Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 107 / 6

Multiply the Dirichlet Laplacian eigenfunction equation by $u$ and use integration by parts. The zero Dirichlet boundary condition removes the boundary term and givesThe denominator is positive because the eigenfunction is nonzero. The numerator is also positive: if it were zero, the given Sobolev inequality for $u\in W_0^{1,2}(\Omega )$ would force $u=0$. Therefore the Dirichlet Laplacian eigenvalue has the positive Rayleigh quotientEvery nonzero Dirichlet eigenfunction in this problem has a positive eigenvalue.

Use a signed power test for a Laplacian eigenfunction. For $\gamma \ge 2$, test the Dirichlet Laplacian eigenfunction equation with $|u{|}^{\gamma -2}u$. Its derivative is $(\gamma -1)|u{|}^{\gamma -2}Du$; at $\gamma =2$ the test function is just $u$. Integration by parts gives the exact energy estimateSet $v=|u{|}^{\gamma /2}$. It belongs to the zero-boundary Sobolev space and obeys the Sobolev chain rule:For $\gamma >2$, the power map is $C^1$ with bounded derivative on the bounded range of $u$. For $\gamma =2$, the absolute value map is Lipschitz, and $D|u|=\mathrm{sgn}(u)Du$ almost everywhere; on the zero set, use the fact that a gradient of a Sobolev function vanishes on a level set. These facts justify the identity even when $u$ changes sign.

Apply the given Sobolev inequality with the Sobolev conjugate exponent $2\kappa =2n/(n-2)$:Testing with a signed power raises the integrability exponent from $\gamma$ to $\gamma \kappa$. Absorbing the factor $1/4$ into $C(n)$ gives precisely the requested inequality.

Put $p_j=2{\kappa }^j$, where $\kappa =n/(n-2)>1$. Since $p_j^2/(p_j-1)\le 2p_j$, part (ii), followed by a $p_j$-th root, gives the Moser iteration stepAfter $N$ steps,The geometric series and its differentiated form giveIn particular the exact convergent product isThe printed hint's equality to a single power of $2\kappa$ is not correct in general; its claimed finiteness is correct and the displayed expression supplies the correction.

On a finite-measure domain, Lp norms converge to the supremum norm for a bounded continuous function. Indeed, the upper bound is $\Vert u{\Vert }_p\le |\Omega {|}^{1/p}\Vert u{\Vert }_{\infty }$; for every $a<\Vert u{\Vert }_{\infty }$, the set $\{|u|>a\}$ has positive measure, giving $\Vert u{\Vert }_p\ge a|\{|u|>a\}{|}^{1/p}$. Let $N\to \infty$ in the iteration and use part (i), which gives $\lambda >0$. The Dirichlet eigenfunction supremum estimate isThe eigenvalue exponent is exactly $n/4$. The constant depends only on $n$: the supplied zero-boundary Sobolev inequality has a dimension-only constant, and every product factor above depends only on $\kappa$.