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.

Sobolev spaces with boundary conditions

Let Template:Math be a bounded domain with smooth boundary. Since Template:Math is contained in a large square in Template:Math, it can be regarded as a domain in Template:Math by identifying opposite sides of the square. The theory of Sobolev spaces on Template:Math can be found in Template:Harvtxt, an account which is followed in several later textbooks such as Template:Harvtxt and Template:Harvtxt.

For Template:Mvar an integer, the (restricted) Sobolev space Template:Math is defined as the closure of Template:Math in the standard Sobolev space Template:Math.

• Template:Math.
• Vanishing properties on boundary: For Template:Math the elements of Template:Math are referred to as "Template:Math functions on Template:Math which vanish with their first Template:Math derivatives on Template:Math." In fact if Template:Math agrees with a function in Template:Math, then Template:Math is in Template:Math. Let Template:Math be such that Template:Math in the Sobolev norm, and set Template:Math. Thus Template:Math in Template:Math. Hence for Template:Math and Template:Math,

$\underset{\mathrm{\Omega }}{\iint }(g(Dh)+(Dg)h)dxdy=\underset{n\to 0}{\lim }\underset{\mathrm{\Omega }}{\iint }(g(D{h}_n)+(Dg)h_n)dxdy=0.$

By Green's theorem this implies

$\underset{\partial \mathrm{\Omega }}{\int }gk=0,$

where

$k=h\cos (𝐧\cdot (a,b)),$

with Template:Math the unit normal to the boundary. Since such Template:Mvar form a dense subspace of Template:Math, it follows that Template:Math on Template:Math.

• Support properties: Let Template:Math be the complement of Template:Math and define restricted Sobolev spaces analogously for Template:Math. Both sets of spaces have a natural pairing with Template:Math. The Sobolev space for Template:Math is the annihilator in the Sobolev space for Template:Math of Template:Math and that for Template:Math is the annihilator of Template:Math. 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 Template:Mvar in Template:Math annihilates Template:Math. By compactness, there are finitely many open sets Template:Math covering Template:Math such that the closure of Template:Math is disjoint from Template:Math and each Template:Math is an open disc about a boundary point Template:Math such that in Template:Math small translations in the direction of the normal vector Template:Math carry Template:Math into Template:Math. Add an open Template:Math with closure in Template:Math to produce a cover of Template:Math and let Template:Math be a partition of unity subordinate to this cover. If translation by Template:Mvar is denoted by Template:Math, then the functions

$g_t={\psi }_{0}g+\sum _{i=1}^N{\psi }_n{\lambda }_{t{n}_i}g$

tend to Template:Mvar as Template:Mvar decreases to Template:Math and still lie in the annihilator, indeed they are in the annihilator for a larger domain than Template:Math, the complement of which lies in Template:Math. Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in Template:Math. These are necessarily smooth functions of compact support in Template:Math.

• Further vanishing properties on the boundary: The characterization in terms of annihilators shows that Template:Math lies in Template:Math if (and only if) it and its derivatives of order less than Template:Mvar vanish on Template:Math. In fact Template:Math can be extended to Template:Math by setting it to be Template:Math on Template:Math. This extension Template:Mvar defines an element in Template:Math using the formula for the norm

$\Vert h{\Vert }_{(k)}^2=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\Vert \underset{x}{\overset{j}{\partial }}\underset{y}{\overset{k-j}{\partial }}h\Vert }^2.$

Moreover Template:Math satisfies Template:Math for g in Template:Math.

• Duality: For Template:Math, define Template:Math to be the orthogonal complement of Template:Math in Template:Math. Let Template:Math be the orthogonal projection onto Template:Math, so that Template:Math is the orthogonal projection onto Template:Math. When Template:Math, this just gives Template:Math. If Template:Math and Template:Math, then

$(f,g)=(f,P_{k}g).$

This implies that under the pairing between Template:Math and Template:Math, Template:Math and Template:Math are each other's duals.

• Approximation by smooth functions: The image of Template:Math is dense in Template:Math for Template:Math. This is obvious for Template:Math since the sum Template:Math + Template:Math is dense in Template:Math. Density for Template:Math follows because the image of Template:Math is dense in Template:Math and Template:Math annihilates Template:Math.
• Canonical isometries: The operator Template:Math gives an isometry of Template:Math into Template:Math and of Template:Math onto Template:Math. In fact the first statement follows because it is true on Template:Math. That Template:Math is an isometry on Template:Math follows using the density of Template:Math in Template:Math: for Template:Math we have:

$\begin{array}{cc}{\Vert P_k(I+\mathrm{\Delta }{)}^{k}f\Vert }_{(-k)}& =\underset{\Vert g{\Vert }_{(-k)}=1}{\sup }\left|{((I+\mathrm{\Delta }{)}^{k}f,g)}_{(-k)}\right|\\ & =\underset{\Vert g{\Vert }_{(-k)}=1}{\sup }|(f,g)|\\ & =\Vert f{\Vert }_{(k)}.\end{array}$

Since the adjoint map between the duals can by identified with this map, it follows that Template:Math is a unitary map.

Application to Dirichlet problem

Invertibility of Template:Math

The operator Template:Math defines an isomorphism between Template:Math and Template:Math. In fact it is a Fredholm operator of index Template:Math. The kernel of Template:Math in Template:Math consists of constant functions and none of these except zero vanish on the boundary of Template:Math. Hence the kernel of Template:Math is Template:Math and Template:Math is invertible.

In particular the equation Template:Math has a unique solution in Template:Math for Template:Mvar in Template:Math.

