Let $(M^3,g)$ be a compact $3$-manifold with boundary and let $\Sigma$ be a surface such that $\partial M \cap \Sigma = \partial \Sigma$ and the intersection is orthogonal ($\Sigma$ is a free-boundary surface). Denote by $\nu$ the unit conormal for $\Sigma$. Let

$$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$
and $$F = \left\{ \phi \in C^{0,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\},$$

for fixed $0 < \alpha < 1$.

I would like to know why, given $(f,h) \in F \times C^{1,\alpha}(\partial \Sigma)$, the problem

$$\begin{cases} -\Delta_{\Sigma} u = f + \frac{1}{\vert \Sigma \vert} \int_{\partial \Sigma} h \, \mathrm{d}L, \quad \text{on } \Sigma \\ \frac{\partial u}{\partial \nu} = -h, \quad \text{on } \partial \Sigma \end{cases}$$

has a solution in $E$. This came up for me in this paper, page 8. I am sorry if the question is silly, but I am not very good at PDE's.

Let $\Sigma$ be a compact surface and denote by $\nu$ the unit conormal for $\partial \Sigma$. Let

$$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$
and $$F = \left\{ \phi \in C^{0,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\},$$

for fixed $0 < \alpha < 1$.

I would like to know why, given $(f,h) \in F \times C^{1,\alpha}(\partial \Sigma)$, the problem

$$\begin{cases} -\Delta_{\Sigma} u = f + \frac{1}{\vert \Sigma \vert} \int_{\partial \Sigma} h \, \mathrm{d}L, \quad \text{on } \Sigma \\ \frac{\partial u}{\partial \nu} = -h, \quad \text{on } \partial \Sigma \end{cases}$$

has a solution in $E$. This came up for me in this paper, page 8. I am sorry if the question is silly, but I am not very good at PDE's.

Let $(M^3,g)$ be a compact $3$-manifold with boundary and let $\Sigma$ be a surface such that $\partial M \cap \Sigma = \partial \Sigma$ and the intersection is orthogonal ($\Sigma$ is a free-boundary surface). Denote by $\nu$ the unit conormal for $\Sigma$. Let

$$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$
and $$F = \left\{ \phi \in C^{0,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\},$$

for fixed $0 < \alpha < 1$.

I would like to know why, given $(f,g) \in F \times C^{1,\alpha}(\partial \Sigma)$, the problem

$$\begin{cases} -\Delta_{\Sigma} u = f + \frac{1}{\vert \Sigma \vert} \int_{\partial \Sigma} g \, \mathrm{d}L, \quad \text{on } \Sigma \\ \frac{\partial u}{\partial \nu} = -g, \quad \text{on } \partial \Sigma \end{cases}$$

has a solution in $E$. This came up for me in a paper. I am sorry if the question is silly, but I am not very good at PDE's.

Let $(M^3,g)$ be a compact $3$-manifold with boundary and let $\Sigma$ be a surface such that $\partial M \cap \Sigma = \partial \Sigma$ and the intersection is orthogonal ($\Sigma$ is a free-boundary surface). Denote by $\nu$ the unit conormal for $\Sigma$. Let

$$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$
and $$F = \left\{ \phi \in C^{0,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\},$$

for fixed $0 < \alpha < 1$.

I would like to know why, given $(f,h) \in F \times C^{1,\alpha}(\partial \Sigma)$, the problem

$$\begin{cases} -\Delta_{\Sigma} u = f + \frac{1}{\vert \Sigma \vert} \int_{\partial \Sigma} h \, \mathrm{d}L, \quad \text{on } \Sigma \\ \frac{\partial u}{\partial \nu} = -h, \quad \text{on } \partial \Sigma \end{cases}$$

has a solution in $E$. This came up for me in this paper, page 8. I am sorry if the question is silly, but I am not very good at PDE's.