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| In [[mathematics]], '''Sobolev spaces for planar domains''' are one of the principal techniques used in the theory of [[partial differential equation]]s for solving the [[Dirichlet boundary condition|Dirichlet]] and [[Neumann boundary condition|Neumann]] boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of [[bounded operator]]s on [[Hilbert space]]. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
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| ==Sobolev spaces with boundary conditions==
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| Let Ω be a bounded domain in '''R'''<sup>2</sup> with smooth boundary. Since Ω is contained in a large square in '''R'''<sup>2</sup>, it can be regarded as a domain in '''T'''<sup>2</sup> by identifying opposite sides of the square. The theory of Sobolev spaces on '''T'''<sup>2</sup> 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}}.
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| For ''k'' an integer, the (restricted) '''Sobolev space''' H<sup>''k''</sup><sub>0</sub>(Ω) is defined as the closure of C<sup>∞</sup><sub>''c''</sub>(Ω) in the standard [[Sobolev space]] H<sup>''k</sup>('''T'''<sup>2</sup>). | |
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| *For ''k'' = 0, H<sup>0</sup><sub>0</sub>(Ω) equals L<sup>2</sup>(Ω).
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| *'''Vanishing properties on boundary:''' For ''k'' > 0 the elements of H<sup>''k''</sup><sub>0</sub>(Ω) are referred to as "L<sup>2</sup> functions on Ω which vanish with their first ''k'' − 1 derivatives on the boundary."<ref>{{harvnb|Bers|John|Schechter|1979|pp=192–193}}</ref> In fact if
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| ::<math>\displaystyle{f \in C^k(\overline{\Omega})}</math>
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| :agrees with a function in H<sup>''k''</sup><sub>0</sub>(Ω), then ''g'' = ∂<sup>α</sup> ''f'' is in C<sup>1</sup>. Let ''f''<sub>''n''</sub> be smooth functions of compact support in Ω tending to ''f'' in the Sobolev norm and set ''g''<sub>''n''</sub> = ∂<sup>α</sup> ''f''<sub>''n''</sub>. Thus ''g''<sub>''n''</sub> tends towards ''g'' in H<sup>1</sup><sub>0</sub>(Ω). Hence for ''h'' in C<sup>∞</sup>('''T'''<sup>2</sup>) and ''D'' = ''a'' ∂<sub>''x''</sub> +b ∂<sub>''y''</sub>,
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| ::<math>\displaystyle{\iint_\Omega g (Dh) + (Dg) h \, dx dy =\lim_{n\rightarrow 0}\iint_\Omega g (Dh_n) + (Dg) h_n \, dx dy =0.}</math>
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| :By [[Green's theorem]] this implies
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| ::<math>\displaystyle{\int_{\partial\Omega} gk =0,}</math>
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| :where ''k'' = ''h'' cos '''n''' ⋅ (''a'',''b'') with '''n''' the unit normal to the boundary. Since such ''k'' form a dense subspace of L<sup>2</sup>(Ω), it follows that ''g'' = 0 on the boundary.
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| *'''Support properties:''' Let Ω<sup>''c''</sup> be the complement of the closure of Ω and define restricted Sobolev spaces analogously for Ω<sup>''c''</sup>. Both sets of spaces have a natural pairing with C<sup>∞</sup>('''T'''<sup>2</sup>). The Sobolev space for Ω is the annihilator in the Sobolev space for '''T'''<sup>2</sup> of C<sup>∞</sup><sub>''c''</sub>(Ω<sup>''c''</sup>) and that for Ω<sup>''c''</sup> is the annihilator of C<sup>∞</sup><sub>''c''</sub>(Ω).<ref>{{harvnb|Chazarain|Piriou|1982}}</ref> 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.
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| :Suppose ''g'' in H<sup>''k''</sup>('''T'''<sup>2</sup>) annihilates C<sup>∞</sup><sub>''c''</sub>(Ω<sup>''c''</sup>). By compactness, there are finitely many opens ''U''<sub>0</sub>, ''U''<sub>1</sub>, ... , ''U''<sub>''N''</sub> covering the closure of Ω such that the closure of ''U''<sub>0</sub> is disjoint from the boundary and each ''U''<sub>''i''</sub> is an open disc about a boundary point ''z''<sub>''i''</sub> such that in ''U''<sub>''i''</sub> small translations in the direction of the normal vector ''n''<sub>''i''</sub> carry the closure of Ω into Ω. Add an open ''U''<sub>''N''+1</sub> with closure in Ω<sup>''c''</sup> to produce a cover of '''T'''<sup>2</sup> and let ψ<sub>''i''</sup> be a [[partition of unity]] subordinate to this cover. If translation by ''n'' is denoted by ''λ''<sub>''n''</sub>, then
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| ::<math>\displaystyle{g_t=\psi_0 \cdot g +\sum_{i=1}^N \psi_n \lambda_{tn_i} g}</math>
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| :tend to ''g'' as ''t'' decreases to 0 and still lie in the annihilator, indeed they are in the annihilator for a larger domain than Ω<sup>''c''</sup>, the complement of which lies in Ω. Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in Ω. These are necessarily smooth functions of compact support in Ω.
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| *'''Further vanishing properties on the boundary:''' The characterization in terms of annihilators shows that a C<sup>''k''</sup> function ''f'' on the closure of Ω lies in H<sup>''k''</sup><sub>0</sub>(Ω) if (and only if) it and its derivatives of order less than ''k'' vanish on the boundary.<ref>{{harvnb|Folland|1995|p=226}}</ref> In fact ''f'' can be extended to '''T'''<sup>2</sup> by setting it to be 0 on Ω<sup>''c''</sup>. The extension ''F'' defines as element in H<sup>''k''</sup>('''T'''<sup>2</sup>) using the formula
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| ::<math>\displaystyle{\|h\|_{(k)}^2=\sum_{j=0}^k {k\choose j} \|\partial_x^j\partial_y^{k-j} h\|^2,}</math>
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| :and satisfies (''F'',''g'') = 0 for ''g'' in C<sup>∞</sup><sub>''c''</sub>(Ω<sup>''c''</sup>).
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| *'''Duality:''' For ''k'' ≥ 0, define H<sup>−''k''</sup>(Ω) to be the orthogonal complement in H<sup>−''k''</sup>('''T'''<sup>2</sup>) of H<sup>−''k''</sup><sub>0</sub>(Ω<sup>''c''</sup>). Let ''P''<sub>''k''</sub> be the orthogonal projection onto H<sup>−''k''</sup>(Ω), so that ''Q''<sub>''k''</sub> = ''I'' - ''P''<sub>''k''</sub> is the orthogonal projection onto H<sup>−''k''</sup><sub>0</sub>(Ω<sup>''c''</sup>). When ''k'' = 0, this just gives H<sup>0</sup>(Ω) = L<sup>2</sup>(Ω). If ''f'' is in H<sup>''k''</sup><sub>0</sub>(Ω) and ''g'' in H<sup>−''k''</sup>('''T'''<sup>2</sup>), then
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| ::<math>\displaystyle{(f,g)=(f,P_kg).}</math>
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| :This implies that under the pairing between H<sup>''k''</sup>('''T'''<sup>2</sup>) and H<sup>−''k''</sup>('''T'''<sup>2</sup>), H<sup>''k''</sup><sub>0</sub>(Ω) and H<sup>−''k''</sup>(Ω) are each other's duals.
