Solution uniqueness and stability of the variational problems for elliptic PDEs

Contents

 1 General procedures for obtaining the solvability and stability conditions
 2 Related theorems
 3 Dirichlet boundary condition
 4 Neumann boundary condition
 5 Mixed boundary conditions
 6 Robin boundary condition
 7 Summary

In the article Variational problems, the equivalence between operator equations, variational equations and minimization of energy functionals are explained. The unique solvability and solution stability of the variational equations are governed by the Lax-Milgram Lemma, where the ellipticity of the bounded bilinear form \(a(\cdot ,\cdot ): X\times X \rightarrow \mathbb {R}\) or its associated bounded linear operator \(A: X \rightarrow X'\) is the key. The ellipticity condition is \begin{equation} a(v,v) = \langle Av,v \rangle \geq c_1^A \lVert v \rVert _X^2 \quad \forall v\in X. \end{equation} The stability condition of the solution \(u\) is \begin{equation} \lVert u \rVert _X \leq \frac {1}{c_1^A} \lVert f \rVert _X'. \end{equation} However, the previous summary is only about the general theory, which is not directly related to any actual PDE with any boundary condition yet. In this article, I’ll present how this general theory is adopted for analyzing the elliptic PDE \begin{equation} \label {eq:elliptic-pde} (Lu)(x) = f(x) \quad \forall x\in \Omega \subset \mathbb {R}^d \end{equation} under different boundary conditions, namely, Dirichlet, Neumann, mixed and Robbin’s boundary conditions. The partial differential operator \(L\) is \begin{equation} L = -\sum _{i,j=1}^d \frac {\partial }{\partial x_j} \left [ a_{ji}(x) \frac {\partial }{\partial x_i} \right ]. \end{equation} If the eigenvalues of the coefficient matrix \(a_{ji}(x)\) have a lower bound \(\lambda _0 > 0\) for all \(x\in \Omega \), \(L\) is called a uniform elliptic partial differential operator.

1 General procedures for obtaining the solvability and stability conditions

1. The variational formulation of the above elliptic PDE always starts from the Green’s first formula \begin{equation} a(u,v) = \left \langle Lu,v \right \rangle _{\Omega } + \left \langle \gamma _1^{\mathrm {int}}u, \gamma _0^{\mathrm {int}}v \right \rangle _{\Gamma }, \end{equation} where \(u\) is the solution and \(v\) is the test function. Because \(u\) is the solution, we substitute \(f\) for \(Lu\) in the Green’s formula and get the variational equation \begin{equation} \label {eq:green-first-formula} a(u,v) = \left \langle f,v \right \rangle + \left \langle \gamma _1^{\mathrm {int}}u, \gamma _0^{\mathrm {int}}v \right \rangle . \end{equation} Also note that the integration domain \(\Omega \) and \(\Gamma \) as subscripts in the above formula are omitted, since they can be determined by the functions involved in the duality pairing \(\left \langle \cdot ,\cdot \right \rangle \).

   The bilinear form \(a(u,v)\) is defined as \begin{equation} a(u,v) = \sum _{i,j=1}^d \int _{\Omega } a_{ji}(x) \frac {\partial u}{\partial x_i} \frac {\partial v}{\partial x_j} dx. \end{equation} Obviously, the linear operator \(A\) induced from \(a(u,v)\) such that \(\left \langle Au,v \right \rangle = a(u,v)\) for any \(u,v\in X\) is not the same as the original partial differential operator \(L\). Therefore, the operator equation \(Au=f\) mentioned in my previous article Variational problems is not the elliptic PDE \(Lu=f\), but another operator equation which is implied by the Green’s first formula in Equation 6.

2. Select appropriate function spaces for \(u\) and \(v\) in the bilinear form.

3. For incorporating boundary conditions into the variational equation, we can

   • either apply the them as constraints, such as the case of Dirichlet boundary condition

   • or substitute them into the boundary integral term, such as the case of Neumann boundary condition.

   During this process, the solution function is decomposed as a homogeneous part and a inhomogeneous part, when the Dirichlet boundary condition is involved. Or even the bilinear form \(a(\cdot ,\cdot )\) will be modified, when the Robin’s boundary condition is handled.

