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Kirk E. Lancaster

Publications and source records attributed to Kirk E. Lancaster.

6 recordsLinked to original sources
arXiv2019

Boundary Continuity of Nonparametric Prescribed Mean Curvature Surfaces

We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of local barrier functions). We prove that if the Dirichlet boundary data $ϕ$ is continuous at such a point (and possibly nowhere else), then the solution of the variational problem is continuous at this point.

math.AP↗
arXiv2018

Bernstein Functions and Radial Limits of Prescribed Mean Curvature Surfaces

The radial limits at a point ${\bf y}$ of the boundary of the domain $Ω\subset {\bf R}^{2}$ of a bounded variational solution $f$ of Dirichlet or contact angle boundary value problems for a prescribed mean curvature equation are studied with an emphasis on the effects of assumptions about the curvatures of the boundary $\partialΩ$ on each side of the point ${\bf y}.$ For example, at a nonconvex corner ${\bf y},$ we previously proved that all nontangential radial limits of $f$ at ${\bf y}$ exist, here we provide sufficient conditions for the tangential radial limits to exist, even when the Dirichlet data $ϕ\in L^{\infty}(\partialΩ)$ has no one-sided limits at ${\bf y}$ or the contact angle $γ\in L^{\infty}(\partialΩ:[0,π])$ is not bounded away from $0$ or $π.$ We also provide a complement to a 1976 Theorem by Leon Simon on least area surfaces.

math.AP↗
arXiv2017

A Generalization of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner"

The principle existence theorem (i.e. Theorem 1) of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner" (Pacific J. Math. Vol. 176, No. 1 (1996), 165-194) is extended to the case of a contact angle $γ$ which is not bounded away from $0$ and $π$ (and depends on position in a bounded domain $Ω\in {\bf R}^{2}$ with a convex corner at ${\cal O}=(0,0)$). The lower bound on the size of "side fans" (i.e. Theorem 2 in the above paper) is extended to case of such contact angles for convex and nonconvex corners.

math.AP↗
arXiv2016

On Cusp Solutions to a Prescribed Mean Curvature Equation

The nonexistence of "cusp solutions" of prescribed mean curvature boundary value problems in $Ω\times{\bf R}$ when $Ω$ is a domain in ${\bf R}^{2}$ is proven in certain cases and an application to radial limits at a corner is mentioned.

math.AP↗
arXiv2015

Radial Limits of Bounded Nonparametric PMC Surfaces

Consider a solution $f\in C^{2}(Ω)$ of a prescribed mean curvature equation \[ {\rm div}\left(\frac{\nabla f}{\sqrt{1+|\nabla f|^{2}}}\right)=2H(x,f) \ \ \ \ {\rm in} \ \ Ω, \] where $Ω\subset \Real^{2}$ is a domain whose boundary has a corner at ${\cal O}=(0,0)\in\partialΩ.$ If $\sup_{x\inΩ} |f(x)|$ and $\sup_{x\inΩ} |H(x,f(x))|$ are both finite and $Ω$ has a reentrant corner at ${\cal O},$ then the radial limits of $f$ at ${\cal O},$ \[ Rf(θ) \myeq \lim_{r\downarrow 0} f(r\cos(θ),r\sin(θ)), \] are shown to exist and to have a specific type of behavior, independent of the boundary behavior of $f$ on $\partialΩ.$ If $\sup_{x\inΩ} |f(x)|$ and $\sup_{x\inΩ} |H(x,f(x))|$ are both finite and the trace of $f$ on one side has a limit at ${\cal O},$ then the radial limits of $f$ at ${\cal O}$ exist and have a specific type of behavior.

math.AP↗