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| *'''Approximation by smooth functions:''' The image of C<sup>∞</sup><sub>c</sub>(Ω) is dense in H<sup>−''k''</sup>(Ω) for ''k'' ≤ 0. This is obvious for ''k'' = 0 since the sum C<sup>∞</sup><sub>c</sub>(Ω) + C<sup>∞</sup><sub>c</sub>(Ω<sup>''c''</sup>) is dense in L<sup>2</sup>('''T'''<sup>2</sup>). Density for ''k'' < follows because the image of L<sup>2</sup>('''T'''<sup>2</sup>) is dense in H<sup>−''k''</sup>('''T'''<sup>2</sup>) and ''P''<sub>''k''</sub> annihilates C<sup>∞</sup><sub>c</sub>(Ω<sup>''c''</sup>).
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| *'''Canonical isometries:''' The operator (I + ∆)<sup>''k''</sup> gives an isometry of H<sup>2''k''</sup><sub>0</sub>(Ω) into H<sup>0</sup>(Ω) and of H<sup>''k''</sup><sub>0</sub>(Ω) onto H<sup>−''k''</sup>(Ω). In fact the first statement follows because it is true on '''T'''<sup>2</sup>. That (I + ∆)<sup>''k''</sup> is an isometry on H<sup>''k''</sup><sub>0</sub>(Ω) follows using the density of C<sup>∞</sup><sub>c</sub>(Ω) in H<sup>−''k''</sup>(Ω): for ''f'', ''g'' smooth of compact support on Ω
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| ::<math>\displaystyle{\|P_k(I+\Delta)^k f\|_{(-k)}=\sup_{\|g\|_{(-k)}=1} |((I+\Delta)^kf,g)_{(-k})|
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| =\sup_{\|g\|_{(-k)}=1} |(f,g)|=\|f\|_{(k)}.}</math>
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| :Since the adjoint map between the duals can by identified with this map, it follows that (I + ∆)<sup>''k''</sup> is a unitary map.
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| ==Application to Dirichlet problem==
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| {{see also|Dirichlet problem}}
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| *'''Invertibility of ∆:''' The operator ∆ defines an isomorphism between H<sup>1</sup><sub>0</sub>(Ω) and H<sup>−1</sup>(Ω). In fact it is a Fredholm operator of index 0. The kernel of ∆ in H<sup>1</sup>('''T'''<sup>2</sup>) consists of constant functions and none of these except zero vanish on the boundary of Ω. Hence the kernel of H<sup>1</sup><sub>0</sub>(Ω) is (0) and ∆ is invertible.
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| :In particular the equation ∆ ''f'' = ''g'' has a unique solution in H<sup>1</sup><sub>0</sub>(Ω) for ''g'' in H<sup>−1</sup>(Ω).
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| *'''Eigenvalue problem:''' Let ''T'' be the operator on L<sup>2</sup>(Ω) defined by
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| ::<math>\displaystyle{T = R_1\Delta^{-1}R_0,}</math>
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| :where ''R''<sub>0</sub> is the inclusion of L<sup>2</sup>(Ω) in H<sup>−1</sup>(Ω) and ''R''<sub>1</sub> of H<sup>1</sup><sub>0</sub> in L<sup>2</sup>(Ω), both compact operators by Rellich's theorem. The operator ''T'' is compact, self-adjoint and with (''Tf'',''f'') > 0 for all ''f''. By the [[spectral theorem]], there is a complete orthonormal set of eigenfunctions ''f''<sub>''n''</sub> in L<sup>2</sup>(Ω) with
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| ::<math>\displaystyle{Tf_n=\mu_nf_n,\,\,\mu_n >0,\,\,\mu_n\rightarrow 0.}</math>
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| :Since μ<sub>''n''</sub> > 0, ''f''<sub>''n''</sub> lies in H<sup>1</sup><sub>0</sub>(Ω). Setting λ<sub>''n''</sub> = 1/ μ<sub>''n''</sub>, the ''f''<sub>''n''</sub> are eigenfunctions of the Laplacian:
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| ::<math>\displaystyle{\Delta f_n = \lambda_n f_n,\,\, \lambda_n>0,\,\, \lambda_n\rightarrow \infty.}</math>
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| ==Sobolev spaces without boundary condition==
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| To determine the regularity properties of the eigenfunctions ''f''<sub>''n''</sub> and solutions of
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| :<math>\displaystyle{\Delta f =u,\,\, u\in H^{-1}(\Omega),\,\, f\in H^1_0(\Omega),}</math>
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| enlargements of the Sobolev spaces H<sup>''k''</sup><sub>0</sub>(Ω) have to be considered. Let C<sup>∞</sup>(Ω<sup>−</sup>) be the space of smooth functions on Ω which with their derivatives extend continuously to the closure of Ω. By [[Borel's lemma]], these are precisely the restrictions of smooth functions on '''T'''<sup>2</sup>. The Sobolev space H<sup>''k''</sup>(Ω) is defined to the Hilbert space completion of this space for the norm
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| :<math>\displaystyle{\|f\|_{(k)}^2 = \sum_{j=0}^k {k\choose j} \|\partial_x^j\partial_y^{k-j}f\|^2.}</math>
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| This norm agrees with the Sobolev norm on C<sup>∞</sup><sub>''c''</sub>(Ω) so that H<sup>''k''</sup><sub>0</sub>(Ω) can be regarded as a closed subspace of H<sup>''k''</sup>(Ω). Unlike H<sup>''k''</sup><sub>0</sub>(Ω). H<sup>''k''</sup>(Ω) is not naturally a subspace of H<sup>''k''</sup>('''T'''<sup>2</sup>), but the map restricting smooth functions from '''T'''<sup>2</sup> to the closure of Ω is continuous for the Sobolev norm so extends by continuity to a map ρ<sub>''k''</sub> of H<sup>''k''</sup>('''T'''<sup>2</sup>) into H<sup>''k''</sup>(Ω).
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| *'''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.
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| *'''Extension theorem:''' The restriction of ρ<sub>''k''</sub> to the orthogonal complement of its kernel defines an isomorphism onto H<sup>''k''</sup>(Ω). The extension map ''E''<sub>''k''</sub> is defined to be the inverse of this map: it is an isomorphism (not necessarily norm preserving) of H<sup>''k''</sup>(Ω) onto the orthogonal complement of H<sup>''k''</sup><sub>0</sub>(Ω<sup>''c''</sup>) such that ρ<sub>''k''</sub> ∘ E<sub>''k''</sub> = I. On C<sup>∞</sup><sub>''c''</sub>(Ω), it agrees with the natural inclusion map. Bounded extension maps ''E''<sub>''k''</sub> of this kind from H<sup>''k''</sup>(Ω) to H<sup>''k''</sup>('''T'''<sup>2</sup>) 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 [[Alberto Calderón|Calderón]] using [[Singular integral operators of convolution type|singular integral operators]] and generalized by {{harvtxt|Stein|1970}}.
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| :It is sufficient to construct an extension ''E'' for a neighbourhood of a closed annulus, since a collar around the boundary is diffeomorphic to an annulus I x '''T''' with I a closed interval in '''T'''. Taking a smooth bump function ψ with 0 ≤ ψ ≤ 1, equal to 1 near the boundary and 0 outside the collar, ''E''( ψ''f'') + (1- ψ)''f'' will provide an extension on Ω. On the annulus, the problem reduces to finding an extension for C<sup>''k''</sup>(I) in C<sup>''k''</sup>('''T'''). Using a partition of unity the task of extending reduces to a neighbourhood of the end points of ''I''. Assuming 0 is the left end point, an extension is given locally by
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| ::<math>\displaystyle{E(f) = \sum_{m=0}^k a_m f(-x/(m+1)).}</math>
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| :Matching the first derivatives of order ''k'' or less at 0, gives
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| ::<math>\displaystyle{\sum_{m=0}^k (-m-1)^{-k} a_m =1.}</math>
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| :This matrix equation is solvable because the determinant is non-zero by [[Vandermonde determinant|Vandermonde's formula]]. It is straightforward to check that the formula for ''E''(''f''), when appropriately modified with bump functions, leads to an extension which is continuous in the above Sobolev norm.<ref>{{harvnb|Folland|1995}}</ref>
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| *'''Restriction theorem:''' The restriction map ρ<sub>''k''</sub> is surjective with ker ρ<sub>''k''</sub> = H<sup>''k''</sup><sub>0</sub>(Ω<sup>''c''</sup>). This is an immediate consequence of the extension theorem and the support properties for Sobolev spaces with boundary condition.