4. Derive the ellipticity condition for the bilinear form \(a(\cdot ,\cdot )\). In this step, the semi-ellipticity lemma is adopted, from which the desired ellipticity condition can be derived by defining an equivalent norm for the involved Sobolev space.

5. With the ellipticity condition, we can finally obtain the solution uniqueness and its stability condition using the Lax-Milgram Lemma.

2 Related theorems

Key theorems are listed here as reference.

Lemma 1 (Lax-Milgram Lemma) Let \(X\) be a Hilbert space and \(X'\) be its dual space with respect to the duality pairing \(\left \langle \cdot ,\cdot \right \rangle \)¹. Let the operator \(A: X \rightarrow X'\) be bounded \begin{equation} \lVert Av \rVert _{X'} \leq c_2^A \lVert v \rVert _X \quad \forall v\in X \end{equation} and \(X\)-elliptic \begin{equation} \left \langle Av,v \right \rangle \geq c_1^A \lVert v \rVert _X^2 \quad \forall v\in X. \end{equation} For the operator equation \(Au=f\) with an arbitrary right hand side \(f\in X'\), there exists a unique solution \(u\) satisfying the stability condition \begin{equation} \lVert u \rVert _X \leq \frac {1}{c_1^A} \lVert f \rVert _{X'}. \end{equation}

Comment 1 \(c_2^A\) is the continuity or boundedness constant of the operator \(A\). When \(c_2^A\) is larger, the continuity of \(A\) becomes worse, since the norm of the image \(Av\) is allowed to vary in a larger domain. \(c_1^A\) is the ellipticity constant of \(A\). When \(c_1^A\) is larger, the solution \(u\) of \(Au=f\) is more stable, because its norm is confined in a smaller domain under a given \(f\). If there is noise or experimental error in the given data \(f\), it will also be amplified by the factor \(\frac {1}{c_1^A}\). According to Understanding about ellipticity of operators, when \(A\) is a square matrix, i.e. a linear map between two finite dimensional spaces, \(c_1^A\) corresponds to the minimum eigenvalue of \(A\). Obviously, the larger the minimum eigenvalue, the easier the linear system can be solved.

Lemma 2 (Semi-ellipticity lemma) Let \(L\) be a uniform elliptic partial differential operator, it satisfies \begin{equation} a(v,v) \geq \lambda _0 \lvert v \rvert _{H^1(\Omega )}^2 \quad \forall v\in H^1(\Omega ), \end{equation} where \(\lambda _0\) is the lower bound of eigenvalues of the coefficient matrix \(a_{ji}(x)\).

Theorem 1 (Norm equivalence of Sobolev) Let \(f: W_2^1(\Omega ) \rightarrow \mathbb {R}\) be a bounded linear functional such that \begin{equation} \lvert f(v) \rvert \leq c_f \lVert v \rVert _{W_2^1(\Omega )} \quad \forall v\in W_2^1(\Omega ). \end{equation} If \(f(c) = 0\), \(c\) being a constant function, is only satisfied for \(c=0\), then \begin{equation} \lVert v \rVert _{W_2^1(\Omega ),f} := \left \{ \lvert f(v) \rvert ^2 + \lvert v \rvert _{W_2^1(\Omega )}^2 \right \}^{\frac {1}{2}} \end{equation} defines an equivalent norm in \(W_2^1(\Omega )\), where \(\lvert \cdot \rvert _{W_2^1(\Omega )}\) is a semi-norm in \(W_2^1(\Omega )\), which is equal to \(\lVert \nabla v \rVert _{L_2(\Omega )}\).

N.B. When \(\Omega \) is a Lipschitz domain, \(W_2^1(\Omega )\) is the same as \(H^1(\Omega )\).

Theorem 2 (Trace theorem) Let \(\Omega \subset \mathbb {R}^d\) be a \(C^{k-1,1}\) domain, i.e. \(\Omega \) is \((k-1)\)-th order Lipschitz continuous. When \(s\in (\frac {1}{2}, k]\), the interior Dirichlet trace operator \(\gamma _0^{\mathrm {int}}: H^s(\Omega ) \rightarrow H^{s-1/2}(\Gamma )\) is bounded such that \begin{equation} \lVert \gamma _0^{\mathrm {int}}v \rVert _{H^{s-1/2}(\Gamma )} \leq c_T \lVert v \rVert _{H^s(\Omega )} \quad \forall v\in H^s(\Omega ). \end{equation}

When \(\Omega \) is Lipschitz continuous, i.e. \(k=1\), if \(s=k\), we have \(\gamma _0^{\mathrm {int}}: H^1(\Omega ) \rightarrow H^{1/2}(\Gamma )\). This case is what we usually deal with in BEM.

