Sobolev spaces for planar domains#Application to smooth Riemann mapping theorem

Let {{math|Ω ⊂ R²}} be a bounded domain with smooth boundary. Since {{math|Ω}} is contained in a large square in {{math|R²}}, it can be regarded as a domain in {{math|T²}} by identifying opposite sides of the square. The theory of Sobolev spaces on {{math|T²}} can be found in {{harvtxt|Bers|John|Schechter|1979}}, an account which is followed in several later textbooks such as {{harvtxt|Warner|1983}} and {{harvtxt|Griffiths|Harris|1994}}.

For {{mvar|k}} an integer, the (restricted) Sobolev space {{math|H{{su|p=k|b=0}}(Ω)}} is defined as the closure of {{math|C{{su|p=∞|b=c}}(Ω)}} in the standard Sobolev space {{math|H^k(T²)}}.

• {{math|H{{su|p=0|b=0}}(Ω) {{=}} L²(Ω)}}.
• Vanishing properties on boundary: For {{math|k > 0}} the elements of {{math|H{{su|p=k|b=0}}(Ω)}} are referred to as "{{math|L²}} functions on {{math|Ω}} which vanish with their first {{math|k − 1}} derivatives on {{math|∂Ω}}."{{harvnb|Bers|John|Schechter|1979|pp=192–193}} In fact if {{math| f  ∈ C^k({{overline|Ω}})}} agrees with a function in {{math|H{{su|p=k|b=0}}(Ω)}}, then {{math|g {{=}} ∂^αf }} is in {{math|C¹}}. Let {{math| f_n ∈ C{{su|p=∞|b=c}}(Ω)}} be such that {{math| f_n → f }} in the Sobolev norm, and set {{math|g_n {{=}} ∂^αf_n }}. Thus {{math|g_n → g}} in {{math|H{{su|p=1|b=0}}(Ω)}}. Hence for {{math|h ∈ C^∞(T²)}} and {{math|D {{=}} a∂_x + b∂_y}},

::$\backslash iint\_\backslash Omega\; \backslash left\; (g\; (Dh)\; +\; (Dg)\; h\; \backslash right\; )\; \backslash ,\; dx\; \backslash ,\; dy\; =\; \backslash lim\_\{n\backslash to\; 0\}\; \backslash iint\_\backslash Omega\; \backslash left\; (g\; (Dh\_n)\; +\; (Dg)\; h\_n\; \backslash right\; )\; \backslash ,\; dx\; \backslash ,\; dy\; =0.$

:By Green's theorem this implies

::$\backslash int\_\{\backslash partial\backslash Omega\}\; gk\; =0,$

:where

::$k\; =\; h\; \backslash cos\; \backslash left\; (\backslash mathbf\{n\}\; \backslash cdot\; (a,\; b)\; \backslash right\; ),$

:with {{math|n}} the unit normal to the boundary. Since such {{mvar|k}} form a dense subspace of {{math|L²(Ω)}}, it follows that {{math|g {{=}} 0}} on {{math|∂Ω}}.

• Support properties: Let {{math|Ω^c}} be the complement of {{math|{{overline|Ω}}}} and define restricted Sobolev spaces analogously for {{math|Ω^c}}. Both sets of spaces have a natural pairing with {{math|C^∞(T²)}}. The Sobolev space for {{math|Ω}} is the annihilator in the Sobolev space for {{math|T²}} of {{math|C{{su|p=∞|b=c}}(Ω^c)}} and that for {{math|Ω^c}} is the annihilator of {{math|C{{su|p=∞|b=c}}(Ω)}}.{{harvnb|Chazarain|Piriou|1982}} 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 {{mvar|g}} in {{math|H^k(T²)}} annihilates {{math|C{{su|p=∞|b=c}}(Ω^c)}}. By compactness, there are finitely many open sets {{math|U₀, U₁, ... , U_N}} covering {{math|{{overline|Ω}}}} such that the closure of {{math|U₀}} is disjoint from {{math|∂Ω}} and each {{math|U_i}} is an open disc about a boundary point {{math|z_i}} such that in {{math|U_i}} small translations in the direction of the normal vector {{math|n_i}} carry {{math|{{overline|Ω}}}} into {{math|Ω}}. Add an open {{math|U_N+1}} with closure in {{math|Ω^c}} to produce a cover of {{math|T²}} and let {{math|ψ_i}} be a partition of unity subordinate to this cover. If translation by {{mvar|n}} is denoted by {{math|λ_n}}, then the functions

::$g\_t=\backslash psi\_0\; g\; +\backslash sum\_\{i=1\}^N\; \backslash psi\_n\; \backslash lambda\_\{tn\_i\}\; g$

:tend to {{mvar|g}} as {{mvar|t}} decreases to {{math|0}} and still lie in the annihilator, indeed they are in the annihilator for a larger domain than {{math|Ω^c}}, the complement of which lies in {{math|Ω}}. Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in {{math|Ω}}. These are necessarily smooth functions of compact support in {{math|Ω}}.

• Further vanishing properties on the boundary: The characterization in terms of annihilators shows that {{math| f  ∈ C^k({{overline|Ω}})}} lies in {{math|H{{su|p= k|b=0}}(Ω)}} if (and only if) it and its derivatives of order less than {{mvar|k}} vanish on {{math|∂Ω}}.{{harvnb|Folland|1995|p=226}} In fact {{math| f }} can be extended to {{math|T²}} by setting it to be {{math|0}} on {{math|Ω^c}}. This extension {{mvar|F}} defines an element in {{math|H^k(T²)}} using the formula for the norm

::$\backslash |h\backslash |\_\{(k)\}^2=\backslash sum\_\{j=0\}^k\; \{k\backslash choose\; j\}\; \backslash left\; \backslash |\backslash partial\_x^j\backslash partial\_y^\{k-j\}\; h\; \backslash right\backslash |^2.$

:Moreover {{math|F}} satisfies {{math|(F, g) {{=}} 0}} for g in {{math|C{{su|p=∞|b=c}}(Ω^c)}}.

• Duality: For {{math|k ≥ 0}}, define {{math|H^−k(Ω)}} to be the orthogonal complement of {{math|H{{su|p=−k|b=0}}(Ω^c)}} in {{math|H^−k(T²)}}. Let {{math|P_k}} be the orthogonal projection onto {{math|H^−k(Ω)}}, so that {{math|Q_k {{=}} I − P_k}} is the orthogonal projection onto {{math|H{{su|p=−k|b=0}}(Ω^c)}}. When {{math|k {{=}} 0}}, this just gives {{math|H⁰(Ω) {{=}} L²(Ω)}}. If {{math| f  ∈ H{{su|p=k|b=0}}(Ω^c)}} and {{math|g ∈ H^−k(T²)}}, then

::$(f,g)=\; (f,\; P\_kg).$

:This implies that under the pairing between {{math|H^k(T²)}} and {{math|H^−k(T²)}}, {{math|H{{su|p=k|b=0}}(Ω^c)}} and {{math|H^−k(Ω)}} are each other's duals.

