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Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

Mozhgan Nora Entekhabi, Kirk Eugene Lancaster

2020
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Taiwanese journal of mathematics
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The influence of the geometry of the domain on the behavior of generalized solutions of Dirichlet problems for elliptic partial differential equations has been an important subject for over a century. We investigate the boundary behavior of variational solutions f of Dirichlet problems for prescribed mean curvature equations in a domain Ω ⊂ R 2 near a point O ∈ ∂Ω under different assumptions about the curvature of ∂Ω on each side of O. We prove that the radial limits at O of f exist under
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... ent assumptions about the Dirichlet boundary data φ, depending on the curvature properties of ∂Ω near O. 599 600 Mozhgan Nora Entekhabi and Kirk Eugene Lancaster solution of (1.1)-(1.2) exists; when H ≡ 0, much of the history (up to 1985) of this topic can be found in Nitsche's book [24] (e.g., §285, 403-418) and, for general H, one might consult [26]. (Appropriate "smallness of φ" conditions can imply the existence of classical solutions when Ω is not convex in the H ≡ 0 case (e.g., [24, §285 & §412] and [17, 25, 27, 28] ) or when ∂Ω does not satisfy appropriate curvature conditions in the general case (e.g., [1, 16, 22] ); however see [24, §411]. In [2], Bourni assumes ∂Ω and φ are (C 1,α ) smooth, ignores the geometry of Ω and characterizes the "graph" of a variational solution which may include portions of the boundary cylinder ∂Ω × R; in comparison, we do not assume any regularity for our boundary data φ and focus on the closure in Ω × R of the graph of f over Ω.) We wish to investigate the effects of the geometry of Ω on the behavior of a variational solution f of (1.1)-(1.2) near a point O ∈ ∂Ω; ∂Ω might be smooth or have a corner at O or φ might be discontinuous at O. For convenience, we assume O = (0, 0). In many cases, the approximate solution is unique since if f, g ∈ C 2 (Ω) both satisfy (1.1) and f = g almost everywhere on ∂Ω, then f = g in Ω (e.g., [12, Theorem 5.1]); see, for example, [7, 645-6]) for a discussion of when Perron and variational solutions exist. Let α and β, α < β < α + 2π, be the angles which the tangent rays to ∂Ω at O make with the positive x-axis such that

doi:10.11650/tjm/201101
fatcat:od2mhz6x7nc63jauep575kpk6i