Theorem 3 (Inverse trace theorem) Let \(\Omega \subset \mathbb {R}^d\) be a \(C^{k-1,1}\) domain. When \(s\in (\frac {1}{2},k]\), the interior Dirichlet trace operator \(\gamma _0^{\mathrm {int}}\) has a continuous right inverse operator \(\mathcal {E}: H^{s-1/2}(\Gamma ) \rightarrow H^s(\Omega )\) such that for all \(w\in H^{s-1/2}(\Gamma )\), \(\gamma _0^{\mathrm {int}}\mathcal {E}w = w\) and \begin{equation} \lVert \mathcal {E}w \rVert _{H^s(\Omega )} \leq c_{IT}\lVert w \rVert _{H^{s-1/2}(\Gamma )}. \end{equation}

In the following sections, we will follow the above procedures to derive the solvability and stability conditions for the elliptic equation with all kinds of boundary conditions, i.e. Dirichlet, Neumann, mixed and Robin boundary conditions.

3 Dirichlet boundary condition

The elliptic PDE in Equation 3 is assigned the Dirichlet boundary condition \(\gamma _0^{\mathrm {int}}u(x) = g(x)\) for \(x\in \Gamma \) with \(g\in H^{1/2}(\Gamma )\). Because the solution \(u\) should satisfy this boundary condition, it should belong to the following space \begin{equation} V_g := \left \{ v\in H^1(\Omega ) \big \vert \gamma _0^{\mathrm {int}}v(x)=g(x), x\in \Gamma \right \}. \end{equation} The test function \(v\) should belong to \begin{equation} V_0 := \left \{ v\in H^1(\Omega ) \big \vert \gamma _0^{\mathrm {int}}v(x)=0, x\in \Gamma \right \}, \end{equation} which is just \(H_0^1(\Omega )\).

Because \(v\) vanishes on \(\Gamma \), the Green’s first formula becomes \begin{equation} a(u,v) = \left \langle f,v \right \rangle . \end{equation} Let \(u_g\in H^1(\Omega )\) be an arbitrary but \(H^1(\Omega )\)-conforming extension of the boundary data \(g\in H^{1/2}(\Gamma )\) into the domain \(\Omega \), i.e. \(u_g = \mathcal {E}g\). Due to the inverse trace theorem 3, we have \begin{equation} \lVert u_g \rVert _{H^1(\Omega )} \leq c_{IT} \lVert g \rVert _{H^{1/2}(\Gamma )}. \end{equation} Then the solution \(u\) can be decomposed into a homogeneous part and an inhomogeneous part: \begin{equation} u = u_0 + u_g \quad u_0\in H_0^1(\Omega ). \end{equation} Such a decomposition is natural, because the boundary values of \(u\) have already been given as the boundary condition, the actual unknown function to be solved should exclude this part. Then the above variational formulation becomes \begin{equation} \begin{aligned} a(u_0 + u_g, v) &= \left \langle f,v \right \rangle \\ a(u_0,v) &= \left \langle f,v \right \rangle - a(u_g,v) \end{aligned}. \end{equation} Now, the two operand functions in the bilinear form \(a(\cdot ,\cdot )\) belong to the same space \(H_0^1(\Omega )\) and the right hand side can be considered as a new bounded linear functional \(\tilde {f}\) applied to \(v\), i.e. \(\left \langle f,v \right \rangle - a(u_g,v) = \tilde {f}(v)\). If we can prove \(a(\cdot ,\cdot )\) or its associated linear operator \(A\) is \(H_0^1(\Omega )\)-elliptic, \(u_0\) can be uniquely solved with the given data \(f\) and the extension \(u_g\)².