• Approximation by smooth functions: The image of {{math|C{{su|p=∞|b=c}}(Ω)}} is dense in {{math|H^−k(Ω)}} for {{math|k ≤ 0}}. This is obvious for {{math|k {{=}} 0}} since the sum {{math|C{{su|p=∞|b=c}}(Ω)}} + {{math|C{{su|p=∞|b=c}}(Ω^c)}} is dense in {{math|L²(T²)}}. Density for {{math|k < 0}} follows because the image of {{math|L²(T²)}} is dense in {{math|H^−k(T²)}} and {{math|P_k}} annihilates {{math|C{{su|p=∞|b=c}}(Ω^c)}}.
• Canonical isometries: The operator {{math|(I + ∆)^k}} gives an isometry of {{math|H{{su|p= 2k|b=0}}(Ω)}} into {{math|H⁰(Ω)}} and of {{math|H{{su|p= k|b=0}}(Ω)}} onto {{math|H^−k(Ω)}}. In fact the first statement follows because it is true on {{math|T²}}. That {{math|(I + ∆)^k}} is an isometry on {{math|H{{su|p= k|b=0}}(Ω)}} follows using the density of {{math|C{{su|p=∞|b=c}}(Ω)}} in {{math|H^−k(Ω)}}: for {{math| f, g ∈ C{{su|p=∞|b=c}}(Ω)}} we have:

::$\backslash begin\{align\}$

\left \|P_k (I+\Delta)^k f \right \|_{(-k)} &= \sup_{\|g\|_{(-k)}=1} \left|\left ((I+\Delta)^kf,g \right )_{(-k)} \right | \\

&= \sup_{\|g\|_{(-k)}=1} |(f,g)| \\

&=\|f\|_{(k)}.

\end{align}

:Since the adjoint map between the duals can by identified with this map, it follows that {{math|(I + ∆)^k}} is a unitary map.

{{see also|Dirichlet problem}}

===Invertibility of {{math|∆}}===

The operator {{math|∆}} defines an isomorphism between {{math|H{{su|p=1|b=0}}(Ω)}} and {{math|H⁻¹(Ω)}}. In fact it is a Fredholm operator of index {{math|0}}. The kernel of {{math|∆}} in {{math|H¹(T²)}} consists of constant functions and none of these except zero vanish on the boundary of {{math|Ω}}. Hence the kernel of {{math|H{{su|p=1|b=0}}(Ω)}} is {{math|(0)}} and {{math|∆}} is invertible.

In particular the equation {{math|∆f {{=}} g}} has a unique solution in {{math|H{{su|p=1|b=0}}(Ω)}} for {{mvar|g}} in {{math|H⁻¹(Ω)}}.

===Eigenvalue problem===

Let {{mvar|T}} be the operator on {{math|L²(Ω)}} defined by

:$T\; =\; R\_1\backslash Delta^\{-1\}R\_0,$

where {{math|R₀}} is the inclusion of {{math|L²(Ω)}} in {{math|H⁻¹(Ω)}} and {{math|R₁}} of {{math|H{{su|p=1|b=0}}(Ω)}} in {{math|L²(Ω)}}, both compact operators by Rellich's theorem. The operator {{mvar|T}} is compact and self-adjoint with {{math|(Tf, f ) > 0}} for all {{mvar|f}}. By the spectral theorem, there is a complete orthonormal set of eigenfunctions {{math|f_n}} in {{math|L²(Ω)}} with

:$Tf\_n\; =\; \backslash mu\_nf\_n,\; \backslash qquad\; \backslash mu\_n\; >0,\; \backslash mu\_n\; \backslash to\; 0.$

Since {{math|μ_n > 0}}, {{math|f_n}} lies in {{math|H{{su|p=1|b=0}}(Ω)}}. Setting {{math|λ_n {{=}} μ_−n}}, the {{math|f_n}} are eigenfunctions of the Laplacian:

:$\backslash Delta\; f\_n\; =\; \backslash lambda\_n\; f\_n,\backslash qquad\; \backslash lambda\_n>0,\; \backslash lambda\_n\backslash to\; \backslash infty.$

To determine the regularity properties of the eigenfunctions {{math| f_n }} and solutions of

:$\backslash Delta\; f\; =u,\backslash qquad\; u\backslash in\; H^\{-1\}(\backslash Omega),\; f\backslash in\; H^1\_0(\backslash Omega),$

enlargements of the Sobolev spaces {{math|H{{su|p=k|b=0}}(Ω)}} have to be considered. Let {{math|C^∞(Ω⁻)}} be the space of smooth functions on {{math|Ω}} which with their derivatives extend continuously to {{math|{{overline|Ω}}}}. By Borel's lemma, these are precisely the restrictions of smooth functions on {{math|T²}}. The Sobolev space {{math|H^k(Ω)}} is defined to the Hilbert space completion of this space for the norm

:$\backslash |f\backslash |\_\{(k)\}^2\; =\; \backslash sum\_\{j=0\}^k\; \{k\backslash choose\; j\}\; \backslash left\; \backslash |\backslash partial\_x^j\backslash partial\_y^\{k-j\}f\; \backslash right\backslash |^2.$

This norm agrees with the Sobolev norm on {{math|C{{su|p=∞|b=c}}(Ω)}} so that {{math|H{{su|p=k|b=0}}(Ω)}} can be regarded as a closed subspace of {{math|H^k(Ω)}}. Unlike {{math|H{{su|p=k|b=0}}(Ω)}}, {{math|H^k(Ω)}} is not naturally a subspace of {{math|H^k(T²)}}, but the map restricting smooth functions from {{math|T²}} to {{math|{{overline|Ω}}}} is continuous for the Sobolev norm so extends by continuity to a map {{math|ρ_k : H^k(T²) → H^k(Ω)}}.

• 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 {{math|ρ_k}} to the orthogonal complement of its kernel defines an isomorphism onto {{math|H^k(Ω)}}. The extension map {{math|E_k}} is defined to be the inverse of this map: it is an isomorphism (not necessarily norm preserving) of {{math|H^k(Ω)}} onto the orthogonal complement of {{math|H{{su|p=k|b=0}}(Ω^c)}} such that {{math|ρ_k ∘ E_k {{=}} I}}. On {{math|C{{su|p=∞|b=c}}(Ω)}}, it agrees with the natural inclusion map. Bounded extension maps {{math|E_k}} of this kind from {{math|H^k(Ω)}} to {{math|H^k(T²)}} 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 {{harvtxt|Stein|1970}}.

:It is sufficient to construct an extension {{mvar|E}} for a neighbourhood of a closed annulus, since a collar around the boundary is diffeomorphic to an annulus {{math|I × T}} with {{mvar|I}} a closed interval in {{math|T}}. Taking a smooth bump function {{mvar|ψ}} with {{math|0 ≤ ψ ≤ 1}}, equal to 1 near the boundary and 0 outside the collar, {{math|E(ψf ) + (1 − ψ) f }} will provide an extension on {{math|Ω}}. On the annulus, the problem reduces to finding an extension for {{math|C^k( I )}} in {{math|C^k(T)}}. Using a partition of unity the task of extending reduces to a neighbourhood of the end points of {{mvar|I}}. Assuming 0 is the left end point, an extension is given locally by

::$E(f)\; =\; \backslash sum\_\{m=0\}^k\; a\_m\; f\; \backslash left\; (-\backslash frac\{x\}\{m+1\}\; \backslash right\; ).$

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

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

:This matrix equation is solvable because the determinant is non-zero by Vandermonde's formula. It is straightforward to check that the formula for {{math|E( f )}}, when appropriately modified with bump functions, leads to an extension which is continuous in the above Sobolev norm.{{harvnb|Folland|1995}}

• Restriction theorem: The restriction map {{math|ρ_k}} is surjective with {{math|ker ρ_k {{=}} H{{su|p=k|b=0}}(Ω^c)}}. This is an immediate consequence of the extension theorem and the support properties for Sobolev spaces with boundary condition.
• Duality: {{math|H^k(Ω)}} 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:

::$\backslash cdots\; \backslash subset\; H^2(\backslash Omega)\; \backslash subset\; H^1(\backslash Omega)\; \backslash subset\; H^0(\backslash Omega)\; \backslash subset\; H^\{-1\}\_0(\backslash Omega)\; \backslash subset\; H^\{-2\}\_0(\backslash Omega)\; \backslash subset\; \backslash cdots$

:The differentiation operators {{math|∂_x, ∂_y}} carry each Sobolev space into the larger one with index 1 less.

