$L^2$ solvability and representation by caloric layer potentials in time-varying domains
$L^2$ solvability and representation by caloric layer potentials in time-varying domains
复制标题
$L^2$ 可解性和时变域中热量层势的表示
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
John L. Lewis
中科院分区：
文献类型：
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作者：
S. Hofmann;John L. Lewis
We consider boundary value problems for the heat equation in time varying graph domains of the form Ω = {(x0, x, t) ∈ IR × IRn−1 × IR : x0 > A(x, t) }, obtaining solvability of the Dirichlet and Neumann problems when the data lies in L(∂Ω). We also prove optimal regularity estimates for solutions to the Dirichlet problem when the data lies in a parabolic Sobolev space of functions having a tangential (spatial) gradient, and 1/2 of a time derivative in L(∂Ω). Furthermore, we obtain representations of our solutions as caloric layer potentials. We prove these results for functions A(x, t) satisfying a minimal regularity condition which is essentially sharp from the point of view of the related singular integral theory. We construct counter examples which show that our results are in the nature of “ best possible. ” 1991 Mathematics Subject Classification. Primary 42B20, 35K05. keywords and phrases. heat equation, Dirichlet problem, Neumann problem, layer potentials, timevarying domains, singular integrals, Rellich inequalities. 1 Supported by an NSF Grant 0. Background and Notation. A longstanding problem concerning solvability of the Dirichlet problem for Laplace’s equation in a Lipschitz domain was resolved by B. Dahlberg [D1], who showed that in such domains harmonic measure, dω, and surface measure, dσ, are mutually absolutely continuous, and furthermore, that the Dirichlet problem is solvable with data in L(dσ) (and consequently with data in L, 2− < p <∞). R. Hunt proposed the problem of finding an analogue of Dahlberg’s result for the heat equation in domains whose boundaries are given locally as graphs of functions A(x, t) which are Lipschitz in the space variable. It was conjectured at one time that A should be Lip 1 2 in the time variable, but subsequent counterexamples of Kaufmann and Wu [KW] showed that this condition does not suffice. Motivated in part by work of Strichartz [Stz] on BMO Sobolev spaces, and in part by work of M. Murray [Mu], Lewis and Murray [LM], made significant progress toward a solution of Hunt’s question, by establishing mutual absolute continuity of caloric measure and a certain parabolic analogue of surface measure in the case that A has 1 2 of a time derivative in BMO(IR) on rectangles, a condition only slightly stronger than Lip 1 2 . Furthermore these authors obtained solvability of the Dirichlet problem with data in L, for p sufficiently large, but unspecified. The regularity condition which Lewis and Murray imposed upon A(x, t) (or, to be more precise, an equivalent formulation of it) was shown by the first named author to be necessary and sufficient for L boundedness of the first parabolic Calderón commutator, thus further clarifying the connection between the results of [LM] and those of [D1]. Still, by analogy to [D1], it remained an open problem to treat the case of boundary value problems with L data in the parabolic setting. It is this issue of L solvability that we address here. To be more specific in this paper we study the Dirichlet and Neumann problems for the heat equation in non cylindrical (i.e. time-varying) graph domains. We treat each of these problems in the case that the data belongs to L with respect to a certain projective Lebesgue measure. We 1 also consider regularity estimates for solutions of the Dirichlet problem when the data belongs to a parabolic Sobolev space having a full spatial derivative and one half of a time derivative in L. Existence of our solutions will be obtained by using the method of layer potentials. In addition we shall give an alternate, simpler proof of recent results of the first author [H2] concerning “ smoothing operators of Calderòn type, ” including the caloric single layer potential. We shall study these problems in graph domains of the form Ω = {(x0, x, t) ∈ IR× IRn−1 × IR : x0 > A(x, t) } (0.1) where n ≥ 2 and A(x, t) is Lipschitz in the space variable, uniformly in time, i.e., |A(x, t)− A(y, t)| ≤ β0 |x− y|, x, y ∈ IRn−1, t ∈ IR, (0.2) and where A(x, t) satisfies a certain half order smoothness condition in the time variable. To describe this condition we follow Fabes and Riviere [FR1] and define a half-order time derivative by IDnA(x, t) = ( τ ‖(ξ, τ)‖ Â(ξ, τ) )̌ (x, t) (0.3) whereˆandˇdenote respectively the Fourier and inverse Fourier transforms on IR, and ξ, τ denote, respectively, the space and time variables on the Fourier transform side. Also ‖z‖ denotes the parabolic “ norm ” of z. We recall that this “ norm ” satisfies the non-isotropic dilation invariance property ‖(δx, δt)‖ ≡ δ‖(x, t)‖. Indeed, ‖(x, t)‖ is defined as the unique positive solution ρ of the equation n−1 ∑ i=1 xi ρ2 + t ρ4 = 1. (0.4) The half order smoothness condition in the time variable which we impose upon A is that IDnA ∈ (parabolic) BMO. We recall that parabolic BMO is the space of all locally integrable 2 functions modulo constants satisfying ‖b‖∗ ≡ sup B 1 |B| ∫ B |b(z)−mBb| dz < ∞. (0.5) Here, z = (x, t) and B denotes the parabolic ball B ≡ Br(z0) ≡ {z ∈ IR : ‖z − z0‖ < r} (0.6) where |B| denotes the Lesbegue n measure of B and mBb ≡ 1 |B| ∫ B b(z)dz. We note that |Br(z0)| ≡ cr where c is a constant and d = n + 1 is the homogeneous dimension of IR endowed with the metric induced by ‖ · ‖. We observe that IR so endowed is a space of homogeneous type in the sense of Coifman and Weiss [CW]. Indeed, there is a polar decomposition z ≡ (x, t) ≡ (ρθ1, . . . , ρθn−1, ρθn), dz ≡ dxdt ≡ ρd−1(1 + θ n)dρ dθ (0.7) where θ = (θ1, . . . , θn), |θ| = 1, and dθ denotes surface area on the unit sphere. Throughout this paper Lp(IRn−1), 1 < p < ∞, denotes, as usual, the space of p th power integrable functions f on IRn−1 with norm, ‖f‖p . To explain the significance of the conditions which we have imposed upon A, we recall a result of the first author [H1], which states that ∥∥∥∥∥ ∆− ∂ ∂t , A ∥∥∥∥∥ op ≈ ‖∇xA‖∞ + ‖IDnA‖∗, where ≈ means the two quantities are bounded by constant multiples of each other. Moreover, ‖ · ‖ denotes the operator norm on L2(IRn−1), and ∇x ≡ ( ∂ ∂x1 , . . . , ∂ ∂xn−1 ). (0.8)