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| *'''Duality:''' H<sup>''k''</sup>(Ω) is naturally the dual of H<sup>−''k''</sup><sub>0</sub>(Ω). Again this is an immediate consequence of the restriction theorem. Thus the Sobolev spaces form a chain:
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| ::<math>\displaystyle{\cdots\subset H^{2}(\Omega) \subset H^{1}(\Omega) \subset H^0(\Omega) \subset H^{-1}_0(\Omega) \subset H^{-2}_0(\Omega) \subset \cdots}</math>
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| :The differentiation operators ∂<sub>''x''</sub>, ∂<sub>''y''</sub> carry each Sobolev space into the larger one with index 1 less.
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| *'''Sobolev embedding theorem:''' H<sup>''k''+2</sup>(Ω) is contained in C<sup>''k''</sup>(Ω<sup>−</sup>). This is an immediate consequence of the extension theorem and the Sobolev embedding theorem for H<sup>''k''+2</sup>('''T'''<sup>2</sup>).
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| *'''Characterization:''' H<sup>''k''</sup>(Ω) consists of ''f'' in L<sup>2</sup>(Ω) = H<sup>0</sup>(Ω) such that all the derivatives ∂<sup>α</sup>''f'' lie in L<sup>2</sup>(Ω) for |α| ≤ ''k''.Here the derivatives are taken within the chain of Sobolev spaces above.<ref>See:
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| *{{harvnb|Agmon|2010}}
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| *{{harvnb|Folland|1995|pp=219–223}}
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| *{{harvnb|Chazarain|Piriou|1982|p=94}}</ref> Since C<sup>∞</sup><sub>''c''</sub>(Ω) is weakly dense in H<sup>''k''</sup>(Ω), this condition is equivalent to the existence of L<sup>2</sup> functions ''f''<sub>α</sub> such that
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| ::<math>\displaystyle{(f_\alpha,\varphi)=(-1)^{|\alpha|} (f,\partial^\alpha \varphi)}</math>
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| :for |α| ≤ ''k'' and φ in C<sup>∞</sup><sub>''c</sub>(Ω).
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|
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| :To prove the characterization, note that if ''f'' is in H<sup>''k''</sup>(Ω), then ∂<sup>α</sup>''f'' lies in H<sup>''k''−|α|</sup>(Ω) and hence in H<sup>0</sup>(Ω) = L<sup>2</sup>(Ω). Conversely the result is well-known for the Sobolev spaces H<sup>''k''</sup>('''T'''<sup>2</sup>): the assumption implies that the (∂<sub>''x''</sub> -i∂<sub>''y''</sub>)<sup>''k''</sup> ''f'' is in L<sup>2</sup>('''T'''<sup>2</sup>) and the corresponding condition on the Fourier coefficients of ''f'' shows that ''f'' lies in H<sup>''k''</sup>('''T'''<sup>2</sup>). Similarly the result an be proved directly for an annulus [−δ,δ] x '''T'''. In fact by the argument on '''T'''<sub>2</sub> the restriction of ''f'' to any smaller annulus [−δ',δ'] x '''T''' lies in H<sup>''k''</sub>: equivalently the restriction of the function ''f''<sub>''R''</sub>(''x'',''y'') = ''f''(''Rx'',''y'') lies in H<sup>''k''</sup> for ''R'' > 1. On the other hand ∂<sup>α</sup>''f''<sub>''R''</sub> tends to ∂<sup>α</sup>''f'' in L<sup>2</sup> as ''R'' tends to 1, so that ''f'' must lie in H<sup>''k''</sup>. The case for a general domain Ω reduces to these two cases since ''f'' can be written as ''f'' = ψ''f'' + (1 − ψ)''f'' with ψ a bump function supported in Ω such that 1 − ψ is supported in a collar of the boundary.
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| *'''Regularity theorem:''' If ''f'' in L<sup>2</sup>(Ω) has both derivatives ∂<sub>''x''</sub>''f'' and ∂<sub>''y''</sub>''f'' in H<sup>''k''</sup>(Ω) then ''f'' lies in H<sup>''k''+1</sup>(Ω). This is an immediate consequence of the characterization of H<sup>''k''</sup>(Ω) above. In fact if this is true even when satisfied at the level of distributions: if there are functions ''g'', ''h'' in H<sup>''k''</sup>(Ω) such that (''g'',φ) = (''f'',φ<sub>''x''</sub>) and (''h'',φ) = (''f'',φ<sub>''y''</sub>) for φ in C<sup>∞</sup><sub>''c''</sub>(Ω), then ''f'' is in H<sup>''k''+1</sup>(Ω).
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| *'''Rotations on an annulus:''' For an annulus ''I'' x '''T''', the extension map to '''T'''<sup>2</sup> is by construction equivariant with respect to rotations in the second variable,
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| ::<math>\displaystyle{R_t f(x,y)=f(x,y+t).}</math>
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| :On '''T'''<sup>2</sup> it is known that if ''f'' is in H<sup>''k''</sup>, then the ''difference quotient'' δ<sub>''h''</sub>''f'' = ''h''<sup>−1</sup>(''R''<sub>''h''</sub>''f'' - ''f'') tends to ∂<sub>''y''</sub> ''f'' in H<sup>''k''−1</sup>; if the difference quotients are bounded in ''H''<sup>''k''</sup> then ∂<sub>''y''</sub> ''f'' lies in H<sup>''k''</sup>. Both assertions are consequences of the formula:
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| ::<math>\displaystyle{\widehat{\delta_h f}(m,n) = h^{-1}(e^{-ihn}-1)\widehat{f}(m,n)=-\int_0^1 in e^{-inht}\, dt \,\,\widehat{f}(m,n).}</math>
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| :These results on '''T'''<sup>2</sup> imply analogous results on the annulus using the extension.
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| ==Regularity for Dirichlet problem==
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| *'''Regularity for dual Dirichlet problem:''' If ∆''u'' = ''f'' with ''u'' in H<sup>1</sup><sub>0</sub>(Ω) and ''f'' in H<sup>''k''−1</sup>(Ω) with ''k'' ≥ 0, then ''u'' lies in H<sup>''k''+1</sup>(Ω).
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| :Take a decomposition ''u'' = ψ''u'' + (1 − ψ)''u'' with ψ supported in Ω and 1 − ψ supported in a collar of the boundary. Standard Sobolev theory for '''T'''<sup>2</sub> can be applied to ψ''u'': elliptic regularity implies that it lies in H<sup>''k''+1</sup>('''T'''<sup>2</sup>) and hence H<sup>''k''+1</sup>(Ω). ''v'' = (1 − ψ)''u'' lies in H<sup>1</sup><sub>0</sub> of a collar, diffeomorphic to an annulus, so it suffices to prove the result with Ω a collar and ∆ replaced by
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| ::<math>\displaystyle{\Delta_1=\Delta -[\Delta,\psi]=\Delta + (p\partial_x + q\partial_y - \Delta \psi)=\Delta +X.}</math>
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| :The proof<ref>{{harvnb|Taylor|2011}}</ref> proceeds by induction on ''k'', proving simultaneously the inequality
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| ::<math>\displaystyle{\|u\|_{(k+1)} \le C\|\Delta_1 u\|_{(k-1)} + C \|u\|_{(k)},}</math>
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| :for some constant ''C'' depending only on ''k''. It is straightforward to establish this inequality for ''k'' = 0, where by density ''u'' can be taken to be smooth of compact support in Ω:
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| ::<math>\displaystyle{\|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)}.}</math>
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| :The collar is diffeomorphic to an annulus. The rotational flow ''R''<sub>''t''</sub> on the annulus induces a flow ''S''<sub>''t''</sub> on the collar with corresponding vector field ''Y'' = ''r'' ∂<sub>''x''</sub> + ''s'' ∂<sub>''y''</sub>. Thus ''Y'' corresponds to the vector field ∂<sub>θ</sub>. The radial vector field on the annulus ''r''∂<sub>''r''</sub> is a commuting vector field which on the collar gives a vector field ''Z'' = ''p'' ∂<sub>''x''</sub> + ''q'' ∂<sub>''y''</sub> proportional to the normal vector field. The vector fields ''Y'' and ''Z'' commute.