Define an equivalent norm for \(H^1(\Omega )\): \begin{equation} \lVert v \rVert _{H^1(\Omega ),\Gamma } := \left \{ \left ( \int _{\Gamma } \gamma _0^{\mathrm {int}}v(x) ds_x \right )^2 + \int _{\Omega } \lvert \nabla v \rvert ^2 dx \right \}^{1/2} = \left \{ \left ( \int _{\Gamma } \gamma _0^{\mathrm {int}}v(x) ds_x \right )^2 + \lvert v \rvert _{H^1(\Omega )}^2 \right \}^{1/2}. \end{equation} With the semi-ellipticity lemma 2, we have \begin{equation} a(v,v) \geq \lambda _0 \lvert v \rvert _{H^1(\Omega )}^2 \quad \forall v\in H_0^1(\Omega ). \end{equation} Because \(\gamma _0^{\mathrm {int}}v(x) = 0\), the first term in the above equivalent norm is zero, hence \(\lvert v \rvert _{H^1(\Omega )} = \lVert v \rVert _{H^1(\Omega ),\Gamma }\) and \begin{equation} a(v,v) \geq \lambda _0 \lVert v \rVert _{H^1(\Omega ),\Gamma } \geq c_1^A \lVert v \rVert _{H^1(\Omega )}. \end{equation} The variational formulation has a unique solution \(u_0\) due to Lax-Milgram Lemma 1.

Considering the boundedness of \(f\) and \(a(\cdot ,\cdot )\) and applying the inverse trace theorem 3 for \(u_g\), the stability condition for \(u_0\) can be obtained. \begin{equation} \begin{aligned} c_1^A \lVert u_0 \rVert _{H^1(\Omega )}^2 &\leq \lvert a(u_0,u_0) \rvert = \lvert \left \langle f,u_0 \right \rangle - a(u_g,u_0) \rvert \\ &\leq \left \{ \lVert f \rVert _{H^{-1}(\Omega )} \lVert u_0 \rVert _{H^1(\Omega )} + c_2^A \lVert u_g \rVert _{H^1(\Omega )} \lVert u_0 \rVert _{H^1(\Omega )} \right \} \\ &\leq \left \{ \lVert f \rVert _{H^{-1}(\Omega )} + c_2^A c_{IT} \lVert g \rVert _{H^{1/2}(\Gamma )} \right \} \lVert u_0 \rVert _{H^1(\Omega )} \end{aligned}. \end{equation} \begin{equation} \lVert u_0 \rVert _{H^1(\Omega )} \leq \frac {1}{c_1^A} \lVert f \rVert _{H^{-1}(\Omega )} + \frac {c_2^A}{c_1^A}c_{IT} \lVert g \rVert _{H^{1/2}(\Gamma )}. \end{equation} Then the stability condition for \(u = u_0 + g_g\) is \begin{equation} \begin{aligned} \lVert u \rVert _{H^1(\Omega )} &\leq \lVert u_0 \rVert _{H^1(\Omega )} + \lVert u_g \rVert _{H^1(\Omega )} \\ &\leq \lVert u_0 \rVert _{H^1(\Omega )} + c_{IT}\lVert g \rVert _{H^{1/2}(\Gamma )}\\ &= \frac {1}{c_1^A} \lVert f \rVert _{H^{-1}(\Omega )} + \left ( 1 + \frac {c_2^A}{c_1^A} \right )c_{IT} \lVert g \rVert _{H^{1/2}(\Gamma )} \end{aligned}. \end{equation}

4 Neumann boundary condition

The elliptic PDE in Equation 3 is assigned the Neumann boundary condition \(\gamma _1^{\mathrm {int}}u(x)=g(x)\) for \(x\in \Gamma \) with \(g(x)\in H^{-1/2}(\Gamma )\), which prescribes the conormal trace at the boundary. A solvability condition inherent in the PDE must be satisfied \begin{equation} \int _{\Omega } f(x) dx + \int _{\Gamma } g(x) ds_x = 0, \end{equation} which associates the volume integral of the source \(f\) in the domain \(\Omega \) and the surface integral of the conormal trace at the boundary \(\Gamma \). This is just another formulation of the Gauss’s divergence theorem we have been familiar with.

