**5. Dense Solvability**

Now we consider the following problems:

$$\begin{split} \mathbb{H}\_{1}(\boldsymbol{\xi}^{(1)},\boldsymbol{\xi}^{\*,(1)}) &= \int\_{\Gamma\_{\mathrm{op}}} \sqrt{\boldsymbol{g}\boldsymbol{H}} \boldsymbol{w} \boldsymbol{\xi}^{(1)} \, d\boldsymbol{\Gamma} + \int\_{\Gamma\_{\mathrm{in}}} \sqrt{\boldsymbol{g}\boldsymbol{H}} \boldsymbol{h} \boldsymbol{\xi}^{(1)} \, d\boldsymbol{\Gamma}, \quad \forall \boldsymbol{\xi}^{\sharp(1)} \in \mathbf{V}^{(1)} \\ \mathbb{H}\_{2}(\boldsymbol{\xi}^{(2)},\boldsymbol{\xi}^{\*,(2)}) &= \int\_{\Gamma\_{\mathrm{in}}} \sqrt{\boldsymbol{g}\boldsymbol{H}} \boldsymbol{h} \boldsymbol{\xi}^{(2)} \, d\boldsymbol{\Gamma}, \quad \forall \boldsymbol{\xi}^{\sharp(2)} \in \mathbf{V}^{(2)} \end{split}$$

with some (possibly non-trivial) functions *h* ∈ *<sup>L</sup>*2(<sup>Γ</sup>*in*), *w* ∈ *<sup>L</sup>*2(<sup>Γ</sup>*op*). Let the functions *ξ*<sup>∗</sup>,(1), *ξ*<sup>∗</sup>,(2) satisfy additional conditions

$$\begin{aligned} \xi^{\*,(1)} &= 0, \quad \text{on } \Gamma\_{op\prime} \\ \xi^{\*,(1)} + \xi^{\*,(2)} &= 0, \quad \text{on } \Gamma\_{in}. \end{aligned}$$

Note that the functions *ξ*∗ = *ξ*<sup>∗</sup>,(*i*) in Ω*i*, *i* = 1, 2, are the solution of a homogenous boundary-value problem in Ω [13] and *ξ*∗ = 0 in Ω. So, we obtain: *ξ*<sup>∗</sup>,(*i*) = 0 in Ω*i*, *i* = 1, 2 and *w* = 0, *h* = 0. Hence, *ker*(*A*∗) = {0}, and this means the *dense solvability* of the problem [26].
