Rodrigo Bañuelos Department of Mathematics Purdue University banuelosmath.purdue.edu

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1 Lifetime of Conditioned Brownian Motion and Related Estimates for Heat Kernels Eigenfunctions and Eigenvalues in Euclidean omains MSRI, January 2, 3, 988 Rodrigo Bañuelos epartment of Mathematics Purdue University banuelosmath.purdue.edu

2 Lecture I Lifetime of Brownian Motion, Conditioned Brownian Motion in Euclidean omains: Some Results, Some Problems. Lecture II Intrinsic Ultracontractivity: Heat kernels, Eigenfunctions, Eigenvalues, Connections to I. Some Results, Some Problems. Basic Question How does the Euclidean geometry of the domain (volume, diameter, inradius) and hyperbolic-quasihyperbolic geometry (growth of hyperbolic and quasihyperbolic distance) affect the above Probabilistic Analytical Objects? Aim To obtain sharp Estimates, often of isoperimetric type.

3 I. Classical Result ( 50 years): R n, vol() <. B t Brownian motion in R n, τ = exit time of B t from. Then That is sup E x (τ ) E 0 (τ ), x = ball of same volume as. = B(0, r), C n r n = vol(), c n = volume of unit ball. Itô applied to f(x) = x 2 gives ( ) 2/n vol(). E 0 (τ ) = n r2 = n c n More ( 20 years): For all x, t > 0, P x {τ > t} P 0 {τ > t}. More general results in C. Bandle [].

4 Proof (Works just as well for symmetric stable!): Set P t (x y) = (2πt) n/2 e x y 2 2t. { } P x {τ > t} = lim P x B jt, j =, 2,..., m m m m = lim... P t/m (x x) P t/m (x j x j )dx... dx m m j=2 m lim... P t/m (x ) P t/m (x j x j )dx... dx m m j=2 { } = lim P 0 B jt, j =, 2,..., m = P 0 {τ > t}. m m (Brascamp-Lieb-Luttinger 974) M. Kac: λ = lim t t log P x{τ > t}. λ λ (Faber Krahn)

5 d (x) = dist(x, c ) = dist(x, ). R = sup x d (x) = radius of largest ball in. R = inradius of. Theorem: Suppose there is a constant C such that ( ) Cap(B(x, 2R ) c ) C R n 2, for all x, n 3 with a similar assumption for n = 2. (Capacity boundary condition.) Then, sup E x (τ ) C 2 R 2 x Proof: ( ) sup P x {B t exits B(x, 2R) before } = P <. x E x (τ ) n (2R ) 2 j=0 P j = 4R2 n P.

6 For C simply connected (S.C), no other assumptions. P 4 π arctan ( 2 ), (Beurling s Thesis). Thus for some universal constant C 0 2 R2 sup E x (τ ) C 0 R 2 x for all S.C. planar domains with R <. Problems : Find the best C 0 and extremal. Or, amongst all S.C. planar with R = find the one (ones) with largest sup E x (τ ). x Problem 2: Let P (2) = sup{p :, S.C., R = }. Find P (2) and the extremal s. It would be of interest if P (2) < 2. Why are Problems and 2 of interest?

7 ϕ = first eigenfunction of in. 2 t τ t τ ϕ (B t τ ) ϕ (x) = ϕ(b s ) db s λ ϕ (B s )ds 0 0 ( τ ) ϕ (x) = λ E x ϕ (B s )ds sup x E x (τ ) λ. 0 For S.C. planar, there is a universal constant e 0 such that e 0 R 2 λ j2 0 2R 2, j0 2 = Problem 3: Find the best e 0 and extremal(s) s. Or, amongst all drums of inradius, find the one(s) that produce the lowest fundamental tone. (Open since before 95; Polya Szegö book.) References to Recent Progress: [7], [9], [24], [26]. In particular, Bañuelos- Carroll [7] disproves a conjecture of R. Osserman and gives a conjecture for a possible extremal drum. Problems 2 and 3 are closely related (as explained in [7]) to famous open problem (open for about 60 years) concerning the schlicht Bloch Landau constant.

