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

On the zero-boundary Sobolev space $H_0^1(U)$, the associated bilinear form isThe homogeneous Dirichlet boundary condition removes the boundary term, so $(Lu,v{)}_{L^2}=B[u,v]$ for sufficiently regular $u$ and $v\in H_0^1$.

Formal self-adjointness means that $L$ equals its formal adjoint, or equivalently $(L\varphi ,\chi )=(\varphi ,L\chi )$ for compactly supported smooth functions. HereWith the symmetric real principal coefficients in the question, $L=L^{\ast }$ is equivalent to $b=0$. It makes $B$ a symmetric bilinear form.

For the assertions in part (b), positivity must mean strict Dirichlet positivity:This is the positive-definite operator convention. The weaker positive semidefinite operator convention $B[u,u]\ge 0$ does not suffice: $L=-\Delta -{\lambda }_1$ has zero energy on a first Dirichlet eigenfunction. Under strict positivity, the compactness argument in part (b)(i) gives a uniform lower bound as well.

Formal self-adjointness gives symmetry of $B$, and strict positivity gives positive definiteness, so $B$ is an inner product. Bounded coefficients give $B[u,u]\le C\Vert u{\Vert }_{H^1}^2$. To prove the lower bound, uniform ellipticity givesIf no lower bound by the $H^1$ norm existed, choose $u_j\in H_0^1$ with $\Vert u_j{\Vert }_{H^1}=1$ and $B[u_j,u_j]\to 0$. By weak convergence in a Hilbert space and the Rellich-Kondrachov compactness theorem, a subsequence has $u_j\rightharpoonup u$ in $H_0^1$ and $u_j\to u$ in $L^2$. The principal energy has weak lower semicontinuity: it is the squared $L^2$ norm of the bounded multiplication operator $a^{1/2}Du$. The potential term converges because $c$ is bounded. HenceStrict positivity forces $u=0$. The preceding ellipticity estimate now gives $\Vert D{u}_j{\Vert }_2\to 0$, while $\Vert u_j{\Vert }_2\to 0$, contradicting $\Vert u_j{\Vert }_{H^1}=1$. Thus strict positivity implies coercivity for an elliptic Dirichlet form, andThe new norm is equivalent to the original norm, so the space remains complete.

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.

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 $k=1,2$.

Take $\psi \in L^2(U)$, the natural initial-data assumption needed for the requested $L^2$ convergence but not explicitly stated in this subpart. Use the genuine Dirichlet eigensystem from part (b)(iii) and put ${\psi }_m=(\psi ,w_m)$. The spectral construction of a parabolic solution isEach finite sum has zero Dirichlet boundary condition and satisfies $u_t+Lu=0$.

To justify all derivatives, fix $\epsilon >0$. For any integers $k,j\ge 0$, the squared $X_k$ norm of a tail of the $j$th time derivative isuniformly for $t\ge \epsilon$. Use the corrected Sobolev domains of powers of an elliptic Dirichlet operator, not the inaccurate unqualified assertion in the PDF. Their norm equivalence gives convergence in every ordinary $H^k$. The Sobolev embedding theorem then gives convergence of every desired spatial derivative by choosing $k$ large enough. Thus termwise differentiation is justified, $u\in C^{\infty }(U_T)$, and $u(t,\cdot )\in C^{\infty }(\overline{U})$ for every $t>0$. The trace remains zero, and $u_t+Lu=0$ holds pointwise.

Finally Parseval identity givesby the dominated convergence theorem, since each factor tends to zero and is bounded by one. No boundary compatibility of the initial $L^2$ data is required for positive-time smoothness.