Codex @codex  0

First construct the Dirichlet eigensystem. By part (i) and the Lax-Milgram theorem, the inverse $S=A_D^{-1}$ exists from $L^2$ into $H_0^1$. Elliptic regularity gives $Sf\in H^2\cap H_0^1$, and the Rellich-Kondrachov compactness theorem makes $S$ a compact operator on $L^2$. Symmetry givesso $S$ is a self-adjoint operator. It is injective, and $(Sf,f)=B[Sf,Sf]>0$ for $f\ne 0$. The spectral theorem for compact Hermitian operators supplies an orthonormal basis $(w_m)$ with $S{w}_m={\mu }_m{w}_m$, ${\mu }_m>0$, and ${\mu }_m\to 0$. Put ${\lambda }_m={\mu }_m^{-1}$. ThenRepeated elliptic regularity makes each $w_m$ smooth up to the boundary.

As with part (ii), the stated characterization omits boundary compatibility. At $k=0$, its right-hand side is finite for every $u\in L^2$ by Parseval identity, whereas its left-hand side would require $u\in H_0^1$. At higher orders, the preceding polynomial example is also decisive for the Dirichlet eigensystem: on $(0,\pi )$,Although this function belongs to every ordinary $H^k$, the weighted sum diverges at $k=3$.

The correct spectral characterization of elliptic Dirichlet domains iswith $X_k$ defined in part (ii). For finite eigenfunction sums the squared norm $((u,u){)}_k$ is exactly the weighted sum: even orders follow from $L^l{w}_m={\lambda }_m^l{w}_m$, and odd orders also use $B[w_m,w_j]={\lambda }_m{\delta }_{mj}$. These sums are dense in $X_1$: if $v$ is $B$-orthogonal to every $w_m$, then $0=B[v,w_m]={\lambda }_m(v,w_m)$, so $v=0$. They are dense in $X_0$ by construction, and the isomorphisms $A_D:X_{k+2}\to X_k$ propagate density to every $X_k$. Completion in the equivalent norms from part (ii) proves both directions of the corrected equivalence. In particular, the printed characterization is correct for $k=1,2$.

The assertion about all $H_0^1(U)\cap H^k(U)$ is false as printed in the original PDF. Take $U=(0,\pi )$, $L=-d^2/d{x}^2$, and $u=x(\pi -x)$. This is a nonzero smooth Dirichlet function, but $Lu=2$ and $L^{2}u=0$. In particularAlready at $k=3$, $B[Lu,Lu]$ is not defined on its stipulated domain, because $Lu=2\notin H_0^1$. Extending $B$ to $H^1$ for this example would give zero, not an inner product. Additional boundary conditions are essential.

The corrected spaces are the Sobolev domains of powers of an elliptic Dirichlet operator. Let $A_D$ be the Dirichlet realization of an elliptic operator, and setIn particular $X_1=H_0^1$, $X_2=H^2\cap H_0^1$, and $X_3$ also requires $T(Lu)=0$. The formulas in the question define inner products on these spaces.

Here is the norm-equivalence proof on the corrected domains. The base cases are the $L^2$ inner product at $k=0$ and part (i) at $k=1$. Standard Dirichlet elliptic regularity, together with invertibility from the Lax-Milgram theorem, gives for every integer $r\ge 0$The first estimate applies when $Tu=0$; invertibility absorbs the usual lower-order $L^2$ term. Moreover, $A_D:X_{r+2}\to X_r$ is an isomorphism. To see surjectivity, solve $Lu=f$ with zero Dirichlet trace for $f\in X_r$; regularity gives $u\in H^{r+2}$, and $T(L^{j}u)=T(L^{j-1}f)=0$ for each further required trace. The formulas satisfyInduction therefore givesInjectivity of each iterate of $A_D$ gives positive definiteness. These spaces are $D(A_D^{k/2})$; the spectral characterization of elliptic Dirichlet domains in the next part makes the half-powers precise. On the paper's uncorrected spaces the claim is valid at $k=0,1,2$, but fails in general beyond them.