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 be a bounded domain with smooth boundary. Since is contained in a large square in , it can be regarded as a domain in by identifying opposite sides of the square. The theory of Sobolev spaces on can be found in , an account which is followed in several later textbooks such as and .
For an integer, the (restricted) Sobolev space is defined as the closure of in the standard Sobolev space .
. Vanishing properties on boundary: For the elements of are referred to as " functions on which vanish with their first derivatives on ." In fact if agrees with a function in , then is in . Let be such that in the Sobolev norm, and set . Thus in . Hence for and ,
By Green's theorem this implies
where
with the unit normal to the boundary. Since such form a dense subspace of , it follows that on .
Support properties: Let be the complement of and define restricted Sobolev spaces analogously for . Both sets of spaces have a natural pairing with . The Sobolev space for is the annihilator in the Sobolev space for of and that for is the annihilator of . 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 in annihilates . By compactness, there are finitely many open sets covering such that the closure of is disjoint from and each is an open disc about a boundary point such that in small translations in the direction of the normal vector carry into . Add an open with closure in to produce a cover of and let be a partition of unity subordinate to this cover. If translation by is denoted by , then the functions
tend to as decreases to and still lie in the annihilator, indeed they are in the annihilator for a larger domain than , the complement of which lies in . Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in . These are necessarily smooth functions of compact support in .
Further vanishing properties on the boundary: The characterization in terms of annihilators shows that lies in if (and only if) it and its derivatives of order less than vanish on . In fact can be extended to by setting it to be on . This extension defines an element in using the formula for the norm
Moreover satisfies for g in .
Duality: For , define to be the orthogonal complement of in . Let be the orthogonal projection onto , so that is the orthogonal projection onto . When , this just gives . If and , then
This implies that under the pairing between and , and are each other's duals.
Approximation by smooth functions: The image of is dense in for . This is obvious for since the sum + is dense in . Density for follows because the image of is dense in and annihilates . Canonical isometries: The operator gives an isometry of into and of onto . In fact the first statement follows because it is true on . That is an isometry on follows using the density of in : for we have:
Since the adjoint map between the duals can by identified with this map, it follows that is a unitary map.
Application to Dirichlet problem
Invertibility of The operator defines an isomorphism between and . In fact it is a Fredholm operator of index . The kernel of in consists of constant functions and none of these except zero vanish on the boundary of . Hence the kernel of is and is invertible.
In particular the equation has a unique solution in for in .
Eigenvalue problem Let be the operator on defined by
where is the inclusion of in and of in , both compact operators by Rellich's theorem. The operator is compact and self-adjoint with for all . By the spectral theorem, there is a complete orthonormal set of eigenfunctions in with
Since , lies in . Setting , the are eigenfunctions of the Laplacian:
Sobolev spaces without boundary condition To determine the regularity properties of the eigenfunctions and solutions of
enlargements of the Sobolev spaces have to be considered. Let be the space of smooth functions on which with their derivatives extend continuously to . By Borel's lemma, these are precisely the restrictions of smooth functions on . The Sobolev space is defined to the Hilbert space completion of this space for the norm
This norm agrees with the Sobolev norm on so that can be regarded as a closed subspace of . Unlike , is not naturally a subspace of , but the map restricting smooth functions from to is continuous for the Sobolev norm so extends by continuity to a map . Invariance under diffeomorphism: Any diffeomorphism between the closures of two smooth domains induces an isomorphism between the Sobolev space. This is a simple consequence of the chain rule for derivatives. Extension theorem: The restriction of to the orthogonal complement of its kernel defines an isomorphism onto . The extension map is defined to be the inverse of this map: it is an isomorphism (not necessarily norm preserving) of onto the orthogonal complement of such that . On , it agrees with the natural inclusion map. Bounded extension maps of this kind from to were constructed first constructed by Hestenes and Lions. For smooth curves the Seeley extension theorem provides an extension which is continuous in all the Sobolev norms. A version of the extension which applies in the case where the boundary is just a Lipschitz curve was constructed by Calderón using singular integral operators and generalized by .