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| :The difference quotients δ<sub>''h''</sub> ''u'' can be formed for the flow ''S''<sub>''t''</sub>. The commutators [δ<sub>''h''</sub>,∆<sub>1</sub>] are second order differential operators from H<sup>''k''+1</sup>(Ω) to H<sup>''k''−1</sup>(Ω). Their operators norms are uniformly bounded for ''h'' near 0; for the computation can be carried out on the annulus where the commutator just replaces the coefficients of ∆<sub>1</sub> by their difference quotients composed with ''S''<sub>''h''</sub>. On the other hand ''v'' = δ<sub>''h''</sub> ''u'' lies in H<sup>1</sup><sub>0</sub>(Ω), so the inequalities for ''u'' apply equally well for ''v'':
| |
| | |
| ::<math>\displaystyle{\|\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)}.}</math>
| |
| | |
| :The uniform boundedness of the difference quotients δ<sub>''h''</sub> ''u'' implies that ''Y'' ''u'' lies in H<sup>''k''+1</sup>(Ω) with
| |
| | |
| ::<math>\displaystyle{\|Y u\|_{(k+1)} \le C\|\Delta_1 u\|_{(k)} + C^\prime \|u\|_{(k+1)}.}</math>
| |
| | |
| :It follows that ''V'' ''u'' lies in H<sup>''k''+1</sup>(Ω) where ''V'' is the vector field
| |
| | |
| ::<math>\displaystyle{V =(r^2+s^2)^{-1/2}Y = a\partial_x + b\partial_y,}</math>
| |
| | |
| :where ''a''<sup>2</sup> + ''b''<sup>2</sup> = 1. Moreover ''Vu'' satisfies a similar inequality to ''Yu''.
| |
| | |
| ::<math>\displaystyle{\|V u\|_{(k+1)} \le C^{\prime\prime}(\|\Delta_1 u\|_{(k)} + \|u\|_{(k+1)}).}</math>
| |
| | |
| :Let ''W'' be the orthogonal vector field
| |
| | |
| ::<math>\displaystyle{W=-b\partial_x + a\partial_y.}</math>
| |
| | |
| :It can also be written as ξ''Z'' for some smooth nowhere vanishing function ξ on a neighbourhood of the collar.
| |
| | |
| :It suffices to show that ''Wu'' lies in H<sup>''k''+1</sup>(Ω). For then
| |
| | |
| ::<math>\displaystyle{(V \pm iW)u =(a\mp i b)(\partial_x \pm i \partial_y)u,}</math>
| |
| | |
| :so that ∂<sub>''x''</sub>''u'' and ∂<sub>''y''</sub>''u'' lie in H<sup>''k''+1</sup>(Ω) and ''u'' must lie in H<sup>''k''+2</sup>(Ω).
| |
| | |
| :To check the result on ''Wu'', it is enough to show that ''VWu'' and ''W''<sup>2</sup>''u'' lie in H<sup>''k''</sup>(Ω). Note that
| |
| | |
| ::<math>\displaystyle{A=\Delta - V^2 - W^2,\,\,\, B=[V,W],}</math>
| |
| | |
| :are vector fields. But then
| |
| | |
| ::<math>\displaystyle{W^2u=\Delta u - V^2u - Au,\,\,\, VWu =WVu + Bu,}</math>
| |
| | |
| :with all terms on the right hand side in H<sup>''k''</sup>(Ω). Moreover the inequalities for ''Vu'' show that
| |
| | |
| ::<math>\|Wu\|_{(k+1)} \le C(\|VWu\|_{(k)} + \|W^2u\|_{(k)})
| |
| \le C\|(\Delta - V^2 -A)u\|_{(k)} + C \|(WV+B)u\|_{(k)} \le C_1 \|\Delta_1 u\|_{(k)} + C_1 \|u\|_{(k+1)}.</math>
| |
| | |
| :Hence
| |
| | |
| ::<math>\displaystyle{\|u\|_{(k+2)} \le C(\|Vu\|_{(k+1)} + \|Wu\|_{(k+1)})
| |
| \le C^\prime \|\Delta_1 u\|_{(k)} + C^\prime \|u\|_{(k+1)}.}</math>
| |
| | |
| *'''Smoothness of eigenfunctions.''' It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of ∆ in H<sup>1</sup><sub>0</sub>(Ω) lie in C<sup>∞</sup>{Ω<sup>−</sup>). Moreover any solution of ∆''u'' = ''f'' with ''f'' in C<sup>∞</sup>{Ω<sup>-</sup>) and ''u'' in H<sup>1</sup><sub>0</sub>(Ω) must have ''u'' in C<sup>∞</sup>{Ω<sup>-</sup>). In both cases by the vanishing properties, the eigenfunctions and ''u'' vanish on the boundary of Ω.
| |
| | |
| *'''Solving the Dirichlet problem.''' The dual Dirichlet problem can be used to solve the Dirichlet problem:
| |
| | |
| ::<math>\displaystyle{\Delta f|_\Omega = 0,\,\, f|_{\partial\Omega} =g,}</math>
| |
| | |
| :with ''g'' a smooth function on ∂Ω. By Borel's lemma ''g'' is the restriction of a function ''G'' in C<sup>∞</sup>(Ω<sup>−</sup>). Let ''F'' be the smooth solution of ∆''F'' = ∆''G'' with ''F'' = 0 on ∂Ω. Then ''f'' = ''G'' − ''F'' solves the Dirichlet problem. By the [[maximal principle]], the solution is unique.<ref>{{harvnb|Folland|1995|p=84}}</ref>
| |
| | |
| ==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.<ref>{{harvnb|Taylor|2011|pp=323–325}}</ref> 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 ''g'' harmonic in the interior of Ω with
| |
| | |
| :<math>\displaystyle{\partial_n g = \kappa - K e^g}</math>
| |
| | |
| on ∂Ω, where κ is the curvature of the boundary curve, ∂<sub>''n''</sub> is the derivative in the direction of the normal to ∂Ω and ''K'' is constant on each boundary component.
| |
| | |
| {{harvtxt|Taylor|2011}} gives a proof of the Riemann mapping theorem for a simply connected domain Ω with smooth boundary. Translating if necessary, it can be assumed that 0 lies in Ω. The solution of the Dirichlet problem shows that there is a unique smooth function ''U''(''z'') on the closure of Ω which is harmonic in Ω and equal to −log |''z''| on the boundary. Define the [[Green's function]] by ''G''(''z'') = log |''z''| + ''U''(''z''). It vanishes on the boundary and is harmonic on Ω away from 0.
