2010/05/21 by John Andersson, Henrik Shahgholian, Andersson, John +3 · 1 citation
Computer Science · Mathematics · #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1005.3882
openalex publication_date 2010/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we are concerned with singular points of solutions to the \it unstable free boundary problem Δu = - χ_\ugt;0\ \hboxin B1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities - that is points at which the second derivatives of the solution are unbounded - as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the nonlinearity χ_\u>0\ . The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in \mathbb R3. A surprising fact in \mathbb R3 is that although \fracu(r\x)supB1|u(r\x)| can converge at singularities to each of the harmonic polynomials xy, x2+y2\over 2-z2 \textrmand z2-x2+y2\over 2, it may \em not converge to any of the non-axially-symmetric harmonic polynomials α((1+ δ)x2 +(1- δ)y2 - 2z2) with δ≠ 1/2. We also prove the existence of stable singularities in \mathbb R3.