A Constructive Existence Theorem for a Nonlinear Elliptic Equation
Alan R. Elcrat · SIAM Journal on Mathematical Analysis · 1971
A constructive existence theorem for the homogeneous Dirichlet problem in ($\mathcal{R}$ two or three dimensions) for the equation $P(u) = \Delta u + \alpha _{ij} u_{xi} u_{xj} + f = 0$ is obtained by placing P in the role of an operator mapping $W_{2,0}^2 (\mathcal{R})$ into $L_2 (\mathcal{R})$ and proving convergence of a Newton sequence for P. The theorem is “local” in the sense that the $L_2 $-norm of f must satisfy a bound which becomes infinite as the diameter of $\mathcal{R}$ shrinks to zero. An essential feature of the proof is an application of Sobolev’s lemma to show that $P(u)$ is an element of $L_2 (\mathcal{R})$ when $u \in W_{2,0}^2 (\mathcal{R})$, and that P satisfies the hypotheses of a theorem of Kantorovich.