Unexpected radial limit

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Entekhabi, Mozhgan. 2017. Unexped radial limit--In Proceedings: 13th Annual Symposium on Graduate Research and Scholarly Projects. Wichita, KS: Wichita State University, p.34

##### Abstract

Consider a bounded solution ƒ of the prescribed mean curvature equation over a bounded domain Ω ⊂ R² which has a corner at which has a corner at (0, 0) of size 2α and assume the mean curvature of the graph of ƒ is bounded. If the corner is non convex/reentrant (i.e α ∈ ( π/2, π ?)), then the radial limits Rƒ(θ) lim= ƒ(r cosθ, r sinθ) exist for all interior directions (e.g. θ ∈ (-α,α) if θ = ± α are tangent rays to δΩ at (0, 0)), no matter how wild is the trace of ƒ on δΩ. If the corner is convex(i.e. α ∈ (0,π)) and some extra conditions are satisfied then the radial limits at (0, 0) from interior directions continue to exist. This generalizes, for example, known results about radial limits of capillary surfaces.

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Research completed in the Department of Mathematics, Statics and Physics