It is sufficient to construct an extension for a neighbourhood of a closed annulus, since a collar around the boundary is diffeomorphic to an annulus with a closed interval in . Taking a smooth bump function with , equal to 1 near the boundary and 0 outside the collar, will provide an extension on . On the annulus, the problem reduces to finding an extension for in . Using a partition of unity the task of extending reduces to a neighbourhood of the end points of . Assuming 0 is the left end point, an extension is given locally by
Matching the first derivatives of order k or less at 0, gives
This matrix equation is solvable because the determinant is non-zero by Vandermonde's formula. It is straightforward to check that the formula for , when appropriately modified with bump functions, leads to an extension which is continuous in the above Sobolev norm.
Restriction theorem: The restriction map is surjective with . This is an immediate consequence of the extension theorem and the support properties for Sobolev spaces with boundary condition. Duality: is naturally the dual of H−k0(Ω). Again this is an immediate consequence of the restriction theorem. Thus the Sobolev spaces form a chain:
The differentiation operators carry each Sobolev space into the larger one with index 1 less.
Sobolev embedding theorem: is contained in . This is an immediate consequence of the extension theorem and the Sobolev embedding theorem for . Characterization: consists of in such that all the derivatives ∂αf lie in for |α| ≤ k. Here the derivatives are taken within the chain of Sobolev spaces above.See: Since is weakly dense in , this condition is equivalent to the existence of functions fα such that
To prove the characterization, note that if is in , then lies in Hk−|α|(Ω) and hence in . Conversely the result is well known for the Sobolev spaces : the assumption implies that the is in and the corresponding condition on the Fourier coefficients of shows that lies in . Similarly the result can be proved directly for an annulus . In fact by the argument on the restriction of to any smaller annulus [−δ',δ'] × T lies in : equivalently the restriction of the function lies in for . On the other hand in as , so that must lie in . The case for a general domain reduces to these two cases since can be written as with ψ a bump function supported in such that is supported in a collar of the boundary.
Regularity theorem: If in has both derivatives and in then lies in . This is an immediate consequence of the characterization of above. In fact if this is true even when satisfied at the level of distributions: if there are functions g, h in such that (g,φ) = (f, φx) and (h,φ) = (f,φy) for φ in , then is in . Rotations on an annulus: For an annulus , the extension map to is by construction equivariant with respect to rotations in the second variable,
On it is known that if is in , then the difference quotient in ; if the difference quotients are bounded in Hk then ∂yf lies in . Both assertions are consequences of the formula:
These results on imply analogous results on the annulus using the extension.
Regularity for Dirichlet problem
Regularity for dual Dirichlet problem If with in and in with , then lies in .
Take a decomposition with supported in and supported in a collar of the boundary. Standard Sobolev theory for can be applied to : elliptic regularity implies that it lies in and hence . lies in of a collar, diffeomorphic to an annulus, so it suffices to prove the result with a collar and replaced by
The proof proceeds by induction on , proving simultaneously the inequality
for some constant depending only on . It is straightforward to establish this inequality for , where by density can be taken to be smooth of compact support in :
The collar is diffeomorphic to an annulus. The rotational flow on the annulus induces a flow on the collar with corresponding vector field . Thus corresponds to the vector field . The radial vector field on the annulus is a commuting vector field which on the collar gives a vector field proportional to the normal vector field. The vector fields and commute.
The difference quotients can be formed for the flow . The commutators are second order differential operators from to . Their operators norms are uniformly bounded for near ; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of by their difference quotients composed with . On the other hand, lies in , so the inequalities for apply equally well for :
The uniform boundedness of the difference quotients implies that lies in with
It follows that lies in where is the vector field
Moreover, satisfies a similar inequality to .
Let be the orthogonal vector field
It can also be written as for some smooth nowhere vanishing function on a neighbourhood of the collar.
It suffices to show that lies in . For then
so that and lie in and must lie in .
To check the result on , it is enough to show that and lie in . Note that
are vector fields. But then
with all terms on the right hand side in . Moreover, the inequalities for show that
Hence
Smoothness of eigenfunctions It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of in lie in . Moreover, any solution of with in and in must have in . In both cases by the vanishing properties, the eigenfunctions and vanish on the boundary of .
Solving the Dirichlet problem The dual Dirichlet problem can be used to solve the Dirichlet problem:
By Borel's lemma is the restriction of a function in . Let be the smooth solution of with on . Then solves the Dirichlet problem. By the maximal principle, the solution is unique.
Application to smooth Riemann mapping theorem