= Sobolev domains of powers of an elliptic Dirichlet operator
{c}
{title2=$X_k=D(A_D^{k/2})$}
For a strictly positive <Dirichlet realization of an elliptic operator> with smooth coefficients and smooth boundary, put $X_0=L^2(U)$. For each integer $k\geq1$,
$$
X_k=\left\{u\in H^k(U):T(L^ju)=0\quad 0\leq j\leq\left\lfloor\frac{k-1}{2}\right\rfloor\right\}.
$$
These are the domains $D(A_D^{k/2})$, with norms equivalent to the indicated <Sobolev space> norms. The <bilinear forms> $((u,v))_{2l}=(L^lu,L^lv)_{L^2}$ and $((u,v))_{2l+1}=B[L^lu,L^lv]$ are <inner products> on $X_{2l}$ and $X_{2l+1}$. The higher boundary conditions are essential: for $L=-d^2/dx^2$ and $u=x(\pi-x)$, $u$ is smooth and zero at the boundary but $L^2u=0$, so $\|L^2u\|_2$ is not a norm on all of $H^4\cap H_0^1$. Repeated <elliptic regularity> proves $A_D:X_{k+2}\to X_k$ is an isomorphism, which gives the norm equivalence inductively from $X_0$ and $X_1=H_0^1$.
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