• Sobolev embedding theorem: {{math|H^k+2(Ω)}} is contained in {{math|C^k(Ω⁻)}}. This is an immediate consequence of the extension theorem and the Sobolev embedding theorem for {{math|H^k+2(T²)}}.
• Characterization: {{math|H^k(Ω)}} consists of {{math| f }} in {{math|L²(Ω) {{=}} H⁰(Ω)}} such that all the derivatives ∂^αf lie in {{math|L²(Ω)}} for |α| ≤ k. Here the derivatives are taken within the chain of Sobolev spaces above.See:
• {{harvnb|Agmon|2010}}
• {{harvnb|Folland|1995|pp=219–223}}
• {{harvnb|Chazarain|Piriou|1982|p=94}} Since {{math|C{{su|p=∞|b=c}}(Ω)}} is weakly dense in {{math|H^k(Ω)}}, this condition is equivalent to the existence of {{math|L²}} functions f_α such that

::$\backslash left(f\_\backslash alpha,\backslash varphi\; \backslash right)=(-1)^$

:To prove the characterization, note that if {{math| f }} is in {{math|H^k(Ω)}}, then {{math|∂^αf }} lies in H^k−|α|(Ω) and hence in {{math|H⁰(Ω) {{=}} L²(Ω)}}. Conversely the result is well known for the Sobolev spaces {{math|H^k(T²)}}: the assumption implies that the {{math|(∂_x − i∂_y)^k f }} is in {{math|L²(T²)}} and the corresponding condition on the Fourier coefficients of {{math| f }} shows that {{math| f }} lies in {{math|H^k(T²)}}. Similarly the result can be proved directly for an annulus {{math|[−δ, δ] × T}}. In fact by the argument on {{math|T₂}} the restriction of {{math| f }} to any smaller annulus [−δ',δ'] × T lies in {{math|H^k}}: equivalently the restriction of the function {{math| f_R (x, y) {{=}} f (Rx, y)}} lies in {{math|H^k}} for {{math|R > 1}}. On the other hand {{math|∂^α f_R → ∂^α f}} in {{math|L²}} as {{math|R → 1}}, so that {{math| f }} must lie in {{math|H^k}}. The case for a general domain {{math|Ω}} reduces to these two cases since {{math| f }} can be written as {{math| f {{=}} ψf + (1 − ψ) f }} with ψ a bump function supported in {{math|Ω}} such that {{math|1 − ψ}} is supported in a collar of the boundary.

• Regularity theorem: If {{math| f }} in {{math|L²(Ω)}} has both derivatives {{math|∂_x f}} and {{math|∂_y f}} in {{math|H^k(Ω)}} then {{math| f }} lies in {{math|H^k+1(Ω)}}. This is an immediate consequence of the characterization of {{math|H^k(Ω)}} above. In fact if this is true even when satisfied at the level of distributions: if there are functions g, h in {{math|H^k(Ω)}} such that (g,φ) = (f, φ_x) and (h,φ) = (f,φ_y) for φ in {{math|C{{su|p=∞|b=c}}(Ω)}}, then {{math| f }} is in {{math|H^k+1(Ω)}}.
• Rotations on an annulus: For an annulus {{math|I × T}}, the extension map to {{math|T²}} is by construction equivariant with respect to rotations in the second variable,

::$R\_t\; f(x,y)=f(x,y+t).$

:On {{math|T²}} it is known that if {{math| f }} is in {{math|H^k}}, then the difference quotient {{math|δ_h f {{=}} h⁻¹(R_h f − f ) → ∂_y f }} in {{math|H^k−1}}; if the difference quotients are bounded in H^k then ∂_yf lies in {{math|H^k}}. Both assertions are consequences of the formula:

::$\backslash widehat\{\backslash delta\_h\; f\}(m,n)\; =\; h^\{-1\}(e^\{-ihn\}-1)\backslash widehat\{f\}(m,n)=-\backslash int\_0^1\; in\; e^\{-inht\}\backslash ,\; dt\; \backslash ,\backslash ,\backslash widehat\{f\}(m,n).$

:These results on {{math|T²}} imply analogous results on the annulus using the extension.

===Regularity for dual Dirichlet problem===

If {{math|∆u {{=}} f}} with {{mvar|u}} in {{math|H{{su|p=1|b=0}}(Ω)}} and {{mvar|f}} in {{math|H^k−1(Ω)}} with {{math|k ≥ 0}}, then {{mvar|u}} lies in {{math|H^k+1(Ω)}}.

Take a decomposition {{math|u {{=}} ψu + (1 − ψ)u}} with {{mvar|ψ}} supported in {{math|Ω}} and {{math|1 − ψ}} supported in a collar of the boundary. Standard Sobolev theory for {{math|T²}} can be applied to {{math|ψu}}: elliptic regularity implies that it lies in {{math|H^k+1(T²)}} and hence {{math|H^k+1(Ω)}}. {{math|v {{=}} (1 − ψ)u}} lies in {{math|H{{su|p=1|b=0}}}} of a collar, diffeomorphic to an annulus, so it suffices to prove the result with {{math|Ω}} a collar and {{math|∆}} replaced by

:$\backslash begin\{align\}$

\Delta_1 &= \Delta -[\Delta,\psi] \\

&=\Delta + \left(p\partial_x + q\partial_y - \Delta \psi \right) \\

&=\Delta +X.

\end{align}

The proof{{harvnb|Taylor|2011}} proceeds by induction on {{mvar|k}}, proving simultaneously the inequality

:$\backslash |u\backslash |\_\{(k+1)\}\; \backslash le\; C\backslash |\backslash Delta\_1\; u\backslash |\_\{(k-1)\}\; +\; C\; \backslash |u\backslash |\_\{(k)\},$

for some constant {{mvar|C}} depending only on {{mvar|k}}. It is straightforward to establish this inequality for {{math|k {{=}} 0}}, where by density {{mvar|u}} can be taken to be smooth of compact support in {{math|Ω}}:

:$\backslash begin\{align\}$

\|u\|_{(1)}^2 &= |(\Delta u, u)| \\

&\le |(\Delta_1 u,u)| +|(Xu,u)| \\

&\le \|\Delta_1 u\|_{(-1)}\|u\|_{(1)} +C^\prime \|u\|_{(1)}\|u\|_{(0)}.

\end{align}

The collar is diffeomorphic to an annulus. The rotational flow {{math|R_t}} on the annulus induces a flow {{math|S_t}} on the collar with corresponding vector field {{math|Y {{=}} r∂_x + s∂_y}}. Thus {{mvar|Y}} corresponds to the vector field {{math|∂_θ}}. The radial vector field on the annulus {{math|r∂_r}} is a commuting vector field which on the collar gives a vector field {{math|Z {{=}} p∂_x + q∂_y}} proportional to the normal vector field. The vector fields {{mvar|Y}} and {{mvar|Z}} commute.

