The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition

We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.


Introduction
We consider the problem x ∈ Ω := {x ∈ R N : |x| > 1}, t > 0, where N ≥ 3, ∆ is the N -dimensional Laplacian (in x), ν is the exterior normal vector to ∂Ω, ∂ t := ∂/∂t, ∂ ν := ∂/∂ν, and (ϕ, ϕ b ) is a pair of measurable functions in Ω and ∂Ω, respectively. Our aim is to study the convergence as ε → 0 of the solution u ε to the solution u of the problem (1.2) For bounded domains this convergence was established in [5] and for the half-space Ω = R N + := R N −1 × R + , N ≥ 2, in [2]. More recently, the following four theorems on the rate of this convergence have been proven in [3].
We see that for the half-space the rate does not depend on the dimension, and we obtain the same rate ε 1/2 also for the exterior of a ball in R 3 , which is a very different domain. The main motivation of this paper is the natural question whether or not other rates may occur. We show that for R N \ B 1 (0) the rate depends on N .
Before we formulate our main results, we introduce some notation. Let Γ D = Γ D (x, y, t) be the Dirichlet heat kernel on Ω. Define for any measurable function φ in Ω. Let P = P (x, y) be the Poisson kernel on B = B(0, 1) := {x ∈ R N : |x| < 1}, that is where c N is a constant to be chosen such that P (x, ·) L 1 (∂B) = 1 for x ∈ B (see (2.28) in [6]). Then P = P (x, y) satisfies as a function of x where δ y is the Dirac measure on ∂B = ∂Ω at y. We denote by K = K(x, y) the Kelvin transform of P as a function of x with respect to B, that is Then it follows from (1.3) and (1.4) that K = K(x, y, t) as a function of x and t satisfies For any nonnegative measurable function ψ on ∂Ω and t > 0, we define We formulate the definition of a solution of (1.1) by the use of the two integral kernels Γ D and K. For simplicity, let ϕ b = ϕ b (x) and g = g(x, t) be continuous functions in ∂Ω and ∂Ω × (0, ∞), respectively. Then the function can be defined for x ∈ Ω and t > 0, and it is a classical solution of the Cauchy problem for the Laplace equation with a nonhomogeneous dynamical boundary condition (1.8) It follows from (1.6) and (1.7) that (1.10) (1.11) x ∈ ∂Ω, t > 0, and w ε is defined as in (1.7) with g ε instead of g, then it follows from (1.8), (1.9) and (1.11) that (1.14) Furthermore, the function u ε := v ε + w ε is a classical solution of (1.1). Motivated by this observation, we formulate the definition of a solution of (1.1) via problem (1.12).
There exists a nonnegative function ϕ ∈ L ∞ (Ω) with (1.15) such that the following holds: Let K be a compact set in Ω × (0, ∞). Then there exists C > 0 such that for any ε ∈ (0, 1) the corresponding solution u ε of (1. (1.20) The rest of this paper is organized as follows. In Section 2 we recall some properties of the Dirichlet heat kernel Γ D and the kernel K. Furthermore, we prepare some useful lemmata. In Section 3, modifying the argument as in [2], we give a proof of Theorem 1.5. In Section 4 we prove Theorem 1.6.

Preliminaries
In this section we recall some properties of the Dirichlet heat kernel Γ D on the exterior Ω of the ball B(0, 1) and obtain some estimates of integral operators S 1 (t), S 2 (t) and F .
By Lemma 2.3 we obtain the following lemma.
Now we are ready to prove Theorem 1.5.
Proof of Theorem 1.6. Let K be a compact set in Ω such that K ⊂ K * × [t 1 , t 2 ] for some compact set K * ⊂ Ω and 0 < t 1 < t 2 < ∞, and let ϕ be as in Lemma 4.2. Then, applying Lemma 4.1 and Lemma 4.2 to τ := t 1 , we see that there exists C * > 0 such that for all (x, t) ∈ K. This implies (1.20), and the proof of Theorem 1.6 is complete.