| |
| The [[harmonic conjugate]] ''V'' of ''U'' is the unique real function on Ω such that ''U'' + ''i'' ''V'' is holomorphic. As such it must satisfy the [[Cauchy-Riemann equations]]:
| |
| | |
| :<math>\displaystyle{U_x=-V_y,\,\, U_y=V_x.}</math>
| |
| | |
| The solution is given by
| |
| | |
| :<math>\displaystyle{V(z)=\int_0^z -U_y dx + V_x dy,}</math>
| |
| | |
| where the integral is taken over any path in the closure of Ω. It is easily verified that ''V''<sub>''x''</sub> and ''V''<sub>''y''</sub> are exist and are given by the corresponding derivatives of ''U''. Thus ''V'' is a smooth function on the closure of Ω, vanishing at 0. By the Cauchy-Riemann ''f'' = ''U'' + ''i'' ''V'' is smooth on the closure of Ω, holomorphic on
| |
| Ω and ''f''(0) = 0. The function ''H'' = arg ''z'' + ''V''(''z'') is only defined up to multiples of 2π, but the function
| |
| | |
| :<math>\displaystyle{F(z) =e^{G(z)+iH(z)}= z e^{f(z)}}</math>
| |
| | |
| is a holomorphic on Ω and smooth on its closure. By construction it satisfies ''F''(0) = 0 and |''F''(''z'')| = 1 for ''z'' on the boundary of Ω. Since ''z'' has [[winding number]] 1, so too does ''F''(''z''). On the other hand ''F''(''z'') = 0 only for ''z'' =0 where there is a simple zero. So by the [[argument principle]] ''F'' assumes every value in the unit disc exactly once and ''F'' ' does not vanish inside Ω. To check that the derivative on the boundary curve is non-zero amounts to computing the derivative of ''e''<sup>''iH''</sup>, i.e. the derivative of ''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 ''G'' vanishes on the boundary and is strictly negative in Ω since |''F''| = ''e''<sup>''G''</sup>. The [[Hopf lemma]] implies that the directional derivative of ''G'' in the direction of the outward normal is strictly positive. So on the boundary curve ''F'' has nowhere vanishing derivative. Since the boundary curve has winding number one, ''F'' defines a diffeomorphism of the boundary curve onto the unit circle. Accordingly ''F'' is a smooth diffeomorphism of the closure of Ω onto the closed unit disk, which restricts to a holomorphic map of Ω onto the open unit disk and a smooth diffeomorphism between the boundaries.
| |
| | |
| Similar arguments can be applied to prove the Riemann mapping theorem for a doubly connected domain Ω bounded by simple smooth curves
| |
| ''C''<sub>''i''</sub> (the inner curve) and ''C''<sub>''o''</sub> (the outer curve). B tranlsating we can assume 1 lies on the outer boundary. Let ''u'' be the smooth solution of the Dirichlet problem with ''U'' = 0 on the outer curve and −1 on the inner curve. By the [[maximum principle]] 0 < ''u''(''z'') < 1 for ''z'' in Ω and so by the [[Hopf lemma]] the normal derivatives of ''u'' are negative on the outer curve and positive on the inner curve. The integral of −''u''<sub>''y''</sub> ''dx'' + ''u''<sub>''y''</sub> ''dx'' over the boundary is zero by Stoke's 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 ''c'' > 0 such that ''U'' = ''cu'' satisfies
| |
| | |
| :<math>\displaystyle{\int_{C} -U_y \,dx + U_x \, dy = 2\pi}</math>
| |
| | |
| on each boundary curve. The harmonic conjugate ''V'' of ''U'' can again be defined by
| |
| | |
| :<math>\displaystyle{V(z)=\int_1^z -u_y \,dx + u_x \, dy}</math>
| |
| | |
| and is well-defined up to multiples of 2π. The function
| |
| | |
| :<math>\displaystyle{F(z)=e^{U(z) + i V(z)}}</math>
| |
| | |
| is smooth on the closure of Ω and holomorphic in Ω. On the outer curve |''F''| = 1 and on the inner curve |''F''| = ''e''<sup>−''c''</sup> = ''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 ''F'' restricts to a diffeonmorphism between the boundary curves and the two concentric circles. Since the images of outer and inner curve have winding number 1 and 0 about any point in the annulus, an application of the argument principle implies that ''F'' assumes every value within the annulus ''r'' < |''z''| < 1 exactly once; since that includes mutliplicities, the complex derivative of ''F'' is nowhere vanishing in Ω. This ''F'' is a smooth diffeomorphism of the closure of Ω onto the closed annulus ''r'' ≤ |''z''| ≤ 1, restricting to a holomorphic map in the interior and a smooth diffeomorphism on both boundary curves.
| |
| | |
| ==Trace map==
| |
| The restriction map τ of C<sup>∞</sup>('''T'''<sup>2</sup>) onto C<sup>∞</sup>('''T''') = C<sup>∞</sup>(1 × '''T''') extends to a continuous map H<sup>''k''</sup>('''T'''<sup>2</sup>) into H<sup>''k'' − ½</sup>('''T''') for ''k'' ≥ 1.<ref>{{harvnb|Chazarain|Piriou|1982}}</ref> In fact
| |
|
| |
| :<math>\displaystyle{\widehat{\tau f}(n)=\sum_m \widehat{f}(m,n),}</math>
| |
| | |
| so the [[Cauchy–Schwarz inequality]] yields
| |
| | |
| :<math>|\widehat{\tau f}(n)|^2 (1+n^2)^{k-1/2}\le \left(\sum_m {(1+n^2)^{k-1/2}\over (1+m^2+n^2)^{k}}\right)\cdot\left(\sum_m |\widehat{f}(m,n)|^2 (1+m^2+n^2)^k\right) \le C_k \sum_m|\widehat{f}(m,n)|^2 (1+m^2+n^2)^k,</math>
| |
| | |
| where, by the [[integral test]],
| |
| | |
| :<math>\displaystyle{C_k=\sup_n \sum_m {(1+n^2)^{k-1/2}\over (1+m^2 +n^2)^k} <\infty,\,\,\, c_k=\inf_n \sum_m {(1+n^2)^{k-1/2}\over (1+m^2 +n^2)^k}>0.}</math>
| |
| | |
| The map τ is onto since a continuous extension map ''E'' can be constructed from H<sup>''k'' − ½</sup>('''T''') to H<sup>''k''</sup>('''T'''<sup>2</sup>).<ref>{{harvnb|Taylor|2011|p=275}}</ref><ref>{{harvnb|Renardy|Rogers|2004|214-218}}</ref> In fact set
| |
| | |
| :<math>\displaystyle{\widehat{Eg}(m,n)=\lambda_{n}^{-1} \widehat{g}(n)\cdot {(1+n^2)^{k-1/2}\over (1+n^2+m^2)^k},}</math>
| |
| | |
| where
| |
| | |
| :<math>\displaystyle{\lambda_n=(1+n^2)^{k-1/2}\sum_m (1+m^2+n^2)^{-k} < C_k.}</math>
| |
| | |
| Thus λ<sub>''n''</sub> > c<sub>''k''</sub>. If ''g'' is smooth, then by construction ''Eg'' restricts to ''g'' on 1 × '''T'''. Moreover ''E'' is a bounded linear map since
| |
| | |
| :<math>\|Eg\|_{(k)}^2=\sum_{m,n}|\widehat{Eg}(m,n)|^2 (1+m^2+n^2)\le c_k^{-2}\sum_{m,n} |\widehat{g}(n)|^2 (1+n^2)^{2k-1} (1+m^2+n^2)^{-k} \le c_k^{-2}C_k \|g\|_{k-1/2}^2.</math>
| |
| | |
| It follows that there is a trace map τ of H<sup>''k''</sup>(Ω) onto H<sup>''k'' − ½</sup>(∂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<sup>''k''</sup> of the collar, which can be identified with H<sup>''k''</sup> of an annulus for which there is a trace map. The invariance under diffeomorphims (or coordinate change) of the half-integer Sobolev spaces on the circle follows from the fact that an equivalent norm on ''H''<sup>''k'' + ½</sup>('''T''') is given by<ref>{{harvnb|Hörmander|1990|pp=240–241}}</ref>
| |
| | |
| :<math>\|f\|_{[k+1/2]}^2 = \|f\|_{(k)}^2 + \int_0^{2\pi}\int_0^{2\pi} {|f^{(k)}(s)-f^{(k)}(t)|^2\over|e^{is} -e^{it}|^{2}}\,ds\, dt.</math> | |
| | |
| It is also a consequence of the properties of τ and ''E'' (the "trace theorem").<ref>{{harvnb|Renardy|Rogers|2004}}</ref> In fact any diffeomorphism ''f'' of '''T''' induces a diffeomorphism ''F'' of '''T'''<sup>2</sup> by acting only on the second factor. Invariance of H<sup>''k''</sup>('''T'''<sup>2</sup>) under the induced map ''F''* therefore implies invariance of H<sup>''k'' − ½</sup>('''T''') under ''f''*, since ''f''* = τ ∘ ''F''* ∘ ''E''.