Eigenvalue problem

Let Template:Mvar be the operator on Template:Math defined by

$T=R_1{\mathrm{\Delta }}^{-1}{R}_0,$

where Template:Math is the inclusion of Template:Math in Template:Math and Template:Math of Template:Math in Template:Math, both compact operators by Rellich's theorem. The operator Template:Mvar is compact and self-adjoint with Template:Math for all Template:Mvar. By the spectral theorem, there is a complete orthonormal set of eigenfunctions Template:Math in Template:Math with

$T{f}_n={\mu }_n{f}_n,{\mu }_n>0,{\mu }_n\to 0.$

Since Template:Math, Template:Math lies in Template:Math. Setting Template:Math, the Template:Math are eigenfunctions of the Laplacian:

$\mathrm{\Delta }{f}_n={\lambda }_n{f}_n,{\lambda }_n>0,{\lambda }_n\to \mathrm{\infty }.$

Sobolev spaces without boundary condition

To determine the regularity properties of the eigenfunctions Template:Math and solutions of

$\mathrm{\Delta }f=u,u\in H^{-1}(\mathrm{\Omega }),f\in H_0^1(\mathrm{\Omega }),$

enlargements of the Sobolev spaces Template:Math have to be considered. Let Template:Math be the space of smooth functions on Template:Math which with their derivatives extend continuously to Template:Math. By Borel's lemma, these are precisely the restrictions of smooth functions on Template:Math. The Sobolev space Template:Math is defined to the Hilbert space completion of this space for the norm

$\Vert f{\Vert }_{(k)}^2=\sum _{j=0}^k(\genfrac{}{}{0ex}{}{k}{j}){\Vert \underset{x}{\overset{j}{\partial }}\underset{y}{\overset{k-j}{\partial }}f\Vert }^2.$

This norm agrees with the Sobolev norm on Template:Math so that Template:Math can be regarded as a closed subspace of Template:Math. Unlike Template:Math, Template:Math is not naturally a subspace of Template:Math, but the map restricting smooth functions from Template:Math to Template:Math is continuous for the Sobolev norm so extends by continuity to a map Template:Math.

• 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 Template:Math to the orthogonal complement of its kernel defines an isomorphism onto Template:Math. The extension map Template:Math is defined to be the inverse of this map: it is an isomorphism (not necessarily norm preserving) of Template:Math onto the orthogonal complement of Template:Math such that Template:Math. On Template:Math, it agrees with the natural inclusion map. Bounded extension maps Template:Math of this kind from Template:Math to Template:Math 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 Template:Harvtxt.

It is sufficient to construct an extension Template:Mvar for a neighbourhood of a closed annulus, since a collar around the boundary is diffeomorphic to an annulus Template:Math with Template:Mvar a closed interval in Template:Math. Taking a smooth bump function Template:Mvar with Template:Math, equal to 1 near the boundary and 0 outside the collar, Template:Math will provide an extension on Template:Math. On the annulus, the problem reduces to finding an extension for Template:Math in Template:Math. Using a partition of unity the task of extending reduces to a neighbourhood of the end points of Template:Mvar. Assuming 0 is the left end point, an extension is given locally by

$E(f)=\sum _{m=0}^k{a}_{m}f(-\frac{x}{m+1}).$

Matching the first derivatives of order k or less at 0, gives

$\sum _{m=0}^k(-m-1{)}^{-k}{a}_m=1.$

This matrix equation is solvable because the determinant is non-zero by Vandermonde's formula. It is straightforward to check that the formula for Template:Math, when appropriately modified with bump functions, leads to an extension which is continuous in the above Sobolev norm.

• Restriction theorem: The restriction map Template:Math is surjective with Template:Math. This is an immediate consequence of the extension theorem and the support properties for Sobolev spaces with boundary condition.
• Duality: Template:Math 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:

$\cdots \subset H^2(\mathrm{\Omega })\subset H^1(\mathrm{\Omega })\subset H^0(\mathrm{\Omega })\subset H_0^{-1}(\mathrm{\Omega })\subset H_0^{-2}(\mathrm{\Omega })\subset \cdots$

The differentiation operators Template:Math carry each Sobolev space into the larger one with index 1 less.

• Sobolev embedding theorem: Template:Math is contained in Template:Math. This is an immediate consequence of the extension theorem and the Sobolev embedding theorem for Template:Math.
• Characterization: Template:Math consists of Template:Math in Template:Math such that all the derivatives ∂^αf lie in Template:Math for |α| ≤ k. Here the derivatives are taken within the chain of Sobolev spaces above. Since Template:Math is weakly dense in Template:Math, this condition is equivalent to the existence of Template:Math functions f_α such that

$(f_{\alpha },\phi )=(-1{)}^{|\alpha |}(f,{\partial }^{\alpha }\phi ),|\alpha |\le k,\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }).$

To prove the characterization, note that if Template:Math is in Template:Math, then Template:Math lies in H^k−|α|(Ω) and hence in Template:Math. Conversely the result is well known for the Sobolev spaces Template:Math: the assumption implies that the Template:Math is in Template:Math and the corresponding condition on the Fourier coefficients of Template:Math shows that Template:Math lies in Template:Math. Similarly the result can be proved directly for an annulus Template:Math. In fact by the argument on Template:Math the restriction of Template:Math to any smaller annulus [−δ',δ'] × T lies in Template:Math: equivalently the restriction of the function Template:Math lies in Template:Math for Template:Math. On the other hand Template:Math in Template:Math as Template:Math, so that Template:Math must lie in Template:Math. The case for a general domain Template:Math reduces to these two cases since Template:Math can be written as Template:Math with ψ a bump function supported in Template:Math such that Template:Math is supported in a collar of the boundary.