8 II. Conditioned Brownian motion: (oob 953, [49]) If h is positive and harmonic in, then the Brownian motion conditioned by h (the oob h process) is determined by the transition probabilities P h t (w, z) = h(w) P t (w, z)h(z) P t (w, z) = transitions for killed B.M. in. The processes has generator L h = 2 + h h. Basic processes: () h(z) = K 0 (z, ξ) = K ξ (z), z, ξ, the Martin (Poisson kernel), (2) h(z) = G (z, w), Green function in. The first gives the Brownian motion conditioned to exit the domain at the point ξ and the second gives Brownian motion conditioned to go from z to w without leaving the domain. The general h processes are mixtures of these. oob used = R n+ + to study boundary behavior of harmonic function. Other beautiful uses in analysis were given in the 70 s by Burkholder Gundy Silverstein.

9 Theorem (M. Cranston, T. McConnell [4] 983) ). (i) For any R 2, sup Ex(τ h ) C 0 area() x h H + () (ii) bounded R 3 and h H + () such that E h z (τ ) =, z. (Question posed by K. L. Chung) Cranston McConnell Basic Estimate: For any R n, For any integer m, define (for any fixed x 0 ) m = {x : 2 m h(x 0 ) < h(x) < 2 m+ h(x 0 )} C m = {x : h(x) = 2 m h(x 0 )}. Then E h x(τ ) 8 m= sup E x (τ m ) x C m An elegant (in Chung s words perspicacious ) proof of the basic estimate can be found in Chung [36]. An analytic proof, which I leaned from T. Wolff and which has been used recently by Aikawa [] [5] and others for various extensions of the basic estimate, can be found in [4].

10 The hyperbolic metric: Assume R 2 is simply connected. The hyperbolic distance between z 0, z is ρ (z 0, z) = ρ 0 (f(z 0 ), f(z)), where 0 is the unit disk and f : 0 conformally. If σ 0 and σ denote the corresponding hyperbolic densities for 0 and, we have ρ 0 (z 0, z) = inf γ 0 σ 0 (γ(t)) γ (t) dt, where the inf is over all curves γ contained in 0 with γ(0) = z 0, γ() = z. with a similar expression for ρ. σ (z) = ( z 2 ) and by the Koebe /4 theorem, σ (z) d (z), where as before d (z) is the distance from z to the boundary of. The minimizing curve is called the hyperbolic geodesic. Note that in the case of 0 we have ( ) + z ρ 0 (0, z) = log log z ( ). d 0 (z) In general this remains true for nice enough domain. That is if the domain is nice enough (so called Hölder domains [3]) we have ρ (z 0, z) C log ( ) + C 2 d (z) We need the hyperbolic metric for arbitrary domains. What is that?

11 Quasi hyperbolic distance: Let F = {Q } = be a Whitney decomposition of : () j= Q j =, (2) Q 0 j Q0 k =, (3) diam(q k ) dist(q k, c ). A Whitney chain from x to y in is a subset {Q, Q 2,..., Q m } of F with x Q, y Q m and Q k Q k+. d W (x, y) = # cubes in minimal chain. = inf γ γ ds d (s) = quasihyperbolic metric. Note: If R 2 is simply connected, this is equivalent to the hyperbolic metric as defined above. Now, a simple application. Assume R n, n 2 has capacity boundary and d W (x 0, x) C d (x) α + C 2, x 0 fixed. Harnack h(x) e cd w(x,x 0 ) h(x 0 ).

