A Singular Initial Value Problem to Construct Density-Equalizing Maps

Abstract

Abstract. The diffusion-based algorithm to produce density-equalizing maps interprets diffusion as an advection process. This algorithm uses the dynamics of a flow that is defined by an initial value problem that turns out to be very singular at the initial time. The singularities appear when the initial density has line or angle discontinuities, which is always the case, for example, in area cartogram maps. This singular initial value problem is analyzed mathematically in this paper and the conclusion is that despite these singularities, it has a unique solution. This justifies the extensive numerical use of this algorithm in the recent years. The techniques presented in this paper use both partial and ordinary differential equations estimates.

Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L data and applications to nonlinear elasticity

“…• Given f ∈ L p (Ω) for some p > 1, we show in Theorem 1 that for every 1 < q < (p + 1)/2 there exists a homeomorphism φ : Ω → Ω which solves the prescribed Jacobian inequality (2) and for which both φ and its inverse φ −1 belong to the Sobolev space W 1,q (Ω; Ω). • For bounded right-hand sides f ∈ L ∞ (Ω), the prescribed Jacobian inequality (2) admits a bi-Lipschitz solution φ : Ω → Ω, see Theorem 3.…”

“…al. [2], and Carlier and Dacorogna [6]. Note also that Ye [17] proved that for right-hand sides f satisfying the above necessary condition and belonging to W m,p (Ω), where m ∈ N and 1 < p < ∞ are such that W m,p (Ω) is an algebra, there exists a solution to (1) in W m+1,p (Ω; Ω).…”

“…In the case´Ω f dx > |Ω|, the change of variables formula excludes the existence of (weak) solutions; in the equality case´Ω f dx = |Ω|, by the change of variables formula the prescribed Jacobian inequality (2) reduces to the more "rigid" case of the prescribed Jacobian equation (1a).…”

A Least-Squares Method for the Solution of the Non-smooth Prescribed Jacobian Equation

“…A special interest is paid to problems for which the data f is non-smooth. Several works in the literature have focused on this prescribed Jacobian equation, starting with the seminal article [16], which has been developed and extended in, e.g., [17,[19][20][21][22][23][24][25]. Existence of classical solutions rely on solutions in Hölder spaces [16].…”