Despite the high degree of symmetry of the domain $D=\{ |x|< R : x=(x_1, x_2)\in \Bbb R^2 \wedge R>0 \}$ where the problem is posed, it cannot be expected that an explicit Green's function exists for the problem at hand, since the properties of $\theta$ influence very much not only the structure but also the very existence of such a function (as we'll see below). However, for large classes of functions $\theta :D \to [0, \pi]$, while an explicit expression of Green's function customarily still lacks, it is possible to prove that such a function exists and that it satisfies interesting pointwise estimate: I'll try to show this, without giving the however complex analytical details, in two steps.
Step 1: the partial differential equation associated to $Q^\theta (\varphi)$ and requirements on $\theta$ ensuring a solvability of the posed problem.
Let's calculate the functional derivative and thus the Euler-Lagrange equation of $Q^\theta(\varphi)$ in order to see what kind of equation we have to deal with. For all admissible variations $h\in C^1_c(D)$, we must have
$$
\begin{split}
\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} Q^\theta(\varphi+\varepsilon h)\right|_{\varepsilon=0} &= 0\\
& \Updownarrow\\
\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} Q^\theta(\varphi+\varepsilon h)\right|_{\varepsilon=0} &= \frac{\mathrm{d}}{\mathrm{d}\varepsilon}\int\limits_D \sin(\theta)^2\big[(\partial_1\varphi + \varepsilon \partial_1h)^2+(\partial_2\varphi + \varepsilon \partial_1h)^2\big]\,\mathrm{d} x\Bigg|_{\varepsilon=0} \\
&=\int\limits_D 2\sin(\theta)^2\nabla\varphi\cdot\nabla h\, \mathrm{d} x =0,
\end{split}\label{1}\tag{E-L}
$$
and assuming $\varphi\in C^2(D)$ from equation \eqref{1} we get the classical formulation of the problem we are dealing with:
$$
\newcommand{\dvg}{\operatorname{\nabla\cdot}}
\int\limits_D 2\sin(\theta)^2\nabla\varphi\cdot\nabla h\, \mathrm{d} x =0 \iff \dvg\big(\sin(\theta)^2 \nabla \phi\big)=0\label{2}\tag{1}
$$
We thus have an homogeneous equation for a differential operator in divergence form, and despite being an extensively studied class, in order to deal with such kind of equations we must do some assumptions: precisely, the following ones.
- Uniform ellipticity: we must assume that
$$
\lambda^{-1} |\xi|^2 \le \sin(\theta)^2\sum_{i=1}^2 \sum_{j=1}^2 \xi_i \xi_j \le \lambda |\xi|^2\quad\forall x\in D
$$
for a fixed $\lambda >0$ and this implies that it must exist a fixed a $\pi > \delta >0 $ such that
$$
\delta \le |\theta(x)| \le \pi-\delta \quad \forall x\in D.
$$
This is a non degeneracy requirement: at the points $x\in D$ where $\sin(\theta)^2=0$, equation \eqref{2} changes its structure (as, for example, it happens for Tricomi's equation) and the analytic study gets too complex.
- Boundedness and measurability for the coefficients: this is really a mild limitation for $\theta :D\to [0, \pi]$, and it is only technical in the sense that these general requirements are used in [1] and [2] in order to work out a general solution to the problem.
Step 2: uniqueness, existence theory and a pointwise estimate for the Green's function $\mathscr{G}$.
Assuptions 1 and 2 are sufficient to develop a complete theory for the Green's function of divergence form uniformly elliptic operators, i.e. the following problem has always a unique solution:
$$
\begin{cases}
-\dvg\Big(\sin\big(\theta(x)\big)^2\nabla\mathscr{G}(x,y)\Big)=\delta(x-y)\\
\left.\mathscr{G}\right|_{x\in\partial D}=0
\end{cases} \quad x,y\in D\label{3}\tag{2}
$$
The "details" of the study are given in the paper [1] and in the course lecture notes [2]: we'll not give an exposition of them here (it would be almost impossible to do so). However, it is worth to note that one of the results proved in these references ([1] §7, theorem 7.1 p. 66, and the corollary at p. 235 of théorème 8.5, [2], ch. 8, pp. 234-235) implies the following estimate:
$$
c^{-1}\le \frac{\mathscr{G}(x,y)}{\mathscr{G}_\Delta(x,y)}\le c\qquad \forall x,y\in D^\prime,
$$
where
- $\mathscr{G}$ is the solution to \eqref{3}, while $\mathscr{G}_\Delta$ is the Green's function for the laplacian on $D$.
- $D^\prime\Subset D$ is any compact subset of $D$,
- $c$ is a positive constant depending on the domain $D$, on $D^\prime$, on the dimension of the ambient space ($n=2$ for this problem) and on the ellipticity constant $\lambda$.
Final notes
- A survey of references [1] and [2] can be found in this Q&A.
References
[1] Walter Littman, Hans Weinberger and Guido Stampacchia (1962), "Regular points for elliptic equations with discontinuous coefficients", Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, serie III, Vol. 17, n° 1-2, pp. 43-77, MR161019, Zbl 0116.30302.
[2] Guido Stampacchia (1966), "Équations elliptiques du second ordre à coefficients discontinus" (notes du cours donné à la 4me session du Séminaire de mathématiques supérieures de l'Université de Montréal, tenue l'été 1965), (in French), Séminaire de mathématiques supérieures 16, Montréal: Les Presses de l'Université de Montréal, pp. 326, ISBN 0-8405-0052-1, MR0251373, Zbl 0151.15501.