12 If = unit disk = {z C: z < } = 0 ([50], [24]), sup Ex(τ h 0 ) = E (τ 0 ) = 4 log x h H + () and for the rectangle R n = [ n, n] [ π/n, π/n], x = n, x 2 = n, E x 2 x (τ Rn ) lim n Ex 2 x (τ Rn ) =. Thus, a long thin rectangles beats the disk. Problem 4: Amongst all domains R 2 of area, find the one(s) that maximizes sup Ex(τ h ). That is, find the best z h H + () constant C in the Cranston-McConnell inequality. We will see shortly that R 2 with area() = and sup Ex(τ h ) <. x h H + () Thus volume is not the determinant factor. Problem 5: Give a geometric characterization for those s for which sup E h x(τ ) <.

13 What is known about problem 4? Theorem: (P. Griffen, T. McConnell, G. Verchota, [50]): Let R 2 be simply connected. Set L (α, β) = Eα(τ β ), α, β. sup L (α, β) = sup L (α, β) () α,β α,β sup L (α, β) < α,β π area(). (2) (3) If is convex symmetric P = perimeter, d = diameter, R = inradius. Then sup L (α, β) Area() 4 α,β π π R (P 2d). In particular, if Then [ R n = n 2, n ] 2 R n E n 2 n 2 [ ] 2,, 2 (τ R ) π 4 πn. Remark: Xu [68] had proved before that there is a universal constant C such that for all convex domains R 2, sup L (α, β) CArea(). α,β

14 Problem 6 (Conjecture) Prove that amongst all planar convex domains of area π, the unit disk minimizes sup L (α, β). α,β That is, R 2 convex, area () = π sup L (α, β) E (τ 0 ). α,β What is known about Problem 5? Necessary and sufficient conditions for finiteness of L(α, β) for domains above the graph of a function in the plane. f = {(x, y): 0 < x <, f(x) < y < }, f non-positive uppersemicontinuous. For z, let A ε (z) = Area of Whitney cubes which intersect the geodesic ( from 2, ) to z and which have length less than ε. 2 A(ε) = sup A ε (z) z Theorem (R. Bañuelos, B. avis [23 ]): sup L (α, β) < if α,β and only if A(ε) C, for all 0 < ε. This theorem easily provides examples of domains of infinite area for which the Cranston-McConnell estimate holds. (The first examples of domains of infinite area with bounded conditional lifetime were given in Xu [68].) More general examples will be given below.

15 The simply connected domain has the Γ condition if there is a fixed point 0 such that for any x there is a curve Γ from 0 to x with the property that dist (z, c ) c dist(x, c ), z Γ. Notice that f has the Γ condition. Theorem(R. Smits [63]: Best known necessary and sufficient condition.) Suppose has the Γ condition. Then sup L(α, β) < α,β if and only if A(ε) C ( big O ), for all 0 < ε. Clearly domains above the graph of a function satisfy the Γ condition. The following are two examples of domains satisfying the conditions of Smits theorem which are not above the graph of a function. The examples are from [63]., (draw a picture). Example. Let k = {(x, y) : cos( π 3 k ) < x <, > y < f k (x) = C k /(x + ) 2 } where C k = (cos(π/3 k + )) 2 /2 k+. Let z j k, j =,..., 2k be the centers of the intervals removed to create the Cantor set. That is, z = 2 ( ) = 2, z 2 = 2 ( ) = 6,.... Set W k = j e 2πizj k k where e iθ = {e iθ z : z }. Finally, set = 0 ( k= W k) where 0 is the unit disk. This domains, (need to check) satisfies the conditions of the theorem

16 and area() 2 k f k (x)dx =. k= Example 2. Let = {(x, y) such that 2 2 < x < and y < x }. Let e iθ be as in the previous example for θ = π 2 4, 3π 4, 5π 4 and 7π 4. From the complex plane we delete the x and y axes except for the portion inside the unit ball. For l = 2,3,... delete the 4l 2 equally spaced line segments between x = l and l, y = l and l, x = l and l + and y = l and l + which are parallel to the axes and don t intersect j e ijπ 4 j =, 3, 5 and 7. The resulting domain satisfies the conditions of the theorem and is not above the graph of a function. The domain is almost the whole plane in the sense that its complement has Lebesgue measure zero Remark: Unfortunately, there are examples of domains of infinite area which do not satisfy the γ condition but which have bounded conditional expected lifetime, [8]. Thus, the geometric necessary and sufficient condition for planar simply connected domains to have the Crason McConnell estimate may be very difficult to find.

