Consider a domain $\mathcal{D}$ which is a right circular cylinder in $\mathbb{R}^3$, with radius 1 and height 1, say. The boundary of $\mathcal{D}$, which I denote by $\partial\mathcal{D}$ consists of two parts:

(a) The unit disk on the plane $z=0$, which I denote by $\mathcal{F}$.

(b) The unit disk on the plane $z=-1$, together with the vertical sides $x^2+y^2=1, z\in(-1,0)$, which I denote by $\mathcal{B}$.

My problem involves establishing existence of minimisers of the Dirichlet energy $J(\Phi)=\int_\mathcal{D} |\nabla\Phi|^2\, dV$ over $H^1(\mathcal{D};\mathbb{R})$, with the constraint $\int_\mathcal{F} \Phi\, dA=0$. The plan is to approach it using the Direct Method of Calculus of Variations, but I have few questions at the moment:

(1) Is the domain $\mathcal{D}$ a Lipschitz domain? I don't really have a good geometric interpretaton of Lipschitz domain, but I suspect the answer is no, since a Lipschitz domain must have zero Lebesgue measure on its boundary but the Lebesgue measure of $\mathcal{F}$ is nonzero.

(2) Is there any variant of Poincare inequality for both Lipschitz/non-Lipschitz domain that also taking into account that the mean of $\Phi$ over part of the boundary, in this case is $\mathcal{F}$ (instead of the usual full domain $\mathcal{D}$) is 0? The reason is that one wants to show that $J(\Phi)$ is coercive over $H^1(\mathcal{D};\mathbb{R})$, and for Dirichlet energy the common way is to estimate it using Poincare inequality; but here I am not working with $H_0^1(\mathcal{D};\mathbb{R})$, and $\Phi$ does not have zero mean over $\mathcal{D}$.

(3) How do I deal with the integral constraint $\int_\mathcal{F} \Phi\, dA=0$. The single way I know is to show that the integral constraint is weakly continuous with respect to the $H^1(\mathcal{D};\mathbb{R})$ norm, but here my integral constraint is defined over $\mathcal{F}$. One sufficient condition (I think) is to require the trace embedding to be compact, but I suspect that claim depends strongly on the domain (whether or not it's Lipschitz). Also, how can we define a trace embedding onto parts of the boundary which has positive measure? I don't even know if this is doable, I have always thought the definition of trace only makes sense because of zero Lebesgue measure of the boundary.

(4) If there exists such trace embedding onto $L^2(\mathcal{F})$, then I think the space $Y$, which consists of functions in $H^1(\mathcal{D};\mathbb{R})$ satisfying the given integral constraint, is a Hilbert space. It is well known that the Dirichlet energy is weakly lower semicontinuous over $H^1(\mathcal{D};\mathbb{R})$, but does this result carries through over the space $Y$?