The difference quotients {{math|δ_hu}} can be formed for the flow {{math|S_t}}. The commutators {{math|[δ_h, ∆₁]}} are second order differential operators from {{math|H^k+1(Ω)}} to {{math|H^k−1(Ω)}}. Their operators norms are uniformly bounded for {{mvar|h}} near {{math|0}}; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of {{math|∆₁}} by their difference quotients composed with {{math|S_h}}. On the other hand, {{math|v {{=}} δ_hu}} lies in {{math|H{{su|p=1|b=0}}(Ω)}}, so the inequalities for {{mvar|u}} apply equally well for {{mvar|v}}:

:$\backslash begin\{align\}$

\|\delta_h u\|_{(k+1)} &\le C\|\Delta_1 \delta_h u\|_{(k-1)} + C \|\delta_h u\|_{(k)} \\

&\le C \|\delta_h \Delta_1 u\|_{(k-1)} + C\|[\delta_h,\Delta_1] u\|_{(k-1)} + C \|\delta_h u\|_{(k)} \\

&\le C\|\Delta_1 u\|_{(k)} + C^\prime \|u\|_{(k+1)}.

\end{align}

The uniform boundedness of the difference quotients {{math|δ_hu}} implies that {{math|Yu}} lies in {{math|H^k+1(Ω)}} with

:$\backslash |Y\; u\backslash |\_\{(k+1)\}\; \backslash le\; C\; \backslash |\backslash Delta\_1\; u\backslash |\_\{(k)\}\; +\; C^\backslash prime\; \backslash |u\backslash |\_\{(k+1)\}.$

It follows that {{math|Vu}} lies in {{math|H^k+1(Ω)}} where {{mvar|V}} is the vector field

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

Moreover, {{math|Vu}} satisfies a similar inequality to {{math|Yu}}.

:$\backslash |V\; u\backslash |\_\{(k+1)\}\; \backslash le\; C^\{\backslash prime\backslash prime\}\; \backslash left\; (\backslash |\backslash Delta\_1\; u\backslash |\_\{(k)\}\; +\; \backslash |u\backslash |\_\{(k+1)\}\; \backslash right).$

Let {{mvar|W}} be the orthogonal vector field

:$W=-b\backslash partial\_x\; +\; a\backslash partial\_y.$

It can also be written as {{math|ξZ}} for some smooth nowhere vanishing function {{mvar|ξ}} on a neighbourhood of the collar.

It suffices to show that {{math|Wu}} lies in {{math|H^k+1(Ω)}}. For then

:$(V\; \backslash pm\; iW)u\; =(a\; \backslash mp\; i\; b)\; \backslash left\; (\backslash partial\_x\; \backslash pm\; i\; \backslash partial\_y\; \backslash right)u,$

so that {{math|∂_xu}} and {{math|∂_yu}} lie in {{math|H^k+1(Ω)}} and {{mvar|u}} must lie in {{math|H^k+2(Ω)}}.

To check the result on {{math|Wu}}, it is enough to show that {{math|VWu}} and {{math|W²u}} lie in {{math|H^k(Ω)}}. Note that

:$\backslash begin\{align\}$

A &=\Delta - V^2 - W^2, \\

B &=[V,W],

\end{align}

are vector fields. But then

:$\backslash begin\{align\}$

W^2u &=\Delta u - V^2u - Au, \\

VWu &= WVu + Bu,

\end{align}

with all terms on the right hand side in {{math|H^k(Ω)}}. Moreover, the inequalities for {{math|Vu}} show that

:$\backslash begin\{align\}$

\|Wu\|_{(k+1)} &\le C \left (\|VWu\|_{(k)} + \left \|W^2u \right \|_{(k)} \right ) \\

&\le C \left \| \left (\Delta - V^2 -A \right )u \right \|_{(k)} + C \|(WV+B)u\|_{(k)} \\

&\le C_1 \|\Delta_1 u\|_{(k)} + C_1 \|u\|_{(k+1)}.

\end{align}

Hence

:$\backslash begin\{align\}$

\|u\|_{(k+2)} &\le C \left (\|Vu\|_{(k+1)} + \|Wu\|_{(k+1)} \right ) \\

&\le C^\prime \|\Delta_1 u\|_{(k)} + C^\prime \|u\|_{(k+1)}.

\end{align}

===Smoothness of eigenfunctions===

It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of {{math|∆}} in {{math|H{{su|p=1|b=0}}(Ω)}} lie in {{math|C^∞(Ω⁻)}}. Moreover, any solution of {{math|∆u {{=}} f}} with {{mvar|f}} in {{math|C^∞(Ω⁻)}} and {{mvar|u}} in {{math|H{{su|p=1|b=0}}(Ω)}} must have {{mvar|u}} in {{math|C^∞(Ω⁻)}}. In both cases by the vanishing properties, the eigenfunctions and {{mvar|u}} vanish on the boundary of {{math|Ω}}.

===Solving the Dirichlet problem===

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

:

By Borel's lemma {{mvar|g}} is the restriction of a function {{mvar|G}} in {{math|C^∞(Ω⁻)}}. Let {{mvar|F}} be the smooth solution of {{math|∆F {{=}} ∆G}} with {{math|F {{=}} 0}} on {{math|∂Ω}}. Then {{math|f {{=}} G − F}} solves the Dirichlet problem. By the maximal principle, the solution is unique.{{harvnb|Folland|1995|p=84}}

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.{{harvnb|Taylor|2011|pp=323–325}} For multiply connected regions with smooth boundary {{harvtxt|Schiffer|Hawley|1962}} 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 {{mvar|g}} such that:

• {{mvar|g}} is harmonic in the interior of {{math|Ω}};
• On {{math|∂Ω}} we have: {{math|∂_ng {{=}} κ − Ke^G}}, where {{mvar|κ}} is the curvature of the boundary curve, {{math|∂_n}} is the derivative in the direction normal to {{math|∂Ω}} and {{mvar|K}} is constant on each boundary component.

{{harvtxt|Taylor|2011}} gives a proof of the Riemann mapping theorem for a simply connected domain {{math|Ω}} with smooth boundary. Translating if necessary, it can be assumed that {{math|0 ∈ Ω}}. The solution of the Dirichlet problem shows that there is a unique smooth function {{math|U(z)}} on {{math|{{overline|Ω}}}} which is harmonic in {{math|Ω}} and equals {{math|−log{{!}}z{{!}}}} on {{math|∂Ω}}. Define the Green's function by {{math|G(z) {{=}} log{{!}}z{{!}} + U(z)}}. It vanishes on {{math|∂Ω}} and is harmonic on {{math|Ω}} away from {{math|0}}. The harmonic conjugate {{mvar|V}} of {{mvar|U}} is the unique real function on {{math|Ω}} such that {{math|U + iV}} is holomorphic. As such it must satisfy the Cauchy–Riemann equations:

:$\backslash begin\{align\}$

U_x &= -V_y, \\

U_y &= V_x.