| |
| | |
| Further consequences of the trace theorem are the two exact sequences<ref>{{harvnb|Chazarain|Piriou|1982}}</ref><ref>{{harvnb|Renardy|Rogers|2004}}</ref>
| |
| | |
| :<math>\displaystyle{(0)\rightarrow H^1_0(\Omega) \rightarrow H^1(\Omega) \rightarrow H^{1/2}(\partial\Omega) \rightarrow (0)}</math>
| |
|
| |
| and
| |
| | |
| :<math>\displaystyle{(0)\rightarrow H^2_0(\Omega) \rightarrow H^2(\Omega) \rightarrow H^{3/2}(\partial\Omega) \oplus H^{1/2}(\partial\Omega) \rightarrow (0),}</math>
| |
| | |
| where the last map takes ''f'' in H<sup>2</sup>(Ω) to ''f''|<sub>∂Ω</sub> and ∂<sub>''n''</sub>''f''|<sub>∂Ω</sub>. There are generalizations of these sequences to H<sup>''k''</sup>(Ω) involving higher powers of the normal derivative in the trace map:
| |
| | |
| :<math>\displaystyle{(0)\rightarrow H^k_0(\Omega) \rightarrow H^k(\Omega) \rightarrow \bigoplus_{j=1}^k H^{j-/2}(\partial\Omega) \rightarrow (0).}</math>
| |
| | |
| The trace map to H<sup>''j'' − ½</sup>(∂Ω) takes ''f'' to ∂<sub>''n''</sub><sup>''k'' − ''j''</sup>''f''|<sub>∂Ω</sub>
| |
| | |
| ==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 ''f'' in H<sup>1</sup>(Ω), the normal derivative ∂<sub>''n''</sub>''f''|<sub>∂Ω</sub> cannot be a priori defined at the level of Sobolev spaces. Instead an alternative formulation of boundary value problems for the Laplacian Δ on a bounded region Ω in the plane is used. It employs '''[[Dirichlet form]]s''', sesqulinear bilinear forms on H<sup>1</sup>(Ω), H<sup>1</sup><sub>0</sub> 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 H<sup>2</sup>(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 ''f''|<sub>∂Ω</sub> is already defined on H<sup>1</sup>(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 Δ on a bounded region Ω in the plane proceeds from the following data:<ref>{{harvnb|Folland|1995|pp=231–248}}</ref>
| |
| | |
| *A closed subspace ''H'' lying between H<sup>1</sup><sub>0</sub>(Ω) and H<sup>1</sup>(Ω)
| |
| *A Dirichlet form for Δ given by a bounded Hermitian bilinear form ''D''(''f'',''g'') defined for ''f'', ''g'' in H<sup>1</sup>(Ω) such that ''D''(''f'',''g'')=(∆ ''f'',''g'') for ''f'', ''g'' in H<sup>1</sup><sub>0</sub>(Ω).
| |
| *''D'' is coercive, i.e. there is a positive constant ''C'' and a non-negative constant λ such that ''D''(''f'',''f'') ≥ ''C'' (''f'',''f'')<sub>(1)</sub> − λ(''f'',''f'').
| |
| | |
| A '''weak solution''' of the boundary value problem given initial data ''f'' in L<sup>2</sup>(Ω) is a function ''u'' in ''H'' such that
| |
| | |
| :<math>\displaystyle{D(f,g)=(u,g)}</math>
| |
| | |
| for all ''g'' in ''H''.
| |
| | |
| For both the Dirichlet and Neumann problem
| |
| | |
| :<math>\displaystyle{D(f,g)=(f_x,g_x) + (f_y,g_y).}</math>
| |
| | |
| For the Dirichlet problem ''H'' = H<sup>1</sup><sub>0</sub>(Ω). In this case
| |
| | |
| :<math>\displaystyle{D(f,g)=(\Delta f,g)}</math>
| |
| | |
| for ''f'', ''g'' in ''H''. By the trace theorem the solution satisfies ''u''|<sub>Ω</sub> = 0 in H<sup>½</sup>(∂Ω).
| |
| | |
| For the Neumann problem ''H'' is taken to be H<sup>1</sup>(Ω).
| |
| | |
| ==Application to Neumann problem==
| |
| The classical Neumann problem on Ω consists in solving the boundary value problem
| |
| | |
| :<math>\displaystyle{\Delta u =f}</math>
| |
| | |
| with ''f'' in C<sup>∞</sup>(Ω<sup>−</sup>) and ''u'' in C<sup>∞</sup>(Ω<sup>−</sup>) and satisfying the Neumann boundary condition
| |
| | |
| :<math>\displaystyle{\partial_n u =0.}</math> | |
| | |
| [[Green's theorem]] implies that for ''u'', ''v'' in C<sup>∞</sup>(Ω<sup>−</sup>)
| |
| | |
| :<math>\displaystyle{(\Delta u,v)= (u_x,v_x) + (u_y,v_y) - (\partial_n u,v)_{\partial \Omega}.}</math>
| |
| | |
| Thus if Δ''u'' = 0 in Ω and satisfies the Neumann boundary conditions, ''u''<sub>''x''</sub>, ''u''<sub>''y''</sub> = 0, and so
| |
| ''u'' is constant in Ω.
| |
| | |
| Hence the Neumann problem has a unique solution up to adding constants.<ref>{{harvnb|Taylor|2011}}</ref>
| |
| | |
| Consider the Hermitian form on H<sup>1</sup>(Ω) defined by
| |
| | |
| :<math>\displaystyle{D(f,g)=(u_x,v_x) + (u_y,v_y).}</math>
| |
| | |
| Since H<sup>1</sup>(Ω) is in duality with H<sup>−1</sup><sub>0</sub>(Ω), there is a unique element ''Lu'' in H<sup>−1</sup><sub>0</sub>(Ω) such that
| |
| | |
| :<math>\displaystyle{D(u,v) =(Lu,v).}</math>
| |
| | |
| The map ''I'' + ''L'' is an isometry of H<sup>1</sup>(Ω) onto H<sup>−1</sup><sub>0</sub>(Ω), so in particular ''L'' is bounded.
