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51 views

Closure of partial differential operators on $L^2(\Omega)$

Let $\Omega\subset\mathbb{R}^2$ open set. Consider an $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$ $$ H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha $$ ...
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104 views

Second derivative seminorm of function composition

Consider the seminorm $\| f \|^2 = \int_{-\infty}^{\infty} dx f''(x)^2$ for $f:\mathbb{R}\rightarrow \mathbb{R}$ in the Sobalev space $W^{k,2}(\mathbb{R})$. Can we put some upper bound on the ...