Existence of a unique solution to an elliptic partial differential equation
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The purpose of this article is to prove the existence of a unique classical solution to the quasilinear elliptic equation -∇ ∙ (α(u)∇u) = ƒ for x ∈ Ω, which satisfies the condition that u(x0) = u0 at a given point x0 ∈ Ω, under the boundary condition n(x) ∙ ∇u(x) = 0 for x ∈ ∂Ω where n(x) is the outward unit normal vector and where 1 / |Ω| ∫Ω ƒ dx = 0. The domain Ω ⊂ ℝN is a bounded, connected, open set with a smooth boundary, and N = 2 or N = 3. The key to the proof lies in obtaining a priori estimates for the solution.
CitationDenny, D. L. (2019). Existence of a unique solution to an elliptic partial differential equation. Electronic Journal of Differential Equations, 2019(110), pp. 1-13.
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