| |
| | |
| In fact
| |
| | |
| :<math>\displaystyle{((L+I)u,v)=(u,v)_{(1)}.}</math>
| |
| | |
| So
| |
| | |
| :<math>\|(L+I)u\|_{(-1)} =\sup_{\|v\|_{(1)}=1} |((L+I)u,v)|= \sup_{\|v\|_{(1)}=1} |(u,v)_{(1)}|=\|u\|_{(1)}.</math>
| |
| | |
| On the other hand any ''f'' in H<sup>−1</sup><sub>0</sub>(Ω) defines a bounded conjugate-linear form on H<sup>1</sup>(Ω) sending ''v'' to (''f'',''v''). By the [[Riesz–Fischer theorem]], there exists ''u'' in H<sup>1</sup>(Ω) such that
| |
| | |
| :<math>\displaystyle{(f,v)=(u,v)_{(1)}.}</math>
| |
| | |
| Hence (''L'' + ''I'')''u'' = ''f'' and so ''L'' + ''I'' is surjective. Define a bounded linear operator ''T'' on L<sup>2</sup>(Ω) by
| |
| | |
| :<math>\displaystyle{T=R_1(I+L)^{-1} R_0,}</math>
| |
| | |
| where ''R''<sub>1</sub> is the map H<sup>1</sup>(Ω) → L<sup>2</sup>(Ω), a compact operator, and ''R''<sub>0</sub> is the map
| |
| L<sup>2</sup>(Ω) → H<sup>−1</sup><sub>0</sub>(Ω), its adjoint, so also compact.
| |
| | |
| The operator ''T'' has the following properties:
| |
| | |
| *''T'' is a contraction since it is a composition of contractions
| |
| *''T'' is compact, since ''R''<sub>0</sub> and ''R''<sub>1</sub> are compact by Rellich's theorem
| |
| *''T'' is self-adjoint, since if ''f'' and ''g'' are in L<sup>2</sup>(Ω), they can be written ''f'' =(''L'' + ''I'')''u'', ''g'' = (''L'' + ''I'')''v'' with ''u'', ''v'' in H<sup>1</sup>(Ω) so
| |
| | |
| ::<math>\displaystyle{(Tf,g)=(u,(I+L)v)=(u,v)_{(1)}=((I+L)u,v)= (f,Tg).}</math>
| |
| | |
| *''T'' has positive spectrum and kernel (0), for
| |
| | |
| ::<math>\displaystyle{(Tf,f)=(u,u)_{(1)}\ge 0,}</math>
| |
| | |
| :and ''Tf'' = 0 implies ''u'' = 0 and hence ''f'' = 0.
| |
| | |
| *There is a complete orthonormal basis ''f''<sub>''n''</sub> of L<sup>2</sup>(Ω) consisting of eigenfunctions of ''T''. Thus
| |
| | |
| ::<math>\displaystyle{Tf_n=\mu_nf_n}</math>
| |
| | |
| :with 0 < μ<sub>''n''</sub> ≤ 1 and μ<sub>''n''</sub> decreasing to 0.
| |
| | |
| *The eigenfunctions all lie in H<sup>1</sup>(Ω) since the image of ''T'' lies in H<sup>1</sup>(Ω).
| |
| | |
| *The ''f''<sub>''n''</sub> are eigenfunctions of ''L'' with
| |
| | |
| ::<math>\displaystyle{Lf_n=\lambda_nf_n,\,\,\,\,\, \lambda_n=\mu_n^{-1} -1.}</math> | |
| | |
| :Thus λ<sub>''n''</sub> are non-negative and increase to ∞.
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| *The eigenvalue 0 occurs with multiplicity one and corresponds to the constant function. For if ''u'' in H<sup>1</sup>(Ω) satisfies ''Lu'' = 0, then
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| ::<math>\displaystyle{(u_x,u_x) + (u_y,u_y)=(Lu,u)=0,}</math>
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| :so ''u'' is constant.
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| ==Regularity for Neumann problem==
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| ===Weak solutions are strong solutions===
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| The first main regularity result shows that a weak solution expressed in terms of the operator ''L'' and the Dirichlet form
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| ''D'' is a strong solution in the classical sense, expressed in terms of the Laplacian Δ and the Neumann boundary conditions. Thus if ''u'' = ''Tf'' with ''u'' in H<sup>1</sup>(Ω) and ''f'' in L<sup>2</sup>(Ω), then ''u'' in fact lies in H<sup>2</sup>(Ω), satisfies Δ''u'' + ''u'' = ''f'' and ∂<sub>''n''</sub>''u''|<sub>∂Ω</sub> = 0. Moreover for some constant ''C'' independent of ''u'',
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| :<math>\displaystyle{\|u\|_{(2)} \le C\|\Delta u\|_{(0)} + C \|u\|_{(1)}.}</math>
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| Note that
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| :<math>\displaystyle{\|u\|_{(1)} \le \|L u\|_{(-1)} + \|u\|_{(0)},}</math>
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| since
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| :<math>\displaystyle{\|u\|_{(1)}^2 =|(L u, u)| +\|u\|^2_{(0)}\le \|L u\|_{(-1)}\|u\|_{(1)} + \|u\|_{(0)}\|u\|_{(1)}.}</math>
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| Take a decomposition ''u'' = ψ''u'' + (1 − ψ)''u'' with ψ supported in Ω and 1 − ψ supported in a collar of the boundary.
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| The operator ''L'' is characterized by
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| :<math>\displaystyle{(Lf,g)=(f_x,g_x)+(f_y,g_y)= (\Delta f,g)_\Omega - (\partial_n f,g)_{\partial\Omega}}</math>
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| for ''f'', ''g'' in C<sup>∞</sup>(Ω<sup>−</sup>). Then
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| :<math>\displaystyle{([L,\psi]f,g)=([\Delta,\psi]f,g),}</math>
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| so that
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| :<math>\displaystyle{[L,\psi]=-[L,1-\psi]=\Delta\psi +2\psi_x\partial_x + 2\psi_y\partial_y.}</math>
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| The function ''v'' = ψ''u'' and ''w'' = (1 − ψ)''u'' are treated separately, ''v'' being essentially subject to usual elliptic regularity considerations for interior points while ''w'' requires special treatment near the boundary using difference quotients. Once the strong properties are established in terms of ∆ and the Neumann boundary conditions, the "bootstrap" regularity results can be proved exactly as for the Dirichlet problem.
