ht in mixed-norm (anisotropic) Sobolev-Slobodevskij spaces $H^{2,(s_1,s_2,s_3)}(G\times S\times I^\circ)$ where $s_j\geq 0$. Regularity results are needed e.g. in considerations of various approximation errors and in convergence analysis of numerical schemes. The regularity analysis remains open.

The variational methods used above apply (after relevant modifications of the assumptions) also to more general problems of the form $$\begin{gathered}
a{{\frac{\partial \psi}{\partial E}}}+ F\cdot\nabla_{\tilde\omega}\psi+\omega\cdot\nabla_x\psi+(\Sigma-K)\psi=f
\nonumber\\
\psi_{|\Gamma_-}=g,\quad \psi(\cdot,\cdot,E_{\rm m})=0, \label{add1}\end{gathered}$$ where $a=a(x,\omega,E)$, $F=(F_1(x,\omega,E),F_2(x,\omega,E),F_3(x,\omega,E))$ and $\nabla_{\tilde{\omega}}$ is the gradient operator with respect to the Riemannian metric on the sphere $S$ induced by the Euclidean metric on ${\mathbb{R}}^3$. Notice that since $\nabla_{\tilde{\omega}}\psi$ is tangent to $S$, we may assume that $\omega\cdot F(x,\omega,E)=0$ for all $(x,\omega,E)$. More explicitly, if $\phi\in C^1(S)$, and if $\tilde{\phi}:{\mathbb{R}}^3\backslash\{0\}\to{\mathbb{R}}$ is given by $\tilde{\phi}(y)=\phi(\frac{y}{{\left\Vert y\right\Vert}})$, then $$\nabla_{\tilde{\omega}}\phi =\nabla_y\tilde{\phi}-(\omega\cdot\nabla_y\tilde\phi)\omega$$ where $\nabla_{\tilde{\omega}} \phi=((\nabla_{\tilde{\omega}} \phi)_1, (\nabla_{\tilde{\omega}} \phi)_2, (\nabla_{\tilde{\omega}} \phi)_3)$ and $\omega=(\omega_1,\omega_2,\omega_3)$ as elements of ${\mathbb{R}}^3$.

In this context the space $H_2$ must be replaced with the space ${{{\mathcal{}}}H}_2$ defined as follows. The space ${{{\mathcal{}}}H}_2$ is the completion of $C^1(G\times S\times I)$ with respect to the inner product$${\left\langle}\psi,v{\right\rangle}_{H_2}+{\left\langle}F\cdot\nabla_{\tilde\omega}\psi,F\cdot\nabla_{\tilde\omega}v{\right\rangle}_{L^2(G\times S\times I)}.$$ This observation is based on the fact that $$2{\left\langle}F\cdot\nabla_{\tilde\omega}\psi,\psi{\right\rangle}_{L^2(G\times S\times I)