\end{align}

The solution is given by

:$V(z)=\backslash int\_0^z\; -U\_y\; dx\; +\; V\_x\; dy,$

where the integral is taken over any path in {{math|{{overline|Ω}}}}. It is easily verified that {{math|V_x}} and {{math|V_y}} exist and are given by the corresponding derivatives of {{mvar|U}}. Thus {{mvar|V}} is a smooth function on {{math|{{overline|Ω}}}}, vanishing at {{math|0}}. By the Cauchy-Riemann {{math| f {{=}} U + iV}} is smooth on {{math|{{overline|Ω}}}}, holomorphic on {{math|Ω}} and {{math| f (0) {{=}} 0}}. The function {{math|H {{=}} arg z + V(z)}} is only defined up to multiples of {{math|2π}}, but the function

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

is a holomorphic on {{math|Ω}} and smooth on {{math|{{overline|Ω}}}}. By construction, {{math|F(0) {{=}} 0}} and {{math|{{!}}F(z){{!}} {{=}} 1}} for {{math|z ∈ ∂Ω}}. Since {{mvar|z}} has winding number {{math|1}}, so too does {{math|F(z)}}. On the other hand, {{math|F(z) {{=}} 0}} only for {{math|z {{=}} 0}} where there is a simple zero. So by the argument principle {{mvar|F}} assumes every value in the unit disc, {{math|D}}, exactly once and {{math|F′}} does not vanish inside {{math|Ω}}. To check that the derivative on the boundary curve is non-zero amounts to computing the derivative of {{math|e^iH}}, i.e. the derivative of {{mvar|H}} 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 {{mvar|G}} vanishes on the boundary and is strictly negative in {{math|Ω}} since {{math|{{!}}F{{!}} {{=}} e^G}}. The Hopf lemma implies that the directional derivative of {{mvar|G}} in the direction of the outward normal is strictly positive. So on the boundary curve, {{mvar|F}} has nowhere vanishing derivative. Since the boundary curve has winding number one, {{mvar|F}} defines a diffeomorphism of the boundary curve onto the unit circle. Accordingly, {{math|F : {{overline|Ω}} → {{overline|D}}}} is a smooth diffeomorphism, which restricts to a holomorphic map {{math|Ω → D}} and a smooth diffeomorphism between the boundaries.

Similar arguments can be applied to prove the Riemann mapping theorem for a doubly connected domain {{math|Ω}} bounded by simple smooth curves {{math|C_i}} (the inner curve) and {{math|C_o}} (the outer curve). By translating we can assume 1 lies on the outer boundary. Let {{mvar|u}} be the smooth solution of the Dirichlet problem with {{math|U {{=}} 0}} on the outer curve and {{math|−1}} on the inner curve. By the maximum principle {{math|0 < u(z) < 1}} for {{mvar|z}} in {{math|Ω}} and so by the Hopf lemma the normal derivatives of {{mvar|u}} are negative on the outer curve and positive on the inner curve. The integral of {{math|−u_ydx + u_ydx}} 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 {{math|c > 0}} such that {{math|U {{=}} cu}} satisfies

:$\backslash int\_\{C\}\; \backslash left\; (-U\_y\; \backslash ,dx\; +\; U\_x\; \backslash ,\; dy\; \backslash right\; )\; =\; 2\backslash pi$

on each boundary curve. The harmonic conjugate {{mvar|V}} of {{mvar|U}} can again be defined by

:$V(z)=\backslash int\_1^z\; -u\_y\; \backslash ,dx\; +\; u\_x\; \backslash ,\; dy$

and is well-defined up to multiples of {{math|2π}}. The function

:$\backslash displaystyle\{F(z)=e^\{U(z)\; +\; i\; V(z)\}\}$

is smooth on {{math|{{overline|Ω}}}} and holomorphic in {{math|Ω}}. On the outer curve {{math|{{!}}F{{!}} {{=}} 1}} and on the inner curve {{math|{{!}}F{{!}} {{=}} e^−c {{=}} r < 1}}. 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 {{mvar|F}} restricts to a diffeomorphism between the boundary curves and the two concentric circles. Since the images of outer and inner curve have winding number {{math|1}} and {{math|0}} about any point in the annulus, an application of the argument principle implies that {{mvar|F}} assumes every value within the annulus {{math|r < {{!}}z{{!}} < 1}} exactly once; since that includes multiplicities, the complex derivative of {{mvar|F}} is nowhere vanishing in {{math|Ω}}. This {{mvar|F}} is a smooth diffeomorphism of {{math|{{overline|Ω}}}} onto the closed annulus {{math|r ≤ {{!}}z{{!}} ≤ 1}}, restricting to a holomorphic map in the interior and a smooth diffeomorphism on both boundary curves.

The restriction map {{math|τ : C^∞(T²) → C^∞(T) {{=}} C^∞(1 × T)}} extends to a continuous map {{math|H^k(T²) → H^{k − 1/2}(T)}} for {{math|k ≥ 1}}.{{harvnb|Chazarain|Piriou|1982}} In fact

:$\backslash displaystyle\{\backslash widehat\{\backslash tau\; f\}(n)=\backslash sum\_m\; \backslash widehat\{f\}(m,n),\}$

so the Cauchy–Schwarz inequality yields

:$\backslash begin\{align\}$

\left |\widehat{\tau f}(n) \right|^2 \left (1+n^2 \right )^{k-\frac{1}{2}} &\le \left(\sum_m \frac{\left (1+n^2 \right)^{k-\frac{1}{2}}}{\left (1+m^2+n^2 \right )^{k}}\right) \left(\sum_m \left |\widehat{f}(m,n) \right |^2 \left (1+m^2+n^2 \right )^k\right) \\

&\le C_k \sum_m \left |\widehat{f}(m,n) \right |^2 \left (1+m^2+n^2 \right )^k,

\end{align}

where, by the integral test,

:$\backslash begin\{align\}$

C_k &= \sup_n \sum_m \frac{\left (1+n^2 \right )^{k-\frac{1}{2}}}{\left (1+m^2 +n^2 \right )^k} < \infty, \\

c_k &= \inf_n \sum_m \frac{\left (1+n^2 \right )^{k-\frac{1}{2}}}{\left (1+m^2 +n^2 \right )^k} > 0.

\end{align}

The map {{mvar|τ}} is onto since a continuous extension map {{mvar|E}} can be constructed from {{math|H^{k − 1/2}(T)}} to {{math|H^k(T²)}}.{{harvnb|Taylor|2011|p=275}}{{harvnb|Renardy|Rogers|2004|pp=214–218}} In fact set

:$\backslash widehat\{Eg\}(m,n)=\backslash lambda\_\{n\}^\{-1\}\; \backslash widehat\{g\}(n)\; \backslash frac\{\backslash left\; (1+n^2\; \backslash right\; )^\{k-\backslash frac\{1\}\{2\}\}\}\{\backslash left\; (1+n^2+m^2\; \backslash right\; )^k\},$

where

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

Thus {{math|c_k < λ_n < C_k}}. If g is smooth, then by construction Eg restricts to g on 1 × T. Moreover, E is a bounded linear map since

:$\backslash begin\{align\}$

\|Eg\|_{(k)}^2 &= \sum_{m,n} \left |\widehat{Eg}(m,n) \right |^2 \left (1+m^2+n^2 \right ) \\

&\le c_k^{-2} \sum_{m,n} \left | \widehat{g}(n) \right |^2 \frac{\left (1+n^2 \right )^{2k-1}}{ \left (1+m^2+n^2 \right )^k} \\

&\le c_k^{-2} C_k \| g \|_{k-\frac{1}{2}}^2.

