Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 2

The zero-boundary Sobolev space isOn it defineIf this quadratic form vanishes, then $\nabla u=0$. The Poincare inequality givesso $u=0$. It is therefore an inner product, and its norm is equivalent to the usual $H^1$ norm.

A function $u\in H_0^1(\Omega )$ is a weak solution whenThis follows from integration by parts and incorporates the homogeneous Dirichlet boundary condition through membership in $H_0^1$.

The functionalis bounded on $H_0^1(\Omega )$ by the Cauchy-Schwarz inequality and the Poincare inequality:The Riesz representation theorem therefore supplies a unique $u\in H_0^1(\Omega )$ satisfyingfor every test function. This is exactly the weak identity from part (a). Uniqueness also follows by testing the homogeneous difference with itself.

After integration by parts, the weak formulation isThe left side is the inner productwhich is positive definite and induces the usual $H^1$ norm. The right side is bounded in this norm. The Riesz representation theorem, equivalently the Lax-Milgram theorem, gives a unique weak solution. This is the weak Dirichlet problem for the massive Laplacian with mass one and the signs multiplied by $-1$.

Combined interior and boundary elliptic regularity for the Dirichlet Laplacian on a smooth bounded domain states that, for every integer $k\ge 0$,when $\Delta u=f$ and $u$ has zero boundary trace. More generally one first has an additional $\Vert u{\Vert }_{L^2}$ term, which uniqueness and the Poincare inequality remove here.

For the shifted equation, write $\Delta u=f+u$. The weak estimate gives $u\in H^1$. Applying the displayed estimate first with an $L^2$ right side gives $u\in H^2$. Repeating,until $u\in H^{k+2}$. The lower-order term is controlled at each stage, yieldingThis is boundary elliptic regularity for the shifted Dirichlet Laplacian.

There is a sign issue in the printed problem. Part (c) constructs the inverse of $\Delta -1$, whereas the second equation printed in part (e) contains $\Delta +1$. The latter operator is invertible with homogeneous Dirichlet data only when $1$ is not a Dirichlet Laplacian eigenvalue. Thus the assertion as printed needs this nonresonance hypothesis; with a minus sign it follows directly from parts (c) and (d).

Under either the intended minus sign or the stated nonresonance condition, let $G_0$ and $G_1$ be the bounded Dirichlet solution operators for the two linear equations. Choose $s>n/2$ and work with $u,v\in H^{s+2}(\Omega )\cap H_0^1(\Omega )$. Since $H^s(\Omega )$ is a Sobolev algebra,DefineElliptic regularity gives, on a ball of radius $R$,and the difference estimate has Lipschitz constant at most $CR$. Choose $R$ small and then ${\epsilon }_0$ so that $C(R^2+{\epsilon }_0\Vert f{\Vert }_{H^s})\le R$. The contraction mapping theorem gives a solution for $0\le \epsilon <{\epsilon }_0$. Repeated elliptic regularity and smoothness of $f$ bootstrap the solution to $C^{\infty }$.