Meanwhile, since only the conormal trace of \(u\) instead of its exact values are fixed at the boundary, if \(u\) is a solution, \(u + \alpha \) with an arbitrary \(\alpha \in \mathbb {R}\) must also be a solution. To ensure solution uniqueness, we need to confine \(u\) within a subspace of \(H^1(\Omega )\): \begin{equation} H_{\ast }^1(\Omega ) := \left \{ v\in H^1(\Omega ) \big \vert \int _{\Omega }v(x)dx = 0 \right \}. \end{equation}

Substitute the boundary condition into the Green’s first formula, we have the variational equation: \begin{equation} a(u,v) = \left \langle f,v \right \rangle + \left \langle g,\gamma _0^{\mathrm {int}}v \right \rangle , \end{equation} where both \(u\) and \(v\) belong to \(H_{\ast }^1(\Omega )\).

Note the condition \(\int _{\Omega } v(x) dx = 0\) in the definition of \(H_{\ast }^1(\Omega )\), we can define an equivalent norm for \(H^1(\Omega )\): \begin{equation} \lVert v \rVert _{H^1(\Omega ),\Omega } := \left \{ \left ( \int _{\Omega } v(x) dx \right )^2 + \lvert v \rvert _{H^1(\Omega )}^2 \right \}^{1/2} = \lvert v \rvert _{H^1(\Omega )}. \end{equation} Therefore, an \(H_{\ast }^1(\Omega )\)-ellipticity condition can be derived from the semi-ellipticity condition as below: \begin{equation} a(v,v) \geq \lambda _0 \lvert v \rvert _{H^1(\Omega )}^2 = \lambda _0 \lVert v \rVert _{H^1(\Omega ),\Omega }^2 \geq \tilde {c}_1^A \lVert v \rVert _{H^1(\Omega )}. \end{equation} Even though the bilinear form \(a(\cdot ,\cdot )\) here has the same formulation as that used in the Dirichlet problem, since the original partial differential operator \(L\) has not changed, the above ellipticity constant \(\tilde {c}_1^A\) is different from \(c_1^A\) in the Dirichlet problem, because \(v\) now belongs to the space \(H_{\ast }^1(\Omega )\) instead of \(H_0^1(\Omega )\).

Using this ellipticity condition and the trace theorem 2, the stability condition for the solution \(u\) can be obtained. \begin{equation} \begin{aligned} \tilde {c}_1^A \lVert u \rVert _{H^1(\Omega )}^2 &\leq \lvert a(u,u) \rvert = \lvert \left \langle f,u \right \rangle + \left \langle g,\gamma _0^{\rm int}u \right \rangle \rvert \\ &\leq \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} \lVert u \rVert _{H^1(\Omega )} + \lVert g \rVert _{H^{-1/2}(\Gamma )} \lVert \gamma _0^{\rm int}u \rVert _{H^{1/2}(\Gamma )} \\ &\leq \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} \lVert u \rVert _{H^1(\Omega )} + \lVert g \rVert _{H^{-1/2}(\Gamma )} c_T \lVert u \rVert _{H^1(\Omega )} \end{aligned}. \end{equation} Therefore, \begin{equation} \lVert u \rVert _{H^1(\Omega )} \leq \frac {1}{\tilde {c}_1^A} \left \{ \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + c_T \lVert g \rVert _{H^{-1/2}(\Gamma )} \right \}. \end{equation}

5 Mixed boundary conditions

The boundary conditions are assigned to two parts comprising \(\Gamma \): \(\gamma _0^{\mathrm {int}}u(x)=g_D(x)\) for \(x\in \Gamma _D\) and \(\gamma _1^{\mathrm {int}}u(x)=g_N(x)\) for \(x\in \Gamma _N\), where \(g_D(x)\in H^{1/2}(\Gamma _D)\) and \(g_N(x)\in H^{-1/2}(\Gamma _N)\).

The variational formulation is still the Green’s first formula. Like what we have done for the Dirichlet problem, we need to make the test function in the variational formulation vanish on the Dirichlet boundary, i.e. \(\Gamma _D\). Therefore, \begin{equation} H_0^1(\Omega ,\Gamma _D) := \left \{ v\in H^1(\Omega ) \big \vert \gamma _0^{\mathrm {int}}v(x) = 0, x\in \Gamma _D \right \}, \end{equation} is selected as the test space for \(v\). The solution space for \(u\) should still be \(H^1(\Omega )\), with the constraint \(\gamma _0^{\mathrm {int}}u(x)=g(x)\) for \(x\in \Gamma _D\).