• Regularity theorem: If Template:Math in Template:Math has both derivatives Template:Math and Template:Math in Template:Math then Template:Math lies in Template:Math. This is an immediate consequence of the characterization of Template:Math above. In fact if this is true even when satisfied at the level of distributions: if there are functions g, h in Template:Math such that (g,φ) = (f, φ_x) and (h,φ) = (f,φ_y) for φ in Template:Math, then Template:Math is in Template:Math.
• Rotations on an annulus: For an annulus Template:Math, the extension map to Template:Math is by construction equivariant with respect to rotations in the second variable,

$R_{t}f(x,y)=f(x,y+t).$

On Template:Math it is known that if Template:Math is in Template:Math, then the difference quotient Template:Math in Template:Math; if the difference quotients are bounded in H^k then ∂_yf lies in Template:Math. Both assertions are consequences of the formula:

$\widehat{{\delta }_{h}f}(m,n)=h^{-1}(e^{-ihn}-1)\hat{f}(m,n)=-\underset{0}{\overset{1}{\int }}in{e}^{-inht}dt\hat{f}(m,n).$

These results on Template:Math imply analogous results on the annulus using the extension.

Regularity for Dirichlet problem

Regularity for dual Dirichlet problem

If Template:Math with Template:Mvar in Template:Math and Template:Mvar in Template:Math with Template:Math, then Template:Mvar lies in Template:Math.

Take a decomposition Template:Math with Template:Mvar supported in Template:Math and Template:Math supported in a collar of the boundary. Standard Sobolev theory for Template:Math can be applied to Template:Math: elliptic regularity implies that it lies in Template:Math and hence Template:Math. Template:Math lies in Template:Math of a collar, diffeomorphic to an annulus, so it suffices to prove the result with Template:Math a collar and Template:Math replaced by

$\begin{array}{cc}{\mathrm{\Delta }}_1& =\mathrm{\Delta }-[\mathrm{\Delta },\psi ]\\ & =\mathrm{\Delta }+(p{\partial }_x+q{\partial }_y-\mathrm{\Delta }\psi )\\ & =\mathrm{\Delta }+X.\end{array}$

The proof proceeds by induction on Template:Mvar, proving simultaneously the inequality

$\Vert u{\Vert }_{(k+1)}\le C\Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(k-1)}+C\Vert u{\Vert }_{(k)},$

for some constant Template:Mvar depending only on Template:Mvar. It is straightforward to establish this inequality for Template:Math, where by density Template:Mvar can be taken to be smooth of compact support in Template:Math:

$\begin{array}{cc}\Vert u{\Vert }_{(1)}^2& =|(\mathrm{\Delta }u,u)|\\ & \le |({\mathrm{\Delta }}_{1}u,u)|+|(Xu,u)|\\ & \le \Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(-1)}\Vert u{\Vert }_{(1)}+C^{\prime }\Vert u{\Vert }_{(1)}\Vert u{\Vert }_{(0)}.\end{array}$

The collar is diffeomorphic to an annulus. The rotational flow Template:Math on the annulus induces a flow Template:Math on the collar with corresponding vector field Template:Math. Thus Template:Mvar corresponds to the vector field Template:Math. The radial vector field on the annulus Template:Math is a commuting vector field which on the collar gives a vector field Template:Math proportional to the normal vector field. The vector fields Template:Mvar and Template:Mvar commute.

The difference quotients Template:Math can be formed for the flow Template:Math. The commutators Template:Math are second order differential operators from Template:Math to Template:Math. Their operators norms are uniformly bounded for Template:Mvar near Template:Math; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of Template:Math by their difference quotients composed with Template:Math. On the other hand, Template:Math lies in Template:Math, so the inequalities for Template:Mvar apply equally well for Template:Mvar:

$\begin{array}{cc}\Vert {\delta }_{h}u{\Vert }_{(k+1)}& \le C\Vert {\mathrm{\Delta }}_1{\delta }_{h}u{\Vert }_{(k-1)}+C\Vert {\delta }_{h}u{\Vert }_{(k)}\\ & \le C\Vert {\delta }_h{\mathrm{\Delta }}_{1}u{\Vert }_{(k-1)}+C\Vert [{\delta }_h,{\mathrm{\Delta }}_1]u{\Vert }_{(k-1)}+C\Vert {\delta }_{h}u{\Vert }_{(k)}\\ & \le C\Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(k)}+C^{\prime }\Vert u{\Vert }_{(k+1)}.\end{array}$

The uniform boundedness of the difference quotients Template:Math implies that Template:Math lies in Template:Math with

$\Vert Yu{\Vert }_{(k+1)}\le C\Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(k)}+C^{\prime }\Vert u{\Vert }_{(k+1)}.$

It follows that Template:Math lies in Template:Math where Template:Mvar is the vector field

$V=\frac{Y}{\sqrt{r^2+s^2}}=a{\partial }_x+b{\partial }_y,a^2+b^2=1.$

Moreover, Template:Math satisfies a similar inequality to Template:Math.

$\Vert Vu{\Vert }_{(k+1)}\le C^{\prime \prime }(\Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(k)}+\Vert u{\Vert }_{(k+1)}).$

Let Template:Mvar be the orthogonal vector field

$W=-b{\partial }_x+a{\partial }_y.$

It can also be written as Template:Math for some smooth nowhere vanishing function Template:Mvar on a neighbourhood of the collar.

It suffices to show that Template:Math lies in Template:Math. For then

$(V\pm iW)u=(a\mp ib)({\partial }_x\pm i{\partial }_y)u,$

so that Template:Math and Template:Math lie in Template:Math and Template:Mvar must lie in Template:Math.

