Radial Limits of Nonparametric PMC Surfaces with Intermediate Boundary Curvature

Mozhgan Nora Entekhabi, Kirk Eugene Lancaster
2020 Taiwanese journal of mathematics  
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
more » ... 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