In the Dirichlet problem, we have decomposed the solution \(u\) as \(u_0 + u_g\), where \(u_g\) is an extension of the boundary data. Now in the mixed boundary problem, the Dirichlet data \(g_D\in H^{1/2}(\Gamma _D)\) are only available on part of the boundary \(\Gamma _D\). So we need to first extend it to the whole boundary \(\Gamma \) as \(\tilde {g}_D\in H^{1/2}(\Gamma )\), then extend \(\tilde {g}_D\) into the domain \(\Omega \) as \(u_{\tilde {g}_D}\in H^1(\Omega )\) based on the inverse trace theorem. The two extensions here are both arbitrary but should be bounded and conforming to corresponding Sobolev spaces. The boundedness conditions are \begin{equation} \lVert \tilde {g}_D \rVert _{H^{1/2}(\Gamma )} \leq c \lVert g_D \rVert _{H^{1/2}(\Gamma _D)} \end{equation} and \begin{equation} \lVert u_{\tilde {g}_D} \rVert _{H^1(\Omega )} \leq c_{IT} \lVert \tilde {g}_D \rVert _{H^{1/2}(\Gamma )}. \end{equation} Now the solution \(u\) can be decomposed as \(u_0 + u_{\tilde {g}_D}\) and the variational form becomes \begin{equation} a(u_0,v) = \left \langle f,v \right \rangle + \left \langle \gamma _1^{\mathrm {int}}u, \gamma _0^{\mathrm {int}}v \right \rangle - a(u_{\tilde {g}_D}, v), \end{equation} where both \(u_0\) and \(v\) belong to the same space \(H_0^1(\Omega ,\Gamma _D)\). The boundary integral on the right hand side can be decomposed as \begin{equation} \left \langle \gamma _1^{\mathrm {int}}u, \gamma _0^{\mathrm {int}}v \right \rangle = \left \langle \gamma _1^{\mathrm {int}}u, \gamma _0^{\mathrm {int}}v \right \rangle _{\Gamma _D} + \left \langle \gamma _1^{\mathrm {int}}u, \gamma _0^{\mathrm {int}}v \right \rangle _{\Gamma _N}. \end{equation} Because on \(\Gamma _D\), \(\gamma _0^{\mathrm {int}}v = 0\) and on \(\Gamma _N\), \(\gamma _1^{\mathrm {int}}u = g_N\), the boundary term is equal to \(\left \langle g_N, \gamma _0^{\mathrm {int}}v \right \rangle _{\Gamma _N}\). Therefore, the variational formulation is \begin{equation} a(u_0,v) = \left \langle f,v \right \rangle + \left \langle g_N, \gamma _0^{\mathrm {int}}v \right \rangle _{\Gamma _N} - a(u_{\tilde {g}_D},v) \quad u_0,v \in H_0^1(\Omega ,\Gamma _D). \end{equation}

Note that the interior Dirichlet trace of \(v\) vanish on \(\Gamma _D\), if we define an equivalent norm as below: \begin{equation} \lVert v \rVert _{H^1(\Omega ),\Gamma _D} := \left \{ \left ( \int _{\Gamma _D} \gamma _0^{\mathrm {int}}v(x) ds_x \right )^2 + \lvert v \rvert _{H^1(\Omega )}^2 \right \}^{1/2}, \end{equation} the above bilinear form is \(H_0^1(\Omega ,\Gamma _D)\)-elliptic: \begin{equation} a(v,v) \geq \lambda _0 \lvert v \rvert _{H^1(\Omega )}^2 = \lambda _0 \lVert v \rVert _{H^1(\Omega ),\Gamma _D} \geq c_1^A \lVert v \rVert _{H^1(\Omega )}^2. \end{equation} Therefore, \(u_0\) can be uniquely solved from the variational equation and \(u\) is also unique.

