Find Greens function associated with quadratic form

Question

Let $D$ be the open disc in $\mathbb{R}^2$ of radius $R$ (for some $R>0$). Let $\theta:D\to[0,\pi]$ be given (constraints on it are postponed for now).

Consider the quadratic form $$ Q^\theta(\varphi) := \int_D \sin(\theta)^2((\partial_1\varphi)^2+(\partial_2\varphi)^2)$$ where $\varphi:D\to[0,2\pi)$ is a function which takes Dirichlet boundary conditions on $\partial D$.

I am trying to figure out what the Greens function $G_D^\theta(x,y)$ associated with $Q^\theta$ is for arbitrary (but nice) $\theta$. Clearly if $\theta$ is the constant function equal to $\pi/2$ then $Q^{\pi/2}$ is simply the quadratic form associated with the Laplacian on $D$ with Dirichlet boundary conditions, in which case we have a closed-form formula for its Greens function $$ G_D^{\pi/2}(x,y) = -\frac{1}{2\pi}\log\left(\|x-y\|\right)+\frac{1}{2\pi}\log(R)+\frac{1}{4\pi}\log\left(1+\frac{1}{R^4}\|x\|^2\|y\|^2-2\frac{1}{R^2}x\cdot y\right)\,. $$

Is there any hope to get a closed form formula for $G_D^\theta$ for general $\theta$? If not that, then at least some asymptotics for large $\|x-y\|$ and $R$? For example what may be said if $\theta$ is very close to being constant? If not the Greens function, it would also be nice to get the complete eigenfunctions and eigenvalues of the operator associated to $Q^\theta$, which are just sines in case $\theta$ is constant, as is well-known.

The Euler-Lagrange equation that $\varphi$ should solve, i.e., the analog of the Laplace equation is I think $$ \Delta \varphi +2\cot(\theta) (\nabla\theta)\cdot\nabla\varphi = 0 $$ but I am not really sure where to go from there.

Answer 1

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.

1. 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.
2. 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.