\end{align}

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{{harvnb|Hörmander|1990|pp=240–241}}

:$\backslash |f\backslash |\_\{[k+\backslash frac\{1\}\{2\}]\}^2\; =\; \backslash |f\backslash |\_\{(k)\}^2\; +\; \backslash int\_0^\{2\backslash pi\}\backslash int\_0^\{2\backslash pi\}\; \backslash frac\{\backslash left\; |f^\{(k)\}(s)-f^\{(k)\}(t)\; \backslash right\; |^2\}\{\backslash left\; |e^\{is\}\; -e^\{it\}\; \backslash right|^2\}\; \backslash ,ds\backslash ,\; dt.$

It is also a consequence of the properties of τ and E (the "trace theorem").{{harvnb|Renardy|Rogers|2004}} 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{{harvnb|Chazarain|Piriou|1982}}{{harvnb|Renardy|Rogers|2004}}

:$(0)\; \backslash to\; H^1\_0(\backslash Omega)\; \backslash to\; H^1(\backslash Omega)\; \backslash to\; H^\{\backslash frac\{1\}\{2\}\}\; (\backslash partial\backslash Omega)\; \backslash to\; (0)$

and

:$(0)\; \backslash to\; H^2\_0(\backslash Omega)\; \backslash to\; H^2(\backslash Omega)\; \backslash to\; H^\{\backslash frac\{3\}\{2\}\}\; (\backslash partial\backslash Omega)\; \backslash oplus\; H^\{\backslash frac\{1\}\{2\}\}(\backslash partial\backslash Omega)\; \backslash to\; (0),$

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:

:$(0)\backslash to\; H^k\_0(\backslash Omega)\; \backslash to\; H^k(\backslash Omega)\; \backslash to\; \backslash bigoplus\_\{j=1\}^k\; H^\{j-\backslash frac\{1\}\{2\}\}(\backslash partial\backslash Omega)\; \backslash to\; (0).$

The trace map to {{math|H^{j − 1/2}(∂Ω)}} takes f to {{math|∂{{su|b=n|p=k − j}}f {{!}}_∂Ω}}

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 {{mvar|f}} in {{math|H¹(Ω)}}, the normal derivative {{math|∂_nf {{!}}_∂Ω}} cannot be a priori defined at the level of Sobolev spaces. Instead an alternative formulation of boundary value problems for the Laplacian {{math|Δ}} on a bounded region {{math|Ω}} in the plane is used. It employs Dirichlet forms, sesqulinear bilinear forms on {{math|H¹(Ω)}}, {{math|H{{su|p=1|b=0}}(Ω)}} 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 {{math|H²(Ω)}}, 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 {{math|f {{!}}_∂Ω}} is already defined on {{math|H¹(Ω)}}, Dirichlet forms do not need to be mentioned explicitly and the operator formulation is more direct. A unified discussion is given in {{harvtxt|Folland|1995}} 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 {{harvtxt|Taylor|2011}}.

The Hilbert space formulation of boundary value problems for the Laplacian {{math|Δ}} on a bounded region {{math|Ω}} in the plane proceeds from the following data:{{harvnb|Folland|1995|pp=231–248}}

• A closed subspace {{math|H{{su|p=1|b=0}}(Ω) ⊆ H ⊆ H¹(Ω)}}.
• A Dirichlet form for {{math|Δ}} given by a bounded Hermitian bilinear form {{math|D( f, g)}} defined for {{math|f, g ∈ H¹(Ω)}} such that {{math|D( f, g) {{=}} (∆f, g)}} for {{math|f, g ∈ H{{su|p=1|b=0}}(Ω)}}.
• {{mvar|D}} is coercive, i.e. there is a positive constant {{mvar|C}} and a non-negative constant {{mvar|λ}} such that {{math|D( f, f ) ≥ C ( f, f )₍₁₎ − λ( f, f )}}.

A weak solution of the boundary value problem given initial data {{mvar|f}} in {{math|L²(Ω)}} is a function u satisfying

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

for all g.

For both the Dirichlet and Neumann problem

:$D(f,g)\; =\; \backslash left\; (f\_x,\; g\_x\; \backslash right\; )\; +\; \backslash left\; (f\_y,\; g\_y\; \backslash right\; ).$

For the Dirichlet problem {{math|H {{=}} H{{su|p=1|b=0}}(Ω)}}. In this case

:$D(f,g)\; =\; (\backslash Delta\; f,g),\; \backslash qquad\; f,\; g\; \backslash in\; H.$

By the trace theorem the solution satisfies {{math|u{{!}}_Ω {{=}} 0}} in {{math|H^1/2(∂Ω)}}.

For the Neumann problem {{mvar|H}} is taken to be {{math|H¹(Ω)}}.

The classical Neumann problem on {{math|Ω}} consists in solving the boundary value problem

:$\backslash begin\{cases\}$

\Delta u =f, & f, u \in C^{\infty}(\Omega^-),\\

\partial_n u =0 & \text{on } \partial \Omega

\end{cases}

Green's theorem implies that for {{math|u, v ∈ C^∞(Ω⁻)}}

:$(\backslash Delta\; u,v)=\; (u\_x,v\_x)\; +\; (u\_y,v\_y)\; -\; (\backslash partial\_n\; u,v)\_\{\backslash partial\; \backslash Omega\}.$

Thus if {{math|Δu {{=}} 0}} in {{math|Ω}} and satisfies the Neumann boundary conditions, {{math|u_x {{=}} u_y {{=}} 0}}, and so {{mvar|u}} is constant in {{math|Ω}}.

Hence the Neumann problem has a unique solution up to adding constants.{{harvnb|Taylor|2011}}

Consider the Hermitian form on {{math|H¹(Ω)}} defined by

:$\backslash displaystyle\{D(f,g)=(u\_x,v\_x)\; +\; (u\_y,v\_y).\}$

Since {{math|H¹(Ω)}} is in duality with {{math|H{{su|p=−1|b=0}}(Ω)}}, there is a unique element {{math|Lu}} in {{math|H{{su|p=−1|b=0}}(Ω)}} such that

:$\backslash displaystyle\{D(u,v)\; =(Lu,v).\}$

The map {{math|I + L}} is an isometry of {{math|H¹(Ω)}} onto {{math|H{{su|p=−1|b=0}}(Ω)}}, so in particular {{mvar|L}} is bounded.

In fact

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

So

:$\backslash |(L+I)u\backslash |\_\{(-1)\}\; =\backslash sup\_\{\backslash |v\backslash |\_\{(1)\}=1\}\; |((L+I)u,v)|=\; \backslash sup\_\{\backslash |v\backslash |\_\{(1)\}=1\}\; |(u,v)\_\{(1)\}|=\backslash |u\backslash |\_\{(1)\}.$

On the other hand, any {{math| f }} in {{math|H{{su|p=−1|b=0}}(Ω)}} defines a bounded conjugate-linear form on {{math|H¹(Ω)}} sending {{mvar|v}} to {{math|( f, v)}}. By the Riesz–Fischer theorem, there exists {{math|u ∈ H¹(Ω)}} such that

:$\backslash displaystyle\{(f,v)=(u,v)\_\{(1)\}.\}$

Hence {{math|(L + I)u {{=}} f }} and so {{math|L + I}} is surjective. Define a bounded linear operator {{mvar|T}} on {{math|L²(Ω)}} by

:$T=R\_1(I+L)^\{-1\}\; R\_0,$

where {{math|R₁}} is the map {{math|H¹(Ω) → L²(Ω)}}, a compact operator, and {{math|R₀}} is the map {{math|L²(Ω) → H{{su|p=−1|b=0}}(Ω)}}, its adjoint, so also compact.