The stability condition for the solution can also be derived similarly as before. \begin{equation} \begin{aligned} c_1^A \lVert u_0 \rVert _{H^1(\Omega )}^2 &\leq \lvert a(u_0,u_0) \rvert = \lvert \left \langle f,u_0 \right \rangle + \left \langle g_N,\gamma _0^{\rm int}u_0 \right \rangle _{\Gamma _N} - a(u_{\tilde {g}_D},u_0) \rvert \\ &\leq \left ( \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + c_2^A \lVert u_{\tilde {g}_D} \rVert _{H^1(\Omega )} \right ) \lVert u_0 \rVert _{H^1(\Omega )} + \lVert g_N \rVert _{H^{-1/2}(\Gamma _N)} \lVert \gamma _0^{\rm int}u_0 \rVert _{\tilde {H}^{1/2}(\Gamma _N)} \end{aligned} \end{equation} Because \(\gamma _0^{\mathrm {int}}u_0 = 0\) on \(\Gamma _D\), \begin{equation} \lVert \gamma _0^{\mathrm {int}}u_0 \rVert _{\tilde {H}^{1/2}(\Gamma _N)} = \lVert \gamma _0^{\mathrm {int}}u_0 \rVert _{H^{1/2}(\Gamma )} \leq c_T \lVert u_0 \rVert _{H^1(\Omega )}. \end{equation} Then \begin{equation} \begin{aligned} c_1^A \lVert u_0 \rVert _{H^1(\Omega )}^2 &\leq \left ( \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + c_2^A \lVert u_{\tilde {g}_D} \rVert _{H^1(\Omega )} \right ) \lVert u_0 \rVert _{H^1(\Omega )} + c_T \lVert g_N \rVert _{H^{-1/2}(\Gamma _N)} \lVert u_0 \rVert _{H^1(\Omega )} \\ \lVert u_0 \rVert _{H^1(\Omega )} &\leq \frac {1}{c_1^A} \left \{ \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + c_2^A \lVert u_{\tilde {g}_D} \rVert _{H^1(\Omega )} + c_T \lVert g_N \rVert _{H^{-1/2}(\Gamma _N)} \right \} \\ &\leq \frac {1}{c_1^A} \left \{ \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + c_2^A c_{IT}\lVert \tilde {g}_D \rVert _{H^{1/2}(\Gamma )} + c_T \lVert g_N \rVert _{H^{-1/2}(\Gamma _N)} \right \} \\ &\leq \frac {1}{c_1^A} \left \{ \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + c_2^A c_{IT} c \lVert g_D \rVert _{H^{1/2}(\Gamma _D)} + c_T \lVert g_N \rVert _{H^{-1/2}(\Gamma _N)} \right \} \end{aligned}. \end{equation} Finally, the stability condition for \(u\) is \begin{equation} \begin{aligned} \lVert u \rVert _{H^1(\Omega )} &\leq \lVert u_0 \rVert _{H^1(\Omega )} + \lVert u_{\tilde {g}_D} \rVert _{H^1(\Omega )} \leq \lVert u_0 \rVert _{H^1(\Omega )} + c_{IT} c \lVert g_D \rVert _{H^{1/2}(\Gamma _D)} \\ &\leq \frac {1}{c_1^A} \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + \left ( 1 + \frac {c_2^A}{c_1^A} \right ) c_{IT} c \lVert g_D \rVert _{H^{1/2}(\Gamma _D)} + \frac {c_T}{c_1^A} \lVert g_N \rVert _{H^{-1/2}(\Gamma _N)} \end{aligned}. \end{equation}

6 Robin boundary condition

Robin boundary condition is \(\gamma _1^{\mathrm {int}}u(x) + \kappa (x)\gamma _0^{\mathrm {int}}u(x) = g(x)\) for \(x\in \Gamma \) with \(g(x)\in H^{-1/2}(\Gamma )\). We also assume that \(\kappa (x) \geq \kappa _0 > 0\) for all \(x\in \Gamma \). Substitute this relation for \(\gamma _1^{\mathrm {int}}u\) in the Green’s first formula, we have \begin{equation} a(u,v) = \left \langle f,v \right \rangle + \left \langle g,\gamma _0^{\mathrm {int}}v \right \rangle - \left \langle \kappa (x)\gamma _0^{\mathrm {int}}u,\gamma _0^{\mathrm {int}}v \right \rangle . \end{equation} We can obtain a new bilinear form \begin{equation} \tilde {a}(u,v) := a(u,v) + \left \langle \kappa (x)\gamma _0^{\mathrm {int}}u,\gamma _0^{\mathrm {int}}v \right \rangle \end{equation} and the variational equation is \begin{equation} \tilde {a}(u,v) = \left \langle f,v \right \rangle + \left \langle g,\gamma _0^{\mathrm {int}}v \right \rangle , \end{equation} where both \(u\) and \(v\) belong to \(H^1(\Omega )\).