17 Key to above results: Conditioned Brownian paths follow hyperbolic geodesics. Theorem (Bañuelos Carroll [8]) Let Q be a geodesic ball of radius. Let z, ξ and γ the geodesic from z to ξants C and C 2 such that C 2 e 2 ρ (γ,q) P ξ x{b t Q for some t < τ } C e 2 ρ (γ,q) ρ (γ, Q) = hyperbolic distance from γ to Q. Also, for any α, β, e ρ(z,γ) dz Eα β (τ ) 4 e ρ(z,γ) dz, 4 where ρ (z, γ) is the hyperbolic distance from the point z γ and γ is the hyperbolic geodesic through α and β. See also A. Ancona [7], [8]. Problem 7: Is there a sharp estimate similar to ( ) for Cartan Hadamard manifolds of curvature between two negative constants?

18 Lecture II Intrinsic Ultracontractivity: Heat kernels, Eigenfunctions, Eigenvalues, Connections to I. Some Results, Some Problems Basic Question: How does the Euclidean geometry of the domain (volume, diameter, inradius) and hyperbolic-quasihyperbolic geometry (growth of hyperbolic and quasihyperbolic distance) affect these Objects? Aim: As in Lecture I, our goal is to describe sharp Estimates

19 Recall: T t = e th a symmetric Markovian semigroup on L 2 (X, dx), (self adjoint, positive preserving, and a contraction on L p, p ) is said to be ultracontractive if T t : L L, t > 0. Equivalent to: T f f(x) = X P t (x, y)f(y)dy where P t (x, y) is a symmetric kernel satisfying 0 P t (x, y) C t x, y, t > 0, C t independent of x and y.

20 Example. The semigroup of Brownian motion in R n. P t (x, y) = x y 2 e 2t (2πt) n/2 (2πt) n/2. Example 2. The Cauchy semigroup in R n. P t (x, y) = c n t ( x y 2 + t 2 ) n+ 2 c n t n (and other symmetric stable processes with 0 < α 2). Example 3. Semigroup T t R n, any domain. of killed Brownian motion in P t (x, y) = irichlet heat kernel for 2 in (2πt) x y 2 e 2t n/2 (2πt) n/2. The Intrinsic Semigroup: Let ϕ be the first eigenfunction for normalized by ϕ 2 dx =, corresponding to λ. Set P t (x, y) = T t f(x) = eλt Pt (x, y) ϕ (x)ϕ (y), P t (x, y)f(y)dµ(y), dµ = ϕ 2 dx, f L 2 (, dµ). Remark: T t to remain forever in. is the semigroup of Brownian motion conditioned

21 (i) P t (x, y) = P t (y, x) (symmetry) (ii) P t (x, y)dµ(y) = (iii) T t f(x) = (i) (iii) T t eλ t ϕ (x) eλ t ϕ (x) P t (x, y)ϕ (y)dy = P t (x, y)f(y)ϕ (y)dy = eλ t ϕ (x) T t (fϕ )(x) is a symmetric Markovian semigroup on L 2 (, dµ). efinition (E.B. avies B. Simon [44]): {T t } is said to be intrinsically ultracontractive (IU) if the intrinsic semigroup { T t } is ultracontractive. That is, Or, T t : L (, dµ) L (, dµ), t > 0. P t (x, y) a t e λ t ϕ (x)ϕ (y), t > 0, a t independent of x, and y. In fact, a t = T t, and a t c t for some non increasing (in t) c t. We will say that is intrinsically ultracontractive (IU) in this case.