To check the result on Template:Math, it is enough to show that Template:Math and Template:Math lie in Template:Math. Note that

$\begin{array}{cc}A& =\mathrm{\Delta }-V^2-W^2,\\ B& =[V,W],\end{array}$

are vector fields. But then

$\begin{array}{cc}{W}^{2}u& =\mathrm{\Delta }u-V^{2}u-Au,\\ VWu& =WVu+Bu,\end{array}$

with all terms on the right hand side in Template:Math. Moreover, the inequalities for Template:Math show that

$\begin{array}{cc}\Vert Wu{\Vert }_{(k+1)}& \le C(\Vert VWu{\Vert }_{(k)}+{\Vert W^{2}u\Vert }_{(k)})\\ & \le C{\Vert (\mathrm{\Delta }-V^2-A)u\Vert }_{(k)}+C\Vert (WV+B)u{\Vert }_{(k)}\\ & \le C_1\Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(k)}+C_1\Vert u{\Vert }_{(k+1)}.\end{array}$

Hence

$\begin{array}{cc}\Vert u{\Vert }_{(k+2)}& \le C(\Vert Vu{\Vert }_{(k+1)}+\Vert Wu{\Vert }_{(k+1)})\\ & \le C^{\prime }\Vert {\mathrm{\Delta }}_{1}u{\Vert }_{(k)}+C^{\prime }\Vert u{\Vert }_{(k+1)}.\end{array}$

Smoothness of eigenfunctions

It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of Template:Math in Template:Math lie in Template:Math. Moreover, any solution of Template:Math with Template:Mvar in Template:Math and Template:Mvar in Template:Math must have Template:Mvar in Template:Math. In both cases by the vanishing properties, the eigenfunctions and Template:Mvar vanish on the boundary of Template:Math.

Solving the Dirichlet problem

The dual Dirichlet problem can be used to solve the Dirichlet problem:

$\{\begin{array}{c}\mathrm{\Delta }f{|}_{\mathrm{\Omega }}=0\\ f{|}_{\partial \mathrm{\Omega }}=g& g\in C^{\mathrm{\infty }}(\partial \mathrm{\Omega })\end{array}$

By Borel's lemma Template:Mvar is the restriction of a function Template:Mvar in Template:Math. Let Template:Mvar be the smooth solution of Template:Math with Template:Math on Template:Math. Then Template:Math solves the Dirichlet problem. By the maximal principle, the solution is unique.

Application to smooth Riemann mapping theorem

The solution to the Dirichlet problem can be used to prove a strong form of the Riemann mapping theorem for simply connected domains with smooth boundary. The method also applies to a region diffeomorphic to an annulus. For multiply connected regions with smooth boundary Template:Harvtxt have given a method for mapping the region onto a disc with circular holes. Their method involves solving the Dirichlet problem with a non-linear boundary condition. They construct a function Template:Mvar such that:

• Template:Mvar is harmonic in the interior of Template:Math;
• On Template:Math we have: Template:Math, where Template:Mvar is the curvature of the boundary curve, Template:Math is the derivative in the direction normal to Template:Math and Template:Mvar is constant on each boundary component.

Template:Harvtxt gives a proof of the Riemann mapping theorem for a simply connected domain Template:Math with smooth boundary. Translating if necessary, it can be assumed that Template:Math. The solution of the Dirichlet problem shows that there is a unique smooth function Template:Math on Template:Math which is harmonic in Template:Math and equals Template:Math on Template:Math. Define the Green's function by Template:Math. It vanishes on Template:Math and is harmonic on Template:Math away from Template:Math. The harmonic conjugate Template:Mvar of Template:Mvar is the unique real function on Template:Math such that Template:Math is holomorphic. As such it must satisfy the Cauchy–Riemann equations:

$\begin{array}{cc}{U}_x& =-V_y,\\ U_y& =V_x.\end{array}$

The solution is given by

$V(z)=\underset{0}{\overset{z}{\int }}-U_{y}dx+V_{x}dy,$

where the integral is taken over any path in Template:Math. It is easily verified that Template:Math and Template:Math exist and are given by the corresponding derivatives of Template:Mvar. Thus Template:Mvar is a smooth function on Template:Math, vanishing at Template:Math. By the Cauchy-Riemann Template:Math is smooth on Template:Math, holomorphic on Template:Math and Template:Math. The function Template:Math is only defined up to multiples of Template:Math, but the function

$F(z)=e^{G(z)+iH(z)}=z{e}^{f(z)}$

is a holomorphic on Template:Math and smooth on Template:Math. By construction, Template:Math and Template:Math for Template:Math. Since Template:Mvar has winding number Template:Math, so too does Template:Math. On the other hand, Template:Math only for Template:Math where there is a simple zero. So by the argument principle Template:Mvar assumes every value in the unit disc, Template:Math, exactly once and Template:Math does not vanish inside Template:Math. To check that the derivative on the boundary curve is non-zero amounts to computing the derivative of Template:Math, i.e. the derivative of Template:Mvar should not vanish on the boundary curve. By the Cauchy-Riemann equations these tangential derivative are up to a sign the directional derivative in the direction of the normal to the boundary. But Template:Mvar vanishes on the boundary and is strictly negative in Template:Math since Template:Math. The Hopf lemma implies that the directional derivative of Template:Mvar in the direction of the outward normal is strictly positive. So on the boundary curve, Template:Mvar has nowhere vanishing derivative. Since the boundary curve has winding number one, Template:Mvar defines a diffeomorphism of the boundary curve onto the unit circle. Accordingly, Template:Math is a smooth diffeomorphism, which restricts to a holomorphic map Template:Math and a smooth diffeomorphism between the boundaries.

