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9 votes
Accepted

Elliptic operators over noncompact manifold

Solutions of the Laplace equation $\Delta u = 0$ include harmonic polynomials, which grow at most polynomially, each $|u(x)| \le C |x|^N$ for some $N>0$ depending on $u$. Define $D[v] = e^{-g(x)} \...
Igor Khavkine's user avatar
2 votes
Accepted

Approximation on $H^1_0(B)$ and cut-off functions

Do it first for the half-space $\{x_n >0\}=\Sigma$. If $u$ vanishes at the boundary then $u(x',x_n)^2=2\int_0^{x_n} uD_n u$ and so ($\Sigma_\delta=\{0 <x_n <\delta\}$) $$ \int_{\Sigma_\delta} ...
Giorgio Metafune's user avatar

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