The operator {{mvar|T}} has the following properties:

• {{mvar|T}} is a contraction since it is a composition of contractions
• {{mvar|T}} is compact, since {{math|R₀}} and {{math|R₁}} are compact by Rellich's theorem
• {{mvar|T}} is self-adjoint, since if {{math| f, g ∈ L²(Ω)}}, they can be written {{math| f {{=}} (L + I)u, g {{=}} (L + I)v}} with {{math|u, v ∈ H¹(Ω)}} so

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

• {{mvar|T}} has positive spectrum and kernel {{math|(0)}}, for

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

:and {{math|Tf {{=}} 0}} implies {{math|u {{=}} 0}} and hence {{math|f {{=}} 0}}.

• There is a complete orthonormal basis {{math| f_n}} of {{math|L²(Ω)}} consisting of eigenfunctions of {{mvar|T}}. Thus

::$Tf\_n=\backslash mu\_nf\_n$

:with {{math|0 < μ_n ≤ 1}} and {{math|μ_n}} decreasing to {{math|0}}.

• The eigenfunctions all lie in {{math|H¹(Ω)}} since the image of {{mvar|T}} lies in {{math|H¹(Ω)}}.
• The {{math| f_n}} are eigenfunctions of {{mvar|L}} with

::$\backslash displaystyle\{Lf\_n=\backslash lambda\_nf\_n,\; \backslash qquad\; \backslash lambda\_n=\backslash mu\_n^\{-1\}\; -1.\}$

:Thus {{math|λ_n}} are non-negative and increase to {{math|∞}}.

• The eigenvalue {{math|0}} occurs with multiplicity one and corresponds to the constant function. For if {{math|u ∈ H¹(Ω)}} satisfies {{math|Lu {{=}} 0}}, then

::$(u\_x,u\_x)\; +\; (u\_y,u\_y)\; =\; (Lu,u)\; =\; 0,$

:so {{mvar|u}} is constant.

The first main regularity result shows that a weak solution expressed in terms of the operator {{mvar|L}} and the Dirichlet form {{mvar|D}} is a strong solution in the classical sense, expressed in terms of the Laplacian {{math|Δ}} and the Neumann boundary conditions. Thus if {{math|u {{=}} Tf }} with {{math|u ∈ H¹(Ω),  f  ∈ L²(Ω)}}, then {{math|u ∈ H²(Ω)}}, satisfies {{math|Δu + u {{=}} f }} and {{math|∂_nu{{!}}_∂Ω {{=}} 0}}. Moreover, for some constant {{mvar|C}} independent of {{mvar|u}},

:$\backslash |u\backslash |\_\{(2)\}\; \backslash le\; C\backslash |\backslash Delta\; u\backslash |\_\{(0)\}\; +\; C\; \backslash |u\backslash |\_\{(1)\}.$

Note that

:$\backslash |u\backslash |\_\{(1)\}\; \backslash le\; \backslash |L\; u\backslash |\_\{(-1)\}\; +\; \backslash |u\backslash |\_\{(0)\},$

since

:$\backslash begin\{align\}$

\|u\|_{(1)}^2 &=|(L u, u)| +\|u\|^2_{(0)} \\

&\le \|L u\|_{(-1)} \|u\|_{(1)} + \|u\|_{(0)} \|u\|_{(1)}.

\end{align}

Take a decomposition {{math|u {{=}} ψu + (1 − ψ)u}} with {{mvar|ψ}} supported in {{math|Ω}} and {{math|1 − ψ}} supported in a collar of the boundary.

The operator {{mvar|L}} is characterized by

:$(Lf,g)=\; \backslash left\; (f\_x,\; g\_x\; \backslash right\; )+\; \backslash left\; (f\_y,\; g\_y\; \backslash right\; )=\; (\backslash Delta\; f,\; g)\_\backslash Omega\; -\; \backslash left\; (\backslash partial\_n\; f,\; g\; \backslash right\; )\_\{\backslash partial\backslash Omega\},\; \backslash qquad\; f,g\; \backslash in\; C^\{\backslash infty\}(\backslash Omega^-).$

Then

:$([L,\backslash psi]f,g)=([\backslash Delta,\backslash psi]f,g),$

so that

:$\backslash displaystyle\{[L,\backslash psi]=-[L,1-\backslash psi]=\backslash Delta\backslash psi\; +2\backslash psi\_x\backslash partial\_x\; +\; 2\backslash psi\_y\backslash partial\_y.\}$

The function {{math|v {{=}} ψu}} and {{math|w {{=}} (1 − ψ)u}} are treated separately, {{mvar|v}} being essentially subject to usual elliptic regularity considerations for interior points while {{mvar|w}} requires special treatment near the boundary using difference quotients. Once the strong properties are established in terms of {{math|∆}} and the Neumann boundary conditions, the "bootstrap" regularity results can be proved exactly as for the Dirichlet problem.

====Interior estimates====

The function {{math|v {{=}} ψu}} lies in {{math|H{{su|p=1|b=0}}(Ω₁)}} where {{math|Ω₁}} is a region with closure in {{math|Ω}}. If {{math| f  ∈ C{{su|p=∞|b=c}}(Ω)}} and {{math|g ∈ C^∞(Ω⁻)}}

:$(Lf,g)\; =\; (\backslash Delta\; f,g)\_\backslash Omega.$

By continuity the same holds with {{math| f }} replaced by {{mvar|v}} and hence {{math|Lv {{=}} ∆v}}. So

:$\backslash Delta\; v=\; Lv=L(\backslash psi\; u)\; =\backslash psi\; Lu\; +[L,\backslash psi]u=\backslash psi(f-u)\; +[\backslash Delta,\backslash psi]u.$

Hence regarding {{mvar|v}} as an element of {{math|H¹(T²)}}, {{math|∆v ∈ L²(T²)}}. Hence {{math|v ∈ H²(T²)}}. Since {{math|v {{=}} φv}} for {{math|φ ∈ C{{su|p=∞|b=c}}(Ω)}}, we have {{math|v ∈ H{{su|p=2|b=0}}(Ω)}}. Moreover,

:$\backslash |v\backslash |\_\{(2)\}^2\; =\; \backslash |\backslash Delta\; v\backslash |^2\; +\; 2\backslash |v\backslash |\_\{(1)\}^2,$

so that

:$\backslash |v\backslash |\_\{(2)\}\; \backslash le\; C\; \backslash left\; (\; \backslash |\backslash Delta(v)\backslash |+\; \backslash |v\backslash |\_\{(1)\}\; \backslash right\; ).$

====Boundary estimates====

The function {{math|w {{=}} (1 − ψ)u}} is supported in a collar contained in a tubular neighbourhood of the boundary. The difference quotients {{math|δ_hw}} can be formed for the flow {{math|S_t}} and lie in {{math|H¹(Ω)}}, so the first inequality is applicable:

:$\backslash begin\{align\}$

\|\delta_h w\|_{(1)} &\le \|L \delta_h w\|_{(-1)} + \|\delta_h w\|_{(0)} \\

&\le \|[L, \delta_h] w\|_{(-1)} +\|\delta_h Lw\|_{(-1)} + \| \delta_h w\|_{(0)} \\

&\le\|[L, \delta_h] w\|_{(-1)} +C\|Lw\|_{(0)} + C\| w\|_{(1)}.

