Sobolev spaces for planar domains

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. 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. 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). The operator ∆ defines an isomorphism between H10(Ω) and H−1(Ω). In fact it is a Fredholm operator of index 0. The kernel of ∆ in H1(T2) consists of constant functions and none of these except zero vanish on the boundary of Ω. Hence the kernel of H10(Ω) is (0) and ∆ is invertible. In particular the equation ∆f = g has a unique solution in H10(Ω) for g in H−1(Ω). Let T be the operator on L2(Ω) defined by where R0 is the inclusion of L2(Ω) in H−1(Ω) and R1 of H10(Ω) in L2(Ω), both compact operators by Rellich's theorem. The operator T is compact and self-adjoint with (Tf, f ) > 0 for all f. By the spectral theorem, there is a complete orthonormal set of eigenfunctions fn in L2(Ω) with Since μn > 0, fn lies in H10(Ω). Setting λn = μ−n, the fn are eigenfunctions of the Laplacian: To determine the regularity properties of the eigenfunctions  fn  and solutions of