Using the semi-ellipticity for the original bilinear form \(a(\cdot ,\cdot )\), we have \begin{equation} \begin{aligned} \tilde {a}(v,v) &\geq \lambda _0 \lvert v \rvert _{H^1(\Omega )}^2 + \left \langle \kappa (x)\gamma _0^{\rm int}u,\gamma _0^{\rm int}v \right \rangle \\ &\geq \lambda _0 \lvert v \rvert _{H^1(\Omega )} + \kappa _0 \lVert \gamma _0^{\rm int}v \rVert _{L_2(\Gamma )}^2 \\ &\geq \min \left \{ \lambda _0,\kappa _0 \right \} \left ( \lvert v \rvert _{H^1(\Omega )}^2 + \lVert \gamma _0^{\rm int}v \rVert _{L_2(\Gamma )}^2 \right ) \end{aligned}. \end{equation} Define an equivalent norm for \(H^1(\Omega )\) as \begin{equation} \lVert v \rVert _{H^1(\Omega ),\Gamma } := \left \{ \lvert v \rvert _{H^1(\Omega )}^2 + \lVert \gamma _0^{\rm int}v \rVert _{L_2(\Gamma )}^2 \right \}^{1/2}, \end{equation} we can obtain the \(H^1(\Omega )\)-ellipticity for \(\tilde {a}(\cdot ,\cdot )\) \begin{equation} \tilde {a}(v,v) \geq c_1^A \lVert v \rVert _{H^1(\Omega )}^2. \end{equation} Therefore, the solution \(u\) for the variational equation is unique.

Because \begin{equation} \begin{aligned} c_1^A \lVert u \rVert _{H^1(\Omega )}^2 &\leq \lvert \tilde {a}(u,u) \rvert = \lvert a(u,u) + \left \langle \kappa (x)\gamma _0^{\rm int}u,\gamma _0^{\rm int}u \right \rangle \rvert \\ &= \lvert \left \langle f,v \right \rangle + \left \langle g,\gamma _0^{\rm int}u \right \rangle \rvert \\ &\leq \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} \lVert u \rVert _{H^1(\Omega )} + \lVert g \rVert _{H^{-1/2}(\Gamma )} c_T \lVert u \rVert _{H^1(\Omega )} \end{aligned}, \end{equation} the stability condition for \(u\) is \begin{equation} \lVert u \rVert _{H^1(\Omega )} \leq \frac {1}{c_1^A} \lVert f \rVert _{\tilde {H}^{-1}(\Omega )} + \frac {c_T}{c_1^A} \lVert g \rVert _{H^{-1/2}(\Gamma )}. \end{equation}

7 Summary

• The actual bilinear form used for proving the solution uniqueness of a PDE with a certain boundary condition (e.g. Robin boundary condition) may not be the original bilinear form in the Green’s first formula.

• The linear operator \(A\) induced from the bilinear form is not the original partial differential operator \(L\).

• When the Dirichlet boundary condition is involved, either on the whole boundary or part of it, the solution to be searched should have zero interior trace at the Dirichlet boundary. This is because the Dirichlet boundary condition is directly enforced as a constraint on the solution \(u\), the unknown part in \(u\) should not contain the Dirichlet data.

• The extension of the Dirichlet boundary data, either from \(\Gamma _D\) to \(\Gamma \) or from \(\Gamma \) to the whole domain \(\Omega \), is arbitrary but should be bounded and space conforming.

¹The concept of duality pairing can be considered as an extension of the inner product. The two operands in a duality pairing respectively belong to a normed space and its dual space, while the two operands in an inner product belong to a same Hilbert space. With the Riesz Representation Theorem in the Hilbert space context, the duality pairing can be defined with respect to an inner product.

²Even though the extension \(u_g\) is arbitrary, the solution \(u\) as the sum of \(u_g\) and \(u_0\) is unique.