22 Consequences of IU Theorem (Well known and easy; see for example Smits [64]): Suppose R n is IU. C > 0 such that t >, ( ) e (λ 2 λ )t sup P t (x, y) Ce (λ 2 λ )t x,y λ 2 = second irichlet eigenvalue. (λ 2 λ ) = spectral gap. (Remark: Ture for any intrinsic ultracontractive semigroup.) Corollary: If R n is IU then (i) ϕ (x) ϕ (y)h(y)dy < h(x) sup x h H + () (ii) sup x h H + () E h x(τ ) < (iii) lim t e λ t P h x {τ > t} = ϕ (x) h(x) In particular, ϕ (y)h(y)dy λ = lim t t log P x h {τ > t}. All follows from ( ) and P h x {τ > t} = h(x) P t (x, y)h(y)dy.

23 Indeed, let t 0 > be such that for all t t 0, and all x, y, 2 e λ t 0 ϕ (x) h(x) 2 eλt Pt (x, y) ϕ (x)ϕ (y) h(y)ϕ (y)dy h(x) 3 2 P t 0 (x, y)h(y)dy. For t t 0 Px h {τ > t} = h(x) P 3 2 e λ t sup x t (x, y)h(y)dy ( ) ϕ (x) ϕ (y)h (y)dy h(x) Remark. There are many other interesting consequences of IU. Here are just two more examples:. (ϕ n, λ n ) = n eigenfunction, n eigenvalue. IU e λnt ϕ n (x) = P t (x, y)ϕ n (y)dy a te λt ϕ(x) Thus for all n, ϕ n (x) a t e λ t e λ nt ϕ (x), n, t > 0. ϕ n (y) ϕ (y)dy. 2. IU P x {τ > t} = P t (x, y)dy a t ϕ(x)e λ t.

24 Problem 8 (The van den Berg 983 Conjecture on irichlet spectral gaps [33]): Let R n be a convex domain of diameter d. Then 3π 2 2d < λ 2 2 λ with the lower bound approached for thin rectangles. First result: SWYY [62] 985: π 2 8d 2 < λ 2 λ. Time to equilibrium: For ε > 0, define { } Tε = inf t > 0: sup P t (x, y) ε x,y Brownian Theorem (R. Smits [64]): For R n convex diameter d, T ε C + 2d2 π 2 log ε. Problem 8 : Amongst all convex domains of fixed diameter the rate to equilibrium is smallest (takes longer time to reach it) for thin rectangles. Remark: equilibrium. For fixed area, there are arbitrarily large rates to For more on this and the intuition comparing time to equilibrium for Brownian motion conditioned to remain forever in to that of reflected Brownian motion in, see R. Smits[64] Spectral gaps and rates to equilibrium on convex domains. L. Saloff-Coste[60]: Precise estimates on the rate at which certain diffusions tend to equilibrium (preprint)..

25 Problem 9: (The hot spots conjecture of J. Rauch (974)). Let R n. Let ψ 2 be any eigenfunction for the Neumann Laplacian in corresponding to µ 2 (the first nontrivial eigenvalue). Then ψ 2 has its maximum (and minimum) on the boundary and only on the boundary of. Progress: Not true for arbitrary domains: (Werner,... etails in a coming paper of Burdzy and Werner). Bañuelos Burdzy (997 preprint): True for several types of planar convex domains. Jerison Naridashvili: Claim to have proved it for all planar convex domains. (No paper yet!) Problem 0 (Conjecture) Prove that the Brownian motion conditioned to remain forever in, convex in R n, also has the hot spots property. Namely, that ϕ 2 /ϕ also has its maximum (and minimum) on the boundary and only on the boundary of.

