Kirk E. Lancaster

Boundary behavior of a nonparametric surface of prescribed mean curvature near a reentrant corner

Let Q be an open set in R2 which is locally convex at each point of its boundary except one, say (0,0). Under certain mild assumptions, the solution of a prescribed mean curvature equation on Q behaves as follows: All radial limits of the solution from directions in Q exist at (0,0), these limits are not identical, and the limits from a certain half-space (H) are identical. In particular, the restriction of the solution to Q n H is the solution of an appropriate Dirichlet problem. 0. Introduction. We consider here the behavior of a generalized solution of the equation for surfaces of prescribed mean curvature at an inner corner of the boundary where the solution is discontinuous. This work is a generalization of the previous work of the second author [8], which dealt with the minimal surface equation. It was shown there that all radial limits exist and that they are constant in directions coming from a half-space. Here we find that the same result holds for (nonparametric) surfaces of prescribed mean curvature. 1. Preliminaries. Let Q be a bounded, open, connected, simply connected subset of R2 with N = (0, 0) c OQ such that Q is locally convex at each point of OQ \ {N}. Let H(x, y, t) be a continuous function on Q x R and q

NONPARAMETRIC MINIMAL SURFACES IN R WHOSE BOUNDARIES HAVE A JUMP DISCONTINUITY

C Co(O9Q). We will make a number of assumptions which will hold throughout this work. ASSUMPTIONS. (A) The equation (1) (p/W)x + (q/W)y = 2H(x, y, z(x, y)) has a solution z = f C C2(Q), where p = zz, q = zy, and W2 = 1 + p2 + q2. (B) f C?(l \ {N}) and f = q on OQ \ {N}

Radial limits of bounded nonparametric prescribed mean curvature surfaces

Let be a domain in R 2 which is locally convex at each point of its boundary except possibly one, say (0,0), be continuous on /{(0,0)} with a jump discontinuity at (0,0) and f be the unique variational solution of the minimal surface equation with boundary values . Then the radial limits of f at (0,0) from all directions in exist. If the radial limits all lie between the lower and upper limits of at (0,0), then the radial limits of f are weakly monotonic; if not, they are weakly increasing and then decreasing (or the reverse). Additionally, their behavior near the extreme directions is examined and a conjecture of the authors is proven.

Bernstein functions and the dirichlet problem

Consider a solution f ∈ C(Ω) of a prescribed mean curvature equation

Theconvergence Rate of Solutions of Quasilinear Elliptic Equations in Slabs

For a nonconvex, symmetric quadrilateral, the nonparametric minimal surface arising from an associated Dirichlet problem can be described in terms of the Weierstrass representation and the stereographic projection of its Gauss map. The Bernstein function—which arises by truncation of the re-entrant corner by a concave arc and by requiring the normal vector to be horizontal there—has the same Gauss map image. This leads to a Riemann–Hilbert problem that can be solved and leads to the existence of this surface.

Numerical Harmonic Analysis on the Hyperbolic Plane

Solutions of Dirichlet problems for quasilinear elliptic equations in unbounded domains inside a slab are considered. The rate at which solutions converge to their limiting functions at infinity is established in terms of properties of the top order coefficients of the operator and the rate at which the boundary values converge to their limiting functions.Our proofs are based on constructing appropriate barrier functions which depend on the behavior of coefficients of the operator and the rate of convergence of boundary value

Boundary behavior near reentrant corners for solutions of certain elliptic equations

Results are reported of a numerical implementation of the hypcrbolic Fourier transform and the geodesic and horocyclic Radon transforms on the hyperbolic plane, and of their inverses. The study is motivated by the hyperbolic geometry approach to the linearized inverse conductivity problem, suggested by C. A. Berenstein and E. Casadio Tarabusi.

A MAXIMUM PRINCIPLE FOR SOLUTIONS OF A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS ON UNBOUNDED DOMAINS

The behavior near a reentrant corner of quasilinear non-uniformly elliptic partial differential equations which satisfy certain conditions is examined. For an elliptic equation satisfying a general maximum principle and having «convex barriers», we show that if a solution has radial limits at the corner, these limits have the same qualitative behavior as those for nonparametric minimal surfaces. For certain equations of mean curvature type, radial limits at the reentrant corner are shown to exist.

On the relationship of continuity and boundary regularity in prescribed mean curvature Dirichlet problems

ABSTRACT For solutions on unbounded domains of boundary value problems for a class of quasilinear elliptic equations which are not uniformly elliptic, we prove that the solutions have the same bounds as those of the boundary data.

Phragmén–Lindelöf theorems in cylinders

In 1976, Leon Simon showed that if a compact subset of the boundary of a domain is smooth and has negative mean curvature, then the nonparametric least area problem with Lipschitz continuous Dirichlet boundary data has a generalized solution which is continuous on the union of the domain and this compact subset of the boundary, even if the generalized solution does not take on the prescribed boundary data. Simon’s result has been extended to boundary value problems for prescribed mean curvature equations by other authors. In this note, we construct Dirichlet problems in domains with corners and demonstrate that the variational solutions of these Dirichlet problems are discontinuous at the corner, showing that Simon’s assumption of regularity of the boundary of the domain is essential.