Similar arguments can be applied to prove the Riemann mapping theorem for a doubly connected domain Template:Math bounded by simple smooth curves Template:Math (the inner curve) and Template:Math (the outer curve). By translating we can assume 1 lies on the outer boundary. Let Template:Mvar be the smooth solution of the Dirichlet problem with Template:Math on the outer curve and Template:Math on the inner curve. By the maximum principle Template:Math for Template:Mvar in Template:Math and so by the Hopf lemma the normal derivatives of Template:Mvar are negative on the outer curve and positive on the inner curve. The integral of Template:Math over the boundary is zero by Stokes' theorem so the contributions from the boundary curves cancel. On the other hand, on each boundary curve the contribution is the integral of the normal derivative along the boundary. So there is a constant Template:Math such that Template:Math satisfies

$\underset{C}{\int }(-U_{y}dx+U_{x}dy)=2\pi$

on each boundary curve. The harmonic conjugate Template:Mvar of Template:Mvar can again be defined by

$V(z)=\underset{1}{\overset{z}{\int }}-u_{y}dx+u_{x}dy$

and is well-defined up to multiples of Template:Math. The function

$F(z)=e^{U(z)+iV(z)}$

is smooth on Template:Math and holomorphic in Template:Math. On the outer curve Template:Math and on the inner curve Template:Math. The tangential derivatives on the outer curves are nowhere vanishing by the Cauchy-Riemann equations, since the normal derivatives are nowhere vanishing. The normalization of the integrals implies that Template:Mvar restricts to a diffeomorphism between the boundary curves and the two concentric circles. Since the images of outer and inner curve have winding number Template:Math and Template:Math about any point in the annulus, an application of the argument principle implies that Template:Mvar assumes every value within the annulus Template:Math exactly once; since that includes multiplicities, the complex derivative of Template:Mvar is nowhere vanishing in Template:Math. This Template:Mvar is a smooth diffeomorphism of Template:Math onto the closed annulus Template:Math, restricting to a holomorphic map in the interior and a smooth diffeomorphism on both boundary curves.

Trace map

The restriction map Template:Math extends to a continuous map Template:Math for Template:Math. In fact

$\widehat{\tau f}(n)=\sum _m\hat{f}(m,n),$

so the Cauchy–Schwarz inequality yields

$\begin{array}{cc}{|\widehat{\tau f}(n)|}^2{(1+n^2)}^{k-\frac{1}{2}}& \le \left(\sum _m\frac{{(1+n^2)}^{k-\frac{1}{2}}}{{(1+m^2+n^2)}^k}\right)\left(\sum _m{|\hat{f}(m,n)|}^2{(1+m^2+n^2)}^k\right)\\ & \le C_k\sum _m{|\hat{f}(m,n)|}^2{(1+m^2+n^2)}^k,\end{array}$

where, by the integral test,

$\begin{array}{cc}{C}_k& =\underset{n}{\sup }\sum _m\frac{{(1+n^2)}^{k-\frac{1}{2}}}{{(1+m^2+n^2)}^k}<\mathrm{\infty },\\ c_k& =\underset{n}{\inf }\sum _m\frac{{(1+n^2)}^{k-\frac{1}{2}}}{{(1+m^2+n^2)}^k}>0.\end{array}$

The map Template:Mvar is onto since a continuous extension map Template:Mvar can be constructed from Template:Math to Template:Math. In fact set

$\widehat{Eg}(m,n)={\lambda }_n^{-1}\hat{g}(n)\frac{{(1+n^2)}^{k-\frac{1}{2}}}{{(1+n^2+m^2)}^k},$

where

${\lambda }_n=\sum _m\frac{{(1+n^2)}^{k-\frac{1}{2}}}{{(1+m^2+n^2)}^k}.$

Thus Template:Math. If g is smooth, then by construction Eg restricts to g on 1 × T. Moreover, E is a bounded linear map since

$\begin{array}{cc}\Vert Eg{\Vert }_{(k)}^2& =\sum _{m,n}{|\widehat{Eg}(m,n)|}^2(1+m^2+n^2)\\ & \le c_k^{-2}\sum _{m,n}{|\hat{g}(n)|}^2\frac{{(1+n^2)}^{2k-1}}{{(1+m^2+n^2)}^k}\\ & \le c_k^{-2}{C}_k\Vert g{\Vert }_{k-\frac{1}{2}}^2.\end{array}$

It follows that there is a trace map τ of H^k(Ω) onto H^{k − 1/2}(∂Ω). Indeed, take a tubular neighbourhood of the boundary and a smooth function ψ supported in the collar and equal to 1 near the boundary. Multiplication by ψ carries functions into H^k of the collar, which can be identified with H^k of an annulus for which there is a trace map. The invariance under diffeomorphisms (or coordinate change) of the half-integer Sobolev spaces on the circle follows from the fact that an equivalent norm on H^{k + 1/2}(T) is given by

$\Vert f{\Vert }_{[k+\frac{1}{2}]}^2=\Vert f{\Vert }_{(k)}^2+\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{2\pi }{\int }}\frac{{|f^{(k)}(s)-f^{(k)}(t)|}^2}{{|e^{is}-e^{it}|}^2}dsdt.$

It is also a consequence of the properties of τ and E (the "trace theorem"). In fact any diffeomorphism f of T induces a diffeomorphism F of T² by acting only on the second factor. Invariance of H^k(T²) under the induced map F* therefore implies invariance of H^{k − 1/2}(T) under f*, since f* = τ ∘ F* ∘ E.

