Variational formulation of an elliptic pde

Question

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, what is the variational formulation of the following problem: $$ \text{div}(y^a\nabla_{x,y}V)=0,\quad\text{on }\mathbb{R}^n\times(0,\infty),$$ $$ V(x,0)=f(x),\quad\forall x\in\mathbb{R}^n,?$$ Is true that the solution is a minimum of the energy: $$ [U]_a=\int_{\mathbb{R}^n\times(0,\infty)}y^a|\nabla U|^2\,dx\,dy?$$ What functional space i have to use? I have no idea on how to proceed. I have think that i have to use the completion of $C^\infty_c(\mathbb{R}^n\times[0,\infty))$ under the norm: $[\cdot]_a^{1/2}$, but $[\cdot]_a^{1/2}$ not seems to be a norm.

Answer 1

One possibility is the class $\mathcal W$ of weakly differentiable functions $u$ such that

• $[u]_a$ is finite,
• $u(\cdot, y)$ is in $L^2(\mathbb R^n \times (0, y_0))$ for some (or, equivalently, for all) $y_0$.

This becomes a Hilbert space when equipped with norm $$[u] + \|u(\cdot, 0)\|_2. $$ Note that one can prove that $u(\cdot, y)$ depends continuously on $y \in [0, \infty)$ with respect to the $L^2(\mathbb R^n)$ norm (after a modification of $u$ on a set of zero Lebesgue measure), so the above definition makes sense.

This is written with all details (but in a much greater generality) in my paper with Jacek Mucha:

• M. Kwaśnicki, J. Mucha, Extension technique for complete Bernstein functions of the Laplace operator, J. Evol. Equ. 18(3) (2018): 1341–1379, DOI:10.1007/s00028-018-0444-4.

There are certainly better references that focus specifically on the extension technique for fractional powers of the Laplace operator.