Nonlinear elliptic boundary value problem

A nonlinear elliptic boundary value problem contains a nonlinear dependence on the unknown function or its derivatives while retaining an elliptic principal part. Small-data existence can often be proved by applying a fixed-point theorem to the inverse of the linear elliptic part.

For $Qu=\Delta u-V(u)$, a subsolution and supersolution satisfy $Q{\phi }^{-}\ge 0$ and $Q{\phi }^{+}\le 0$, respectively. An ordered pair additionally satisfies ${\phi }^{-}\le {\phi }^{+}$ throughout the domain, with boundary data $\psi$ between their boundary traces. This interior condition is separate from boundary ordering when $V$ is non-increasing: for $V(s)=-{\lambda }_{1}s$, a positive first Dirichlet Laplacian eigenfunction $e_1$ and $-e_1$ both solve the homogeneous equation and vanish on the boundary, but taking ${\phi }^{-}=e_1$ and ${\phi }^{+}=-e_1$ produces reversed interior barriers.

For smooth non-increasing $V$ and an ordered subsolution and supersolution, begin with $u_0={\phi }^{-}$ and solve successive Poisson equations with forcing $V(u_k)$ and the fixed boundary data. The weak maximum principle for elliptic operators applied to the negatives of consecutive differences givesThe global Schauder estimate and the Hölder interpolation inequality bound the iterates uniformly in $C^{2,\alpha }$. Their bounded monotone sequence converges, and compact embedding of Hölder spaces permits passing to the equation. The uniform Hölder seminorm bounds retain $C^{2,\alpha }$ regularity of the limit.

Suppose the inverse $S:C^{0,\alpha }\to C_0^{2,\alpha }$ of a linear elliptic boundary value problem is bounded by $K$, and a nonlinear map satisfiesThen $T(v)=S(Q(v)+f)$ maps a sufficiently small closed ball into itself and is a contraction mapping when $\delta$ and $\Vert f{\Vert }_{C^{0,\alpha }}$ are sufficiently small. The Banach fixed-point theorem gives a solution of $Lu=Q(u)+f$ with zero Dirichlet boundary condition, unique within that ball. Invertibility follows from the Fredholm alternative for an elliptic Dirichlet problem when the homogeneous null space is zero.

In three dimensions the map $N(u)=|Du{|}^{2}u$ sends $H^2$ into $L^2$, because $H^2\hookrightarrow W^{1,4}\cap L^{\infty }$. On an $H^2$ ball of radius $R$ it is locally Lipschitz with constant $O(R^2)$.