26 Problem : Give a geometric characterization of IU. Sufficient Conditions: Let f : [0, ] (, 0]. f = {(x, y): 0 < x <, f(x) < y < }. First Group:.. B. avies B. Simon (984) [44]: f Lipschitz f IU Fabes Garofalo Salsa (986) : f Lipschitz f IU eblassie (987) [48]: f Lipschitz with small constant IU C. Kenig J. Pipher (988) [52]: f Lipschitz IU 5.. Burgess avis (990) [47]: f bounded f IU R. Bass K. Burdzy (990) [3]: f L p, p > f IU R. Bañuelos (990) [3]: R n with capacity boundary condition and d w (x 0, x) C d (x) + C 2, 0 < α < 2. α () Then IU. (2) α 2, s.t. d W (x 0, x) and not IU. d (x) α Theorem (Bañuelos avis [23]). f is IU if and only if lim A(ε) = 0. ε 0 Problem 2 (Conjecture): Let R 2 be s.c. with the Γ condition. is IU lim ε 0 A(ε) = 0.

27 Probabilistic characterization of IU (Bass Burdzy [3]): IU For each t > 0 compact K t such that for all x, P x {B t K t τ > t} > a t where a t is independent of x. (This was used in Bass Burdzy [3] and Bañuelos avis [23].) Analytic: (Log Sobolev, E. B. avies [43]) T t = e ta. Suppose ε > 0 β(ε) such that for all u 0, u om(a), u 2 log udx εq(u, u) + β(ε) u u 2 2 log u 2 2, X Q(u, u) = Au, u. Then T t u L e CM(t) u M(t) = t t ( ε Converse true with β(ε) = M ) 0 β(ε)dε <.

28 Since P t (x, y) For T t (2πt) n/2 we have u 0, u C 0 (), = e ta, A = 2 + ϕ ϕ and Q(u, u) = Au, u = 2 u 2 ϕ 2 (x)dx. Suppose we can prove: 0 u C 0 () (2) u 2 log dx ε u 2 dx + β(ε) u 2 L ϕ 2 (dx). Adding () and (2) gives for all u C 0 (), u 2 log ( ) u dx 2ε u 2 dx + (cn log ϕ ε + β(ε) + 2) u L 2 (dx) + u 2 L 2 (dx) log u 2 L 2 (dx).

29 Need to have good estimates on log. Often Harnack is ϕ (x) enough: ϕ (x) > ϕ(x 0 )e Cd w(x,x 0 ) or log ( ) ϕ (x) C d w (x, x 0 ) + C 2. Suppose: log (ab ap u 2 log d W (x, x 0 ) ϕ (x) C d (x) + C α 2 C ) p + bq q ϕ (x) dx C ε C ε C 3 d (x) α + C 4, 0 < α < 2. ε d(x) 2 + C 2 ε α 2 α. u 2 d(x) dx + C 2 2 β(ε) (u) 2 dx u 2 dx + C 2 β(ε) u 2. (By Ancona [6], provided has capacity boundary condition.) Remark: The log-sobolev approach to (IU) leads to many interesting and difficult questions concerning the sharp decay at the boundary of irichlet eigenfunctions. There is now a large body of research concerning this topic. Here we mention only two of the earlier results and give references to many of the more recent ones. First, a non-sharp result.

30 Theorem (B. avies B. Simon [44]): Let f : C 2 [0, ] R, f 0, f L (R + ), f M, f (x)/f(x) 0 as x. Let ϕ (x, y) be the first eigenfunction for f = {(x, y) : f(x) < y < f(x)}. Then C e C 2 x f(x) ϕ (x, 0) C 3 e C 4 x f(x). Here are the two examples of sharp results: Theorem (R. Bañuelos B. avis [6]): Suppose f (x) M, f 0, f L. C e π 2 x dt f(t) π 24 x f (t) 2 f(t) dt ϕ (x, 0) C 2 e π 2 x dt f(t). Theorem: (Bañuelos [20]) R 2, area() <, s.c. Fix z 0. C 2 e 2ρ (z 0,z) ϕ (z) C e 2ρ (z 0,z). Remark: The last theorem was motivated by the fact that for these domains ρ ((, 0), (x, 0)) π 4 x ds f(s). Many other sharp eigenfunction estimates in: Bañuelos [20], Lapidus Pang [55], Bañuelos van den Berg [2], Lindeman Pang Zhao [56], Cranston Li [40], van den Berg Bolthausen [32]. Gradient estimates: Cranston-Zhao [9], Bañuelos-Pang [27], Athanasopoulos- Caffarelli-Salsa [0],....

