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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.

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.

The zero-boundary problem $\Delta u+cu=f$ is uniquely solvable if $c$ has a sufficiently small positive part depending only on the bounded regular domain. Write $c=c_{-}+c_{+}$ with $c_{-}\le 0$. The supremum norm barrier for an elliptic Dirichlet problem gives $\Vert u{\Vert }_{\infty }\le C_{\Omega }\Vert c_{+}u{\Vert }_{\infty }$ for a homogeneous solution. If $C_{\Omega }\Vert c_{+}{\Vert }_{\infty }<1$, its kernel is zero and the Fredholm alternative for an elliptic Dirichlet problem gives existence for every forcing and boundary datum.