness of the requirement on the common angle in its statement. To prove Theorem 1.1 we will need the following result.
Recall that ϕ(x) := G D (x, x 0 ) ∧ 1. The following lemma is known to the experts, but we provide its proof for completeness.

Lemma 3.2. Suppose that D is a bounded Lipschitz domain in n with n ≥ 2.
There is a constant c ≥ 1 such that Proof. It is well-known that D is intrinsic ultracontractive (cf. [8]) and so for every t > 0, there is a constant c t ≥ 1 such that For the definition of intrinsic ultracontractivity and its equivalent characterizations, see Davies and Simon [9]. By (3.4), we have Thus there is a constant c 1 > 0 such that for every x ∈ D.
This proves the Lemma.
Proof of Theorem 1.1. Since ϕ 1 is bounded on D, we have To establish the upper bound on When n ≥ 3, we have by (2.3), Thus we have When n = 2, we have by (2.4), The theorem is now proved in view of Lemma 3.2.
The following result says that Theorem 1.1 is sharp.

Remark 3.4.
Note that the circular cone (θ ) with angle θ = cos −1 (1/ n) has Lipschitz constant 1/ n − 1 at its vertex. So if D is a bounded Lipschitz domain in n with Lipschitz constant strictly less than 1/ n − 1, then D satisfies interior cone condition with common angle θ ∈ cos −1 (1/ n), π . We point out that this is only a sufficient condition. The aforementioned interior cone condition can be satisfied in some bounded Lipschitz domains with Lipschitz constant larger than 1/ n − 1. A smooth domain with an inward sharp cone is such an example.

Semilinear elliptic equations
We start with some technical lemmas for general bounded domains and then proceed to present the proof of Theorem 1.5.
Let Ω be the set of continuous functions from [0, ∞) to n , and let X t (ω) = ω(t), t ≥ 0. Endow Ω with the Borel sigma-field (Ω). Letˆ t denote the canonical sigma-field σ{ω(s) : 0 ≤ s ≤ t}. For x ∈ n let x denote the probability measure on (Ω, (Ω)) under which X is a Brownian motion starting from x. Let { t } denote the usual augmentation of the filtration {ˆ t } with respect to the family of measures { x , x ∈ n } (see p. 45 of [20]). For a positive harmonic function h in D and x ∈ D, we denote by h x the h-transform of x under h (see [3] or [7]). Let x ( h x ) denote expectation with respect to x (respectively, h x ). For any set A ⊂ n we denote The following is a well known result. We provide a proof here for the reader's convenience. This result in fact holds for more general potentials q ≥ 0, for example when q is in some Kato class (see [7]).
The converse is true if q is bounded.
Proof. The proof is along the lines of [6]. Suppose that v is given by (4.1). Then for x ∈ D, by the Markov property of X , For the converse, assume q ≥ 0 is bounded. Suppose now that v satisfies (4.2). Then v is a weak solution to the following equation (cf. As q ≥ 0 is bounded, it is well known that solutions to equation (4.3) are continuous on D and C 1 in D (see, e.g., [14]). Furthermore, the solution of (4.3) enjoys the maximum principle and therefore is unique. This proves the Lemma.

Lemma 4.2.
There exists a constant γ = γ(D) > 0 such that for every h ∈ + and p > 1 − 2γ, we have where C 1 depends only on h, p and D.
Proof. We will be mainly using the notation in [3], page 200-201. Let l k = {x : h(x) = 2 k } for any k ∈ . Note that there exists k 0 such that l k = for k ≥ k 0 . Define S −1 = 0 and let From [3], one has that: (a) there exists a constant γ(D) > 0 such that for any k, v k ≤ c 0 2 2kγ(D) where m denotes the Lebesgue on d . Therefore the family of functions {G D (x, ·)h(·) p , x ∈ D} is uniformly integrable over D. Since D is Lipschitz domain D, for each y ∈ D, it is known (see [7]) that x → G D (x, y) can be extended to be a continuous function on D\{ y} by setting G D (x, y) = 0 for x ∈ ∂ D. So the above particularly implies that the function x → D G D (x, y)h( y) p d y is continuous on D and vanishes on ∂ D. On the other hand, by using the triangle inequality, the family of functions |G D (x, ·) − G D ( y, ·)| h(·) p : x, y ∈ D is uniformly integrable on D. Therefore the function (x, y) → For each g ∈ B h,p , as |g| ≤ h p , the functions in B h,p are continuous in D, uniformly bounded, and converge uniformly to zero as x → ∂ D. For any x, y in D and g ∈ B h,p , is relatively compact in C ∞ (D). Since h ∈ + and by Lemma 4.1, we have On the other hand, for any u ∈ Λ and x ∈ D, since u p−1 ≤ h p−1 , where the last inequality follows from Lemma 4.2. By continuity we have T (u)(x) ≥ exp(−C 1 )h(x) on D. Thus we have shown that An application of the Dominated Convergence Theorem implies that T (u n )(x) → T (u)(x) for all x ∈ D and by (4.7), the convergence holds in the uniform norm. We have shown that T : Λ → Λ is continuous. (4.9) Hence for p ≥ 0, by assumption (1.7), Observe that for any q ∈ , (4.10) For h ∈ + , define Fix c 1 > 0 and a corresponding c 2 > 0. Note that for a suitable constant c 3 > 0, which depends only on By If q is chosen sufficiently large, the last expression above is unbounded over D. This proves (4.13) and thus p 0 > −∞.
For every p > p 0 , using (4.10) and a fixed point argument very similar to the one used in (i), we have p H = for p > p 0 . Note that the only modifications in the fixed point argument of (i) are the following.
(a) We have min{1, e c 1 (p−1) }u p−1 T (u) ∈ B h,p for u ∈ Λ rather than u p−1 T (u) ∈ B h,p in the case of p ≥ 1.
(b) u p−1 ≤ h p−1 for the case of p ≥ 1 is now replaced by (4.10) with q = p − 1.
(iii) Now we show that for every p < p 0 , Then This contradicts the definition of p 0 in (4.12). Hence p + = for every p < p 0 .

C 1,1 domain case
In this section we give a proof of Theorem 1.6.
Proof of Theorem 1.6. (i) As any bounded C 1,1 domain satisfies the hypothesis of Theorem 1.5, the results follows directly from Theorem 1.5(1).
(ii). It is well known that for a bounded C 1,1 domain D, the Euclidean boundary ∂ D is the same as the minimal Martin boundary for ∆ in D. So for any h ∈ + , there is a finite positive measure µ on ∂ D such that where K D (x, z) is the Martin kernel for ∆ in D. It is a direct consequence of (2.1) and (2.4) that |x − z| n for x ∈ D and z ∈ ∂ D. Note that for each fixed z ∈ ∂ D, x → K D (x, z) is a positive harmonic function in D.