Further consequences of the trace theorem are the two exact sequences

$(0)\to H_0^1(\mathrm{\Omega })\to H^1(\mathrm{\Omega })\to H^{\frac{1}{2}}(\partial \mathrm{\Omega })\to (0)$

and

$(0)\to H_0^2(\mathrm{\Omega })\to H^2(\mathrm{\Omega })\to H^{\frac{3}{2}}(\partial \mathrm{\Omega })\oplus H^{\frac{1}{2}}(\partial \mathrm{\Omega })\to (0),$

where the last map takes f in H²(Ω) to f|_∂Ω and ∂_nf|_∂Ω. There are generalizations of these sequences to H^k(Ω) involving higher powers of the normal derivative in the trace map:

$(0)\to H_0^k(\mathrm{\Omega })\to H^k(\mathrm{\Omega })\to \underset{j=1}{\overset{k}{⨁}}{H}^{j-\frac{1}{2}}(\partial \mathrm{\Omega })\to (0).$

The trace map to Template:Math takes f to Template:Math

Abstract formulation of boundary value problems

The Sobolev space approach to the Neumann problem cannot be phrased quite as directly as that for the Dirichlet problem. The main reason is that for a function Template:Mvar in Template:Math, the normal derivative Template:Math cannot be a priori defined at the level of Sobolev spaces. Instead an alternative formulation of boundary value problems for the Laplacian Template:Math on a bounded region Template:Math in the plane is used. It employs Dirichlet forms, sesqulinear bilinear forms on Template:Math, Template:Math or an intermediate closed subspace. Integration over the boundary is not involved in defining the Dirichlet form. Instead, if the Dirichlet form satisfies a certain positivity condition, termed coerciveness, solution can be shown to exist in a weak sense, so-called "weak solutions". A general regularity theorem than implies that the solutions of the boundary value problem must lie in Template:Math, so that they are strong solutions and satisfy boundary conditions involving the restriction of a function and its normal derivative to the boundary. The Dirichlet problem can equally well be phrased in these terms, but because the trace map Template:Math is already defined on Template:Math, Dirichlet forms do not need to be mentioned explicitly and the operator formulation is more direct. A unified discussion is given in Template:Harvtxt and briefly summarised below. It is explained how the Dirichlet problem, as discussed above, fits into this framework. Then a detailed treatment of the Neumann problem from this point of view is given following Template:Harvtxt.

The Hilbert space formulation of boundary value problems for the Laplacian Template:Math on a bounded region Template:Math in the plane proceeds from the following data:

• A closed subspace Template:Math.
• A Dirichlet form for Template:Math given by a bounded Hermitian bilinear form Template:Math defined for Template:Math such that Template:Math for Template:Math.
• Template:Mvar is coercive, i.e. there is a positive constant Template:Mvar and a non-negative constant Template:Mvar such that Template:Math.

A weak solution of the boundary value problem given initial data Template:Mvar in Template:Math is a function u satisfying

$D(f,g)=(u,g)$

for all g.

For both the Dirichlet and Neumann problem

$D(f,g)=(f_x,g_x)+(f_y,g_y).$

For the Dirichlet problem Template:Math. In this case

$D(f,g)=(\mathrm{\Delta }f,g),f,g\in H.$

By the trace theorem the solution satisfies Template:Math in Template:Math.

For the Neumann problem Template:Mvar is taken to be Template:Math.

Application to Neumann problem

The classical Neumann problem on Template:Math consists in solving the boundary value problem

$\{\begin{array}{cc}\mathrm{\Delta }u=f,& f,u\in C^{\mathrm{\infty }}({\mathrm{\Omega }}^{-}),\\ {\partial }_{n}u=0& \text{on }\partial \mathrm{\Omega }\end{array}$

Green's theorem implies that for Template:Math

$(\mathrm{\Delta }u,v)=(u_x,v_x)+(u_y,v_y)-({\partial }_{n}u,v{)}_{\partial \mathrm{\Omega }}.$

Thus if Template:Math in Template:Math and satisfies the Neumann boundary conditions, Template:Math, and so Template:Mvar is constant in Template:Math.

Hence the Neumann problem has a unique solution up to adding constants.

Consider the Hermitian form on Template:Math defined by

$D(f,g)=(u_x,v_x)+(u_y,v_y).$

Since Template:Math is in duality with Template:Math, there is a unique element Template:Math in Template:Math such that

$D(u,v)=(Lu,v).$

The map Template:Math is an isometry of Template:Math onto Template:Math, so in particular Template:Mvar is bounded.

In fact

$((L+I)u,v)=(u,v{)}_{(1)}.$

So

$\Vert (L+I)u{\Vert }_{(-1)}=\underset{\Vert v{\Vert }_{(1)}=1}{\sup }|((L+I)u,v)|=\underset{\Vert v{\Vert }_{(1)}=1}{\sup }|(u,v{)}_{(1)}|=\Vert u{\Vert }_{(1)}.$

On the other hand, any Template:Math in Template:Math defines a bounded conjugate-linear form on Template:Math sending Template:Mvar to Template:Math. By the Riesz–Fischer theorem, there exists Template:Math such that

$(f,v)=(u,v{)}_{(1)}.$

Hence Template:Math and so Template:Math is surjective. Define a bounded linear operator Template:Mvar on Template:Math by

$T=R_1(I+L{)}^{-1}{R}_0,$

where Template:Math is the map Template:Math, a compact operator, and Template:Math is the map Template:Math, its adjoint, so also compact.

The operator Template:Mvar has the following properties:

• Template:Mvar is a contraction since it is a composition of contractions
• Template:Mvar is compact, since Template:Math and Template:Math are compact by Rellich's theorem
• Template:Mvar is self-adjoint, since if Template:Math, they can be written Template:Math with Template:Math so

$(Tf,g)=(u,(I+L)v)=(u,v{)}_{(1)}=((I+L)u,v)=(f,Tg).$

• Template:Mvar has positive spectrum and kernel Template:Math, for

$(Tf,f)=(u,u{)}_{(1)}\ge 0,$

and Template:Math implies Template:Math and hence Template:Math.

