Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 105 / 3 / b / ii

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.