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| '''Interior estimates.''' The function ''v'' = ψ''u'' lies in H<sup>1</sup><sub>0</sub>(Ω<sub>1</sub>) where Ω<sub>1</sub> is a region with closure in Ω. If''f'' is in C<sup>∞</sup><sub>''c''</sub>(Ω) and ''g'' in C<sup>∞</sup>(Ω<sup>−</sup>)
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| :<math>\displaystyle{(Lf,g) = (\Delta f,g)_\Omega.}</math>
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| By continuity the same holds with ''f'' replaced by ''v'' and hence ''Lv'' = ∆''v''. So
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| :<math>\displaystyle{\Delta v= Lv=L(\psi u) =\psi Lu +[L,\psi]u=\psi(f-u) +[\Delta,\psi]u.}</math>
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| Hence regarding ''v'' as an element of H<sup>1</sup>('''T'''<sup>2</sup>), ∆''v'' lies in L<sup>2</sup>('''T'''<sup>2</sup>). Hence ''v'' lies in H<sup>2</sup>('''T'''<sup>2</sup>). Since ''v'' = φ ''v'' for a smooth function of compact support in Ω, ''v'' lies in H<sup>2</sup><sub>0</sub>(Ω). Moreover
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| :<math>\displaystyle{\|v\|_{(2)}^2=\|\Delta v\|^2 + 2\|v\|_{(1)}^2,}</math> | |
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| so that
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| :<math>\displaystyle{\|v\|_{(2)} \le C( \|\Delta(v)\|+ \|v\|_{(1)}).}</math>
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| '''Boundary estimates. ''' The function ''w'' = (1 − ψ)''u'' is supported in a collar contained in a tubular neighbourhood of the boundary. The difference quotients δ<sub>''h''</sub> ''w'' can be formed for the flow ''S''<sub>''t''</sub> and lie in H<sup>1</sup>(Ω), so the first inequality is applicable:
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| :<math>\|\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)}.</math>
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| The commutators [''L'',δ<sub>''h''</sub>] are uniformly bounded as operators from H<sup>1</sup>(Ω) to H<sup>−1</sup><sub>0</sub>(Ω). This is equivalent to checking the inequality
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| :<math>\displaystyle{|([L,\delta_h]g,h)|\le A\|g\|_{(1)}\|h\|_{(1)},}</math>
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| for ''g'', ''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 δ<sub>''h''</sub> with a differential operator is obtained by applying the difference operator to the coefficients after having applied ''R''<sub>''h''</sub> to the function:<ref>{{harvnb|Folland|1995|pp=255–260}}</ref>
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| :<math>\displaystyle{[\delta^h,\sum a_\alpha \partial^\alpha]=(\delta^h(a_\alpha)\circ R_h) \partial^\alpha.}</math>
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| Hence the difference quotients δ<sub>''h''</sub> ''w'' are uniformly bounded, and therefore ''Y'' ''w'' lies in H<sup>1</sup>(Ω) with
| |
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| :<math>\displaystyle{\|Y w\|_{(1)} \le C\|L w\|_{(0)} + C^\prime \|w\|_{(1)}.}</math>
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| Hence ''V'' ''w'' lies in H<sup>1</sup>(Ω) and ''Vw'' satisfies a similar inequality to ''Yw'':
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| :<math>\displaystyle{\|V w\|_{(1)} \le C^{\prime\prime}(\|L w\|_{(0)} + \|w\|_{(1)}).}</math>
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| Let ''W'' be the orthogonal vector field. As for the Dirichlet problem, to show that ''w'' lies in H<sup>2</sup>(Ω), it suffices to show that ''Ww'' lies in H<sup>1</sup>(Ω).
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| To check this, it is enough to show that ''VWw'' and ''W''<sup>2</sup>''u'' lie in L<sup>2</sup>(Ω). As before
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| :<math>\displaystyle{A=\Delta - V^2 - W^2,\,\,\, B=[V,W],}</math>
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| are vector fields. On the other hand (''Lw'',φ) = (∆''w'',φ) for φ smooth of compact support in Ω, so that ''Lw'' and ∆''w'' define the same distribution on Ω. Hence
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| :<math>\displaystyle{(W^2w,\varphi)=(L w - V^2w - Au,\varphi),\,\,\,\,\, (VWw,\varphi) =(WVw + Bw,\varphi).}</math>
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| Since the terms on the right hand side are pairings with functions in L<sup>2</sup>(Ω), the regularity criterion shows that ''Ww'' lies in H<sup>2</sup>(Ω). Hence ''Lw'' = ∆''w'' since both terms lie in L<sup>2</sup>(Ω) and have the same inner products with φ's.
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| Moreover the inequalities for ''Vw'' show that
| |
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| :<math>\|Ww\|_{(1)} \le C(\|VWw\|_{(0)} + \|W^2w\|_{(0)})
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| \le C\|(\Delta - V^2 -A)w\|_{(0)} + C \|(WV+B)w\|_{(0)} \le C_1 \|L w\|_{(0)} + C_1 \|w\|_{(1)}.</math>
| |
| | |
| Hence
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| :<math>\displaystyle{\|w\|_{(2)} \le C(\|V w\|_{(1)} + \|W w\|_{(1)}) \le C^\prime \|\Delta w\|_{(0)} + C^\prime \|w\|_{(1)}.}</math>
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| It follows that ''u'' = ''v'' + ''w'' lies in H<sup>2</sup>(Ω). Moreover
| |
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| :<math>\|u\|_{(2)} \le C(\|\Delta v\| +\|\Delta w\| + \|v\|_{(1)} + \|w\|_{(1)})\le C^\prime(\|\psi\Delta u\| + \|(1-\psi)\Delta u\| + 2\|[\Delta,\psi]u\| +\|u\|_{(1)}) \le C^{\prime\prime} (\|\Delta u\| + \|u\|_{(1)}).</math> | |
| | |
| '''Neumann boundary conditions.''' Since ''u'' lies in H<sup>2</sup>(Ω), Green's theorem is applicable by continuity. Thus for ''v'' in H<sup>1</sup>(Ω),
| |
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| :<math>(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}.</math>
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| Hence the Neumann boundary conditions are satisfied:
| |
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| :<math>\displaystyle{\partial_n u|_{\partial \Omega} =0,}</math>
| |
| | |
| where the left hand side is regarded as an element of H<sup>½</sup>(∂Ω) and hence L<sup>2</sup>(∂Ω).
| |
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| ===Regularity of strong solutions===
| |
| The main result here states that if ''u'' lies in H<sup>''k''+1</sup> with ''k'' ≥ 1, ∆''u'' in H<sup>''k''</sup> and ∂<sub>''n''</sub>''u'' = 0 on ∂Ω, then ''u'' lies in H<sup>''k''+2</sup> and
| |
| | |
| :<math>\displaystyle{\|u\|_{(k+2)} \le C\|\Delta u\|_{(k)} + C \|u\|_{(k+1)},}</math>
| |
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| for some constant independent of ''u''.
| |
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| Like the ocrresponding result for the Dirichlet problem, this is proved by induction on ''k'' ≥ 1. For ''k'' = 1, ''u'' is also a weak solution of the Neumann problem so satisfies the estimate above for ''k'' = 0. The Neumann boundary condition can be written
| |
| | |
| :<math>\displaystyle{Zu|_{\partial\Omega}=0.}</math>
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| | |
| Since ''Z'' commutes with the vector field ''Y'' corresponding to the period flow ''S''<sub>''t''</sub>, the inductive method of proof used for the Dirichlet problem works equally well in this case: for the difference quotients δ<sup>''h''</sup> preserve the boundary condition when expressed in terms of ''Z''.<ref>{{harvnb|Taylor|2011|p=348}}</ref>
| |
|
| |
| '''Smoothness of eigenfunctions.''' It follows by induction from the regularity theorem for the Neumann problem that the eigenfunctions of ''D'' in H<sup>1</sup>(Ω) lie in C<sup>∞</sup>{Ω<sup>−</sup>). Moreover any solution of ''Du'' = ''f'' with ''f'' in C<sup>∞</sup>{Ω<sup>-</sup>) and ''u'' in H<sup>1</sup>(Ω) must have ''u'' in C<sup>∞</sup>{Ω<sup>-</sup>). In both cases by the vanishing properties, the normal derivatives of the eigenfunctions and ''u'' vanish on ∂Ω.
| |
| | |
| '''Solving the associated Neumann problem.''' The method above can be used to solve the associated Neumann boundary value problem:
| |
| | |
| :<math>\displaystyle{\Delta f|_\Omega = 0,\,\, \partial_n f|_{\partial\Omega} =g,}</math>
| |
| | |
| with ''g'' a smooth function on ∂Ω. By Borel's lemma ''g'' is the restriction of a function ''G'' in C<sup>∞</sup>(Ω<sup>−</sup>). Let ''F'' be a smooth function such that ∂<sub>''n''</sub>''F'' = ''G'' near the boundary. Let ''u'' be the solution of ∆''u'' = −∆''F'' with ∂<sub>''n''</sub>''u'' = 0. Then ''f'' = ''u'' + ''F'' solves the boundary value problem.<ref>{{harvnb|Folland|1995|p=85}}</ref>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| [[Category:Partial differential equations]]
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| [[Category:Harmonic analysis]]
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| [[Category:Operator theory]]
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| [[Category:Functional analysis]]
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