Singular Minimal Surfaces

Stable and singular solutions of the equation $Delta u = 1/u$

“…Recall that stable solutions satisfy the stability inequality (6). Proof: Let u 0 be a maximal solution with data ϕ 0 , and let ϕ t = ϕ 0 + t for t > 0.…”

“…Notice that the restriction on p allows for p ≥ n as long as n ≤ 6. For u > 0 smooth, we use the stability inequality (6) with the test function ζu −q . Then for ǫ > 0,…”

“…A degree theoretic program for obtaining such sequences of solutions is outlined in Simon's paper. For more on equation (1), see the survey paper by Dierkes [6]. Our goal is to apply a similar program to the equation ∆u = 1 u .…”

Three dimensional mathematical modeling of violin plate surfaces: An approach based on an ensemble of contour lines

“…The one dimensional catenary curve is well known, and the problem describing it lies at the origins of the calculus of variations [ 10 ]. A catenary surface however is a difficult mathematical beast [ 11 ]. The principles related to its specification were known to Lagrange [ 12 ], and the relevant nonlinear second order partial differential equation characterizing a catenary surface was known to Poisson [ 13 ].…”

“…The principles related to its specification were known to Lagrange [ 12 ], and the relevant nonlinear second order partial differential equation characterizing a catenary surface was known to Poisson [ 13 ]. It is where , , , g is the gravitational constant, and ϵ is the density of the surface [ 11 ]. At the boundary, u = 0.…”