\end{align}

The commutators {{math|[L, δ_h]}} are uniformly bounded as operators from {{math|H¹(Ω)}} to {{math|H{{su|p=−1|b=0}}(Ω)}}. This is equivalent to checking the inequality

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

for {{mvar|g}}, {{mvar|h}} 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 {{math|δ_h}} with a differential operator is obtained by applying the difference operator to the coefficients after having applied {{math|R_h}} to the function:{{harvnb|Folland|1995|pp=255–260}}

:$\backslash left\; [\; \backslash delta^h,\; \backslash sum\; a\_\backslash alpha\; \backslash partial^\backslash alpha\; \backslash right\; ]\; =\; \backslash left\; (\backslash delta^h(a\_\backslash alpha)\backslash circ\; R\_h\; \backslash right\; )\; \backslash partial^\backslash alpha.$

Hence the difference quotients {{math|δ_hw}} are uniformly bounded, and therefore {{math|Yw ∈ H¹(Ω)}} with

:$\backslash |Y\; w\backslash |\_\{(1)\}\; \backslash le\; C\backslash |L\; w\backslash |\_\{(0)\}\; +\; C^\backslash prime\; \backslash |w\backslash |\_\{(1)\}.$

Hence {{math|Vw ∈ H¹(Ω)}} and {{math|Vw}} satisfies a similar inequality to {{math|Yw}}:

:$\backslash |V\; w\backslash |\_\{(1)\}\; \backslash le\; C^\{\backslash prime\backslash prime\}\; \backslash left\; (\; \backslash |L\; w\backslash |\_\{(0)\}\; +\; \backslash |w\backslash |\_\{(1)\}\; \backslash right\; ).$

Let {{mvar|W}} be the orthogonal vector field. As for the Dirichlet problem, to show that {{math|w ∈ H²(Ω)}}, it suffices to show that {{math|Ww ∈ H¹(Ω)}}.

To check this, it is enough to show that {{math|VWw, W²u ∈ L²(Ω)}}. As before

:$\backslash begin\{align\}$

A &= \Delta - V^2 - W^2\\

B &= [V,W]

\end{align}

are vector fields. On the other hand, {{math|(Lw, φ) {{=}} (∆w, φ)}} for {{math|φ ∈ C{{su|p=∞|b=c}}(Ω)}}, so that {{math|Lw}} and {{math|∆w}} define the same distribution on {{math|Ω}}. Hence

:$\backslash begin\{align\}$

\left (W^2w, \varphi \right ) &= \left (L w - V^2w - Au,\varphi \right ),\\

(VWw,\varphi) &=(WVw + Bw,\varphi).

\end{align}

Since the terms on the right hand side are pairings with functions in {{math|L²(Ω)}}, the regularity criterion shows that {{math|Ww ∈ H²(Ω)}}. Hence {{math|Lw {{=}} ∆w}} since both terms lie in {{math|L²(Ω)}} and have the same inner products with {{mvar|φ}}'s.

Moreover, the inequalities for {{math|Vw}} show that

:$\backslash begin\{align\}$

\|Ww\|_{(1)} & \le C \left (\|VWw\|_{(0)} + \left \|W^2w \right \|_{(0)} \right ) \\

&\le C \left \| \left ( \Delta - V^2 -A \right )w \right \|_{(0)} + C \|(WV+B)w\|_{(0)} \\

&\le C_1 \|L w\|_{(0)} + C_1 \|w\|_{(1)}.

\end{align}

Hence

:$\backslash begin\{align\}$

\|w\|_{(2)} &\le C \left ( \|V w\|_{(1)} + \|W w\|_{(1)} \right ) \\

&\le C^\prime \|\Delta w \|_{(0)} + C^\prime \| w \|_{(1)}.

\end{align}

It follows that {{math|u {{=}} v + w ∈ H²(Ω)}}. Moreover,

:$\backslash begin\{align\}$

\|u\|_{(2)} &\le C \left ( \|\Delta v\| +\|\Delta w\| + \|v\|_{(1)} + \|w\|_{(1)} \right ) \\

&\le C^\prime \left ( \|\psi\Delta u\| + \|(1-\psi)\Delta u\| + 2\|[\Delta, \psi] u\| +\|u\|_{(1)} \right ) \\

&\le C^{\prime\prime} \left (\|\Delta u\| + \|u\|_{(1)} \right ).

\end{align}

====Neumann boundary conditions====

Since {{math|u ∈ H²(Ω)}}, Green's theorem is applicable by continuity. Thus for {{math|v ∈ H¹(Ω)}},

:$\backslash begin\{align\}$

(f,v) &= (Lu,v) + (u,v) \\

&= (u_x,v_x) + (u_y,v_y) +(u,v) \\

&= ((\Delta+I)u,v) +(\partial_{n} u,v)_{\partial\Omega} \\

&= (f,v) + (\partial_{n} u,v)_{\partial\Omega}.

\end{align}

Hence the Neumann boundary conditions are satisfied:

:$\backslash partial\_n\; u|\_\{\backslash partial\; \backslash Omega\}\; =\; 0,$

where the left hand side is regarded as an element of {{math|H^1/2(∂Ω)}} and hence {{math|L²(∂Ω)}}.

The main result here states that if {{math|u ∈ H^k+1 (k ≥ 1), ∆u ∈ H^k}} and {{math|∂_nu{{!}}_∂Ω {{=}} 0}}, then {{math|u ∈ H^k+2}} and

:$\backslash |u\backslash |\_\{(k+2)\}\; \backslash le\; C\backslash |\backslash Delta\; u\backslash |\_\{(k)\}\; +\; C\; \backslash |u\backslash |\_\{(k+1)\},$

for some constant independent of {{mvar|u}}.

Like the corresponding result for the Dirichlet problem, this is proved by induction on {{math|k ≥ 1}}. For {{math|k {{=}} 1}}, {{mvar|u}} is also a weak solution of the Neumann problem so satisfies the estimate above for {{math|k {{=}} 0}}. The Neumann boundary condition can be written

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

Since {{mvar|Z}} commutes with the vector field {{mvar|Y}} corresponding to the period flow {{math|S_t}}, the inductive method of proof used for the Dirichlet problem works equally well in this case: for the difference quotients {{math|δ^h}} preserve the boundary condition when expressed in terms of {{mvar|Z}}.{{harvnb|Taylor|2011|p=348}}

====Smoothness of eigenfunctions====

It follows by induction from the regularity theorem for the Neumann problem that the eigenfunctions of {{mvar|D}} in {{math|H¹(Ω)}} lie in {{math|C^∞(Ω⁻)}}. Moreover, any solution of {{math|Du {{=}} f }} with {{math| f }} in {{math|C^∞(Ω⁻)}} and {{mvar|u}} in {{math|H¹(Ω)}} must have {{mvar|u}} in {{math|C^∞(Ω⁻)}}. In both cases by the vanishing properties, the normal derivatives of the eigenfunctions and {{mvar|u}} vanish on {{math|∂Ω}}.

====Solving the associated Neumann problem====

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

:

By Borel's lemma {{mvar|g}} is the restriction of a function {{math|G ∈ C^∞(Ω⁻)}}. Let {{mvar|F}} be a smooth function such that {{math|∂_nF {{=}} G}} near the boundary. Let {{mvar|u}} be the solution of {{math|∆u {{=}} −∆F}} with {{math|∂_nu {{=}} 0}}. Then {{math| f {{=}} u + F}} solves the boundary value problem.{{harvnb|Folland|1995|p=85}}

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