• There is a complete orthonormal basis Template:Math of Template:Math consisting of eigenfunctions of Template:Mvar. Thus

$T{f}_n={\mu }_n{f}_n$

with Template:Math and Template:Math decreasing to Template:Math.

• The eigenfunctions all lie in Template:Math since the image of Template:Mvar lies in Template:Math.
• The Template:Math are eigenfunctions of Template:Mvar with

$L{f}_n={\lambda }_n{f}_n,{\lambda }_n={\mu }_n^{-1}-1.$

Thus Template:Math are non-negative and increase to Template:Math.

• The eigenvalue Template:Math occurs with multiplicity one and corresponds to the constant function. For if Template:Math satisfies Template:Math, then

$(u_x,u_x)+(u_y,u_y)=(Lu,u)=0,$

so Template:Mvar is constant.

Regularity for Neumann problem

Weak solutions are strong solutions

The first main regularity result shows that a weak solution expressed in terms of the operator Template:Mvar and the Dirichlet form Template:Mvar is a strong solution in the classical sense, expressed in terms of the Laplacian Template:Math and the Neumann boundary conditions. Thus if Template:Math with Template:Math, then Template:Math, satisfies Template:Math and Template:Math. Moreover, for some constant Template:Mvar independent of Template:Mvar,

$\Vert u{\Vert }_{(2)}\le C\Vert \mathrm{\Delta }u{\Vert }_{(0)}+C\Vert u{\Vert }_{(1)}.$

Note that

$\Vert u{\Vert }_{(1)}\le \Vert Lu{\Vert }_{(-1)}+\Vert u{\Vert }_{(0)},$

since

$\begin{array}{cc}\Vert u{\Vert }_{(1)}^2& =|(Lu,u)|+\Vert u{\Vert }_{(0)}^2\\ & \le \Vert Lu{\Vert }_{(-1)}\Vert u{\Vert }_{(1)}+\Vert u{\Vert }_{(0)}\Vert u{\Vert }_{(1)}.\end{array}$

Take a decomposition Template:Math with Template:Mvar supported in Template:Math and Template:Math supported in a collar of the boundary.

The operator Template:Mvar is characterized by

$(Lf,g)=(f_x,g_x)+(f_y,g_y)=(\mathrm{\Delta }f,g{)}_{\mathrm{\Omega }}-{({\partial }_{n}f,g)}_{\partial \mathrm{\Omega }},f,g\in C^{\mathrm{\infty }}({\mathrm{\Omega }}^{-}).$

Then

$([L,\psi ]f,g)=([\mathrm{\Delta },\psi ]f,g),$

so that

$[L,\psi ]=-[L,1-\psi ]=\mathrm{\Delta }\psi +2{\psi }_x{\partial }_x+2{\psi }_y{\partial }_y.$

The function Template:Math and Template:Math are treated separately, Template:Mvar being essentially subject to usual elliptic regularity considerations for interior points while Template:Mvar requires special treatment near the boundary using difference quotients. Once the strong properties are established in terms of Template:Math and the Neumann boundary conditions, the "bootstrap" regularity results can be proved exactly as for the Dirichlet problem.

Interior estimates

The function Template:Math lies in Template:Math where Template:Math is a region with closure in Template:Math. If Template:Math and Template:Math

$(Lf,g)=(\mathrm{\Delta }f,g{)}_{\mathrm{\Omega }}.$

By continuity the same holds with Template:Math replaced by Template:Mvar and hence Template:Math. So

$\mathrm{\Delta }v=Lv=L(\psi u)=\psi Lu+[L,\psi ]u=\psi (f-u)+[\mathrm{\Delta },\psi ]u.$

Hence regarding Template:Mvar as an element of Template:Math, Template:Math. Hence Template:Math. Since Template:Math for Template:Math, we have Template:Math. Moreover,

$\Vert v{\Vert }_{(2)}^2=\Vert \mathrm{\Delta }v{\Vert }^2+2\Vert v{\Vert }_{(1)}^2,$

so that

$\Vert v{\Vert }_{(2)}\le C(\Vert \mathrm{\Delta }(v)\Vert +\Vert v{\Vert }_{(1)}).$

Boundary estimates

The function Template:Math is supported in a collar contained in a tubular neighbourhood of the boundary. The difference quotients Template:Math can be formed for the flow Template:Math and lie in Template:Math, so the first inequality is applicable:

$\begin{array}{cc}\Vert {\delta }_{h}w{\Vert }_{(1)}& \le \Vert L{\delta }_{h}w{\Vert }_{(-1)}+\Vert {\delta }_{h}w{\Vert }_{(0)}\\ & \le \Vert [L,{\delta }_h]w{\Vert }_{(-1)}+\Vert {\delta }_{h}Lw{\Vert }_{(-1)}+\Vert {\delta }_{h}w{\Vert }_{(0)}\\ & \le \Vert [L,{\delta }_h]w{\Vert }_{(-1)}+C\Vert Lw{\Vert }_{(0)}+C\Vert w{\Vert }_{(1)}.\end{array}$

The commutators Template:Math are uniformly bounded as operators from Template:Math to Template:Math. This is equivalent to checking the inequality

$\left|([L,{\delta }_h]g,h)\right|\le A\Vert g{\Vert }_{(1)}\Vert h{\Vert }_{(1)},$

for Template:Mvar, Template:Mvar smooth functions on a collar. This can be checked directly on an annulus, using invariance of Sobolev spaces under dffeomorphisms and the fact that for the annulus the commutator of Template:Math with a differential operator is obtained by applying the difference operator to the coefficients after having applied Template:Math to the function:

$[{\delta }^h,\sum a_{\alpha }{\partial }^{\alpha }]=({\delta }^h(a_{\alpha })\circ R_h){\partial }^{\alpha }.$

Hence the difference quotients Template:Math are uniformly bounded, and therefore Template:Math with

$\Vert Yw{\Vert }_{(1)}\le C\Vert Lw{\Vert }_{(0)}+C^{\prime }\Vert w{\Vert }_{(1)}.$

Hence Template:Math and Template:Math satisfies a similar inequality to Template:Math:

$\Vert Vw{\Vert }_{(1)}\le C^{\prime \prime }(\Vert Lw{\Vert }_{(0)}+\Vert w{\Vert }_{(1)}).$

Let Template:Mvar be the orthogonal vector field. As for the Dirichlet problem, to show that Template:Math, it suffices to show that Template:Math.

