Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 107 / 3

The global Schauder estimate on a bounded $C^{2,\alpha }$ domain isHere $C$ depends on $n,\alpha$, the domain's boundary regularity and $\Vert c{\Vert }_{C^{0,\alpha }}$, but not on the particular solution or data. The $C^0$ term is needed before uniqueness has been established. The given $\psi$ is an extension of the boundary data to the closure.

For $c\le 0$, choose $d$ with $\Omega \subset \{|x_1|<d\}$ and set $M={\sup }_{\partial \Omega }|\psi |$, $F={\sup }_{\Omega }|f|$. The hinted functionis nonnegative, dominates $|\psi |$ on the boundary, and satisfiessince $x_1+d>0$. Therefore $(\Delta +c)(u-b)=f-(\Delta +c)b\ge 0$ and $(\Delta +c)(-u-b)=-f-(\Delta +c)b\ge 0$. Both functions are nonpositive on the boundary. The weak maximum principle for elliptic operators gives $|u|\le b$, hence the supremum norm barrier for an elliptic Dirichlet problem:Unlike the full Schauder estimate, this last constant depends only on a slab containing $\Omega$, and is independent of how negative $c$ is.

Let $L=\Delta +c$ with real $c$, and letThe Fredholm alternative for an elliptic Dirichlet problem says that $N$ is finite-dimensional. If $N=0$, all the prescribed data admit a unique solution. In general the exact boundary compatibility in the self-adjoint Fredholm alternative isThese conditions are necessary and sufficient; when they hold, all solutions form $u_0+N$. Thus in the second branch some data are incompatible, while compatible data have nonunique solutions. The outward normal derivative fixes the displayed sign.

To prove the alternative, put $X=\{w\in C^{2,\alpha }(\overline{\Omega }):w{|}_{\partial \Omega }=0\}$ and $Y=C^{0,\alpha }(\overline{\Omega })$. The standard zero-boundary Poisson equation theorem makes $\Delta :X\to Y$ an isomorphism. Multiplication by $c$, followed by ${\Delta }^{-1}$, is a compact operator $K:X\to X$: the embedding $C^{2,\alpha }\hookrightarrow C^{0,\alpha }$ is compact, and multiplication is bounded. Writing $u=\psi +w$ givesThe Fredholm alternative for a compact operator implies finite kernel and cokernel of equal dimension, and invertibility exactly when the kernel vanishes.

For the explicit compatibility conditions, use the self-adjoint Dirichlet realization of $L$ on $L^2$. It has compact resolvent: a sufficiently large positive shift of $-L$ is coercive, its inverse gains two derivatives, and the Rellich-Kondrachov compactness theorem makes that inverse compact. The Fredholm solvability condition for a self-adjoint operator is $F\perp N$. Standard boundary elliptic regularity upgrades the resulting weak solution to $C^{2,\alpha }$ for these data and boundary. Thus it gives the same kernel and solvability as the preceding Hölder-space formulation.

Finally Green second identity, with $z=0$ on the boundary and $Lz=0$, givesConsequently $\int zF=\int zf+\int \psi {\partial }_{\nu }z$, proving both necessity and sufficiency with the stated sign. If using complex data and real $c$, replace $z$ by $\overline{z}$ in the pairings.

Use the supremum norm barrier for an elliptic Dirichlet problem to obtain a domain-only constant $C_{\Omega }=e^{2d}$. Split the coefficient as $c=c_{-}+c_{+}$, where $c_{-}\le 0$ and $c_{+}=\max (c,0)$. Both parts are Hölder continuous functions. ChooseIf $u$ solves the homogeneous zero-boundary problem for $\Delta +c$, thenPart (a), applied to this nonpositive zeroth-order coefficient, givesHence $u=0$. The Fredholm alternative for an elliptic Dirichlet problem now gives existence and uniqueness for every forcing and boundary datum in the stated Hölder spaces. This proves the small positive zeroth-order perturbation of a Dirichlet problem without imposing a bound on the negative part of $c$. The value of $ϵ$ need not be optimal.