In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
Sobolev spaces with boundary conditions Let Ω ⊂ R2 be a bounded domain with smooth boundary. Since Ω is contained in a large square in R2, it can be regarded as a domain in T2 by identifying opposite sides of the square. The theory of Sobolev spaces on T2 can be found in Bers, John & Schechter (1979), an account which is followed in several later textbooks such as Warner (1983) and Griffiths & Harris (1994). For k an integer, the (restricted) Sobolev space Hk0(Ω) is defined as the closure of C∞c(Ω) in the standard Sobolev space Hk(T2).
H00(Ω) = L2(Ω). Vanishing properties on boundary: For k > 0 the elements of Hk0(Ω) are referred to as "L2 functions on Ω which vanish with their first k − 1 derivatives on ∂Ω." In fact if f ∈ Ck(Ω) agrees with a function in Hk0(Ω), then g = ∂ αf is in C1. Let fn ∈ C∞c(Ω) be such that fn → f in the Sobolev norm, and set gn = ∂ αfn . Thus gn → g in H10(Ω). Hence for h ∈ C∞(T2) and D = a∂x + b∂y,
∬ Ω ( g ( D h ) + ( D g ) h ) d x d y = lim n → 0 ∬ Ω ( g ( D h n ) + ( D g ) h n ) d x d y = 0. {\displaystyle \iint _{\Omega }\left(g(Dh)+(Dg)h\right)\,dx\,dy=\lim _{n\to 0}\iint _{\Omega }\left(g(Dh_{n})+(Dg)h_{n}\right)\,dx\,dy=0.}
By Green's theorem this implies
∫ ∂ Ω g k = 0 , {\displaystyle \int _{\partial \Omega }gk=0,}
where
k = h cos ( n ⋅ ( a , b ) ) , {\displaystyle k=h\cos \left(\mathbf {n} \cdot (a,b)\right),}
with n the unit normal to the boundary. Since such k form a dense subspace of L2(Ω), it follows that g = 0 on ∂Ω. Support properties: Let Ωc be the complement of Ω and define restricted Sobolev spaces analogously for Ωc. Both sets of spaces have a natural pairing with C∞(T2). The Sobolev space for Ω is the annihilator in the Sobolev space for T2 of C∞c(Ωc) and that for Ωc is the annihilator of C∞c(Ω). In fact this is proved by locally applying a small translation to move the domain inside itself and then smoothing by a smooth convolution operator. Suppose g in Hk(T2) annihilates C∞c(Ωc). By compactness, there are finitely many open sets U0, U1, ... , UN covering Ω such that the closure of U0 is disjoint from ∂Ω and each Ui is an open disc about a boundary point zi such that in Ui small translations in the direction of the normal vector ni carry Ω into Ω. Add an open UN+1 with closure in Ωc to produce a cover of T2 and let ψi be a partition of unity subordinate to this cover. If translation by n is denoted by λn, then the functions
g t = ψ 0 g + ∑ i = 1 N ψ n λ t n i g {\displaystyle g_{t}=\psi _{0}g+\sum _{i=1}^{N}\psi _{n}\lambda _{tn_{i}}g}
tend to g as t decreases to 0 and still lie in the annihilator, indeed they are in the annihilator for a larger domain than Ωc, the complement of which lies in Ω. Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in Ω. These are necessarily smooth functions of compact support in Ω. Further vanishing properties on the boundary: The characterization in terms of annihilators shows that f ∈ Ck(Ω) lies in H k0(Ω) if (and only if) it and its derivatives of order less than k vanish on ∂Ω. In fact f can be extended to T2 by setting it to be 0 on Ωc. This extension F defines an element in Hk(T2) using the formula for the norm
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