31 Estimates for eigenfunctions, Green functions, Poisson kernels and Intrinsic Ultracontractivity for symmetric stable processes can be found in the recent work of Zhen Qing Chen and Renming Song, [33], [34]. Neumann: Julian Edwards (Florida International University) Eigenfunction decay for the Neumann Laplacian on Horn-like domains, (preprint) and references there.

32 References Note: Not included in this list are many references to applications of conditioned Brownian motion to boundary behavior of harmonic and caloric functions (Martin boundary, boundary Harnack principle, etc.), nor to conditional gage theorems and their applications to Harnack inequalities and potential theory for Schrödinger operators. Some of these topics will be discussed in the lectures of Professor Richard Bass. [] H. Aikawa, ensities with the mean value property for harmonic functions in Lipschitz domains, Proc. AMS (997) [2] H. Aikawa, Norm estimates of Green operators, perturbation of green functions, Cranston-McConnell inequality and integrability of superharmonic functions, (preprint). [3] H. Aikawa, Norm Estimates of Green operators, perturbation of Green functions and integrability of superharmonic functions, (preprint). [4] H. Aikawa, Generalized Cranston McConnell inequalities for discontinuous superharmonic functions, to appear in Potential Analysis. [5] H. Aikawa and M. Murata, Generalized Cranston McConnell inequalities and Martin boundaries on unbounded domains, J.

33 Analyse math. 69 (996) [6] A, Ancona, On strong barriers and an inequality of Hardy for domains in R n, J. London Math. Soc., 34 (986) [7] A, Ancona, Negatively curved manifolds, elliptic operators and the martin boundary, Ann. of Math., 25 (987) [8] A, Ancona, Théorie du potential sur les graphes et les variétés, Lecture Notes in Math 427 (990) 2. [9] A, Ancona, First eigenvalues and comparison of Green s functions for elliptic operators on manifolds or domains, J. Analyse Math., (to appear). [0] I. Athanasopoulos, L. Caffarelli and S. Salsa, Caloric functions in Lipschitz domains and regularity of solutions to phase transition problems, Ann. of Math., 43 (996) [] C. Bandle, Isoperimetric Inequalities and Applications, Pitman, London 980. [2] R. Bañuelos, On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains, Prob. Th. Rel. Fields 76 (987) 29. [3] R. Bañuelos, Intrinsic ultracontractivity and eigenvalue estimates for Schrödinger operators. J. Functional Anal. 00 (99) [4] R. Bañuelos, Lifetime and heat kernel estimates in nonsmooth domains. In: Partial ifferential Equations with Mini-

34 mal Smoothness and Applications, Springer, New York, 992. [5] R. Bañuelos and K. Burdzy, On the hot spots conjecture of J. Rauch, (preprint). [6] R. Bañuelos and B. avis, Sharp Estimates for irichlet eigenfucntions in horn-shape regions. Comm. Math. Phy. 50(992) (See also correction in Comm. Math. Phy V 62) [7] R. Bañuelos and T. Carroll, Brownian motion and the fundamental frequency of a drum, uke Math. J., 75 (994) [8] R. Bañuelos and T. Carroll, Conditioned Brownian motion and hyperbolic geodesics in simply connected domains, Mich. Math. J., 40 (993) [9] R. Bañuelos, T. Carroll and E. Housworth, Inradius and integral means for Green s functions and conformal mappings, to appear Proc. AMS. [20] R. Bañuelos, Sharp L 2 bounds for eigenfunctions in simply connected planar domains, J. iff. Equations 25 (996) [2] R.Bañuelos and M. van den Berg, irichlet eigenfunctions for horn-shape regions and Laplacians on cross sections. J. London Math. Soc. 53 (996)

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