ABSTRACT
Let Ω be a smooth convex unbounded domain in RN . In this chapter we consider both the parabolic problem with homogeneous Neumann boundary conditions
Dtu(t, x)−Au(t, x) = 0, t > 0, x ∈ Ω,
∂u
∂ν (t, x) = 0, t > 0, x ∈ ∂Ω,
u(0, x) = f(x), x ∈ Ω
(12.0.1)
and the elliptic problem λu(x)−Au(x) = f(x), x ∈ Ω,
∂u
∂ν (x) = 0, x ∈ ∂Ω,
(12.0.2)
when f ∈ Cb(Ω). Here, A is given, as usual, by
Aϕ(x) = N∑
qij(x)Dijϕ(x) + N∑ j=1
bj(x)Diϕ(x) + c(x)ϕ(x), x ∈ Ω,
(12.0.3)
on smooth functions and ν(x) denotes the outer unit normal to ∂Ω at x ∈ ∂Ω. Under suitable assumptions on the coefficients of the operator A, we show
that the Cauchy-Neumann problem (12.0.1) admits a unique classical solution u (i.e., a function u ∈ C([0,+∞)×Ω)∩C0,1((0,+∞)×Ω)∩C1,2((0,+∞)×Ω) solving the Cauchy-Neumann problem (12.0.1) pointwise) which is bounded in (0, T )×Ω for any T > 0). This will allow us to associate a semigroup {T (t)} of bounded operators with the Cauchy-Neumann problem (12.0.1) by setting T (t)f = u(t, ·) for any t > 0. As in Chapter 2, the solution to the problem (12.0.1) is obtained by ap-
proximating our problem with a sequence of Cauchy-Neumann problems in (convex) bounded domains Ωn (n ∈ N). The Neumann boundary condition
case
gives some problems. Indeed, differently to what happens in the case considered in Chapter 2, it is not immediate to show that the solutions un to the Cauchy-Neumann problems in the bounded sets Ωn converge to a solution to the problem (12.0.1). To overcome such a difficulty, we prove an a priori gradient estimate for the functions un, with constants being independent of n. This forces us to assume stronger hypotheses on the coefficients than those in Chapter 2. In particular, we have to assume some dissipativity and growth conditions on the coefficients of the operator A. Once this gradient estimate and a suitable maximum principle for bounded
classical solutions to the Cauchy-Neumann problem (12.0.1) is proved, one can show that the sequence {un} converges to a bounded classical solution u to the problem (12.0.1) and that T (t)f satisfies the following estimate:
||DT (t)f ||∞ ≤ CT√ t ||f ||∞, t ∈ (0, T ), (12.0.4)
for any f ∈ Cb(Ω), any T > 0 and some positive constant CT , independent of f . Moreover, one can also show that
||DT (t)f ||∞ ≤ C||f ||C1 b (Ω), t > 0, (12.0.5)
for any f ∈ C1ν (Ω) and C, as above, is a positive constant independent of f . The gradient estimates
||Dun(t, ·)||∞ ≤ CT√ t ||f ||∞, t ∈ (0, T ), f ∈ Cb(Ωn)
and ||Dun(t, ·)||∞ ≤ C||f ||C1ν(Ωn), t ∈ (0, T ), f ∈ C
1 ν (Ωn)
are proved by using the Bernstein method as we did in Chapter 6. Here c0 = supΩ c and CT and C are two positive constants independent, respectively, of t ∈ (0, T ) and of t > 0. Moreover, they are independent of n as well. Concerning the elliptic problem (12.0.2), we show that, for any λ > c0 :=
supΩ c(x) and any f ∈ Cb(Ω), it admits a unique solution u ∈ D(A), where
D(A) =
{ u ∈ Cb(Ω) ∩
⋂ 1≤p<+∞
W 2,p(Ω ∩B(R)) for any R > 0 :
Au ∈ Cb(Ω), ∂u
∂ν (x) = 0 for any x ∈ ∂Ω
} .