To check this, it is enough to show that Template:Math. As before

$\begin{array}{cc}A& =\mathrm{\Delta }-V^2-W^2\\ B& =[V,W]\end{array}$

are vector fields. On the other hand, Template:Math for Template:Math, so that Template:Math and Template:Math define the same distribution on Template:Math. Hence

$\begin{array}{cc}(W^{2}w,\phi )& =(Lw-V^{2}w-Au,\phi ),\\ (VWw,\phi )& =(WVw+Bw,\phi ).\end{array}$

Since the terms on the right hand side are pairings with functions in Template:Math, the regularity criterion shows that Template:Math. Hence Template:Math since both terms lie in Template:Math and have the same inner products with Template:Mvar's.

Moreover, the inequalities for Template:Math show that

$\begin{array}{cc}\Vert Ww{\Vert }_{(1)}& \le C(\Vert VWw{\Vert }_{(0)}+{\Vert W^{2}w\Vert }_{(0)})\\ & \le C{\Vert (\mathrm{\Delta }-V^2-A)w\Vert }_{(0)}+C\Vert (WV+B)w{\Vert }_{(0)}\\ & \le C_1\Vert Lw{\Vert }_{(0)}+C_1\Vert w{\Vert }_{(1)}.\end{array}$

Hence

$\begin{array}{cc}\Vert w{\Vert }_{(2)}& \le C(\Vert Vw{\Vert }_{(1)}+\Vert Ww{\Vert }_{(1)})\\ & \le C^{\prime }\Vert \mathrm{\Delta }w{\Vert }_{(0)}+C^{\prime }\Vert w{\Vert }_{(1)}.\end{array}$

It follows that Template:Math. Moreover,

$\begin{array}{cc}\Vert u{\Vert }_{(2)}& \le C(\Vert \mathrm{\Delta }v\Vert +\Vert \mathrm{\Delta }w\Vert +\Vert v{\Vert }_{(1)}+\Vert w{\Vert }_{(1)})\\ & \le C^{\prime }(\Vert \psi \mathrm{\Delta }u\Vert +\Vert (1-\psi )\mathrm{\Delta }u\Vert +2\Vert [\mathrm{\Delta },\psi ]u\Vert +\Vert u{\Vert }_{(1)})\\ & \le C^{\prime \prime }(\Vert \mathrm{\Delta }u\Vert +\Vert u{\Vert }_{(1)}).\end{array}$

Neumann boundary conditions

Since Template:Math, Green's theorem is applicable by continuity. Thus for Template:Math,

$\begin{array}{cc}(f,v)& =(Lu,v)+(u,v)\\ & =(u_x,v_x)+(u_y,v_y)+(u,v)\\ & =((\mathrm{\Delta }+I)u,v)+({\partial }_{n}u,v{)}_{\partial \mathrm{\Omega }}\\ & =(f,v)+({\partial }_{n}u,v{)}_{\partial \mathrm{\Omega }}.\end{array}$

Hence the Neumann boundary conditions are satisfied:

${\partial }_{n}u{|}_{\partial \mathrm{\Omega }}=0,$

where the left hand side is regarded as an element of Template:Math and hence Template:Math.

Regularity of strong solutions

The main result here states that if Template:Math and Template:Math, then Template:Math and

$\Vert u{\Vert }_{(k+2)}\le C\Vert \mathrm{\Delta }u{\Vert }_{(k)}+C\Vert u{\Vert }_{(k+1)},$

for some constant independent of Template:Mvar.

Like the corresponding result for the Dirichlet problem, this is proved by induction on Template:Math. For Template:Math, Template:Mvar is also a weak solution of the Neumann problem so satisfies the estimate above for Template:Math. The Neumann boundary condition can be written

$Zu{|}_{\partial \mathrm{\Omega }}=0.$

Since Template:Mvar commutes with the vector field Template:Mvar corresponding to the period flow Template:Math, the inductive method of proof used for the Dirichlet problem works equally well in this case: for the difference quotients Template:Math preserve the boundary condition when expressed in terms of Template:Mvar.

Smoothness of eigenfunctions

It follows by induction from the regularity theorem for the Neumann problem that the eigenfunctions of Template:Mvar in Template:Math lie in Template:Math. Moreover, any solution of Template:Math with Template:Math in Template:Math and Template:Mvar in Template:Math must have Template:Mvar in Template:Math. In both cases by the vanishing properties, the normal derivatives of the eigenfunctions and Template:Mvar vanish on Template:Math.

Solving the associated Neumann problem

The method above can be used to solve the associated Neumann boundary value problem:

$\{\begin{array}{c}\mathrm{\Delta }f{|}_{\mathrm{\Omega }}=0\\ {\partial }_{n}f{|}_{\partial \mathrm{\Omega }}=g& g\in C^{\mathrm{\infty }}(\partial \mathrm{\Omega })\end{array}$

By Borel's lemma Template:Mvar is the restriction of a function Template:Math. Let Template:Mvar be a smooth function such that Template:Math near the boundary. Let Template:Mvar be the solution of Template:Math with Template:Math. Then Template:Math solves the boundary value problem.

References

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