On the role of energy convexity in the web function approximation

Abstract

For a given p > 1 and an open bounded convex set Ω ⊂ℝ2,we consider the minimization problem for the functional Jp (u) = ∫Ω(1/p|∇ μ|p - u)over W01,pp Ω.Since the energy of the unique minimizer u p may not be computed explicitly, we restrict the minimization problem to the subspace of web functions, which depend only on the distance from the boundary δΩ. In this case, a representation formula for the unique minimizer v p is available. Hence the problem of estimating the error one makes when approximating J p (u p ) by J p (v p ) arises. When Ω varies among convex bounded sets in the plane, we find an optimal estimate for such error, and we show that it is decreasing and infinitesimal with p. As p → ∞, we also prove that u p-v p converges to zero in W01,m(Ω) for all m<∞. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in J p increases. © Birkhäuser Verlag, Basel 2005

Overdetermined Boundary Value Problems for the -Laplacian

“…Notice that the radial solution on a ball is a web-function in the sense of [3], i.e. a function, whose value depends only on the distance to ∂Ω.…”

A Symmetry Problem for the Infinity Laplacian: Fig. 1.

“…Following [26], by web functions in the sequel we mean continuous functions depending only on d ∂Ω (the name comes from the fact that, in case of planar polygons, level lines of the distance functions recall the pattern of a spider web). To the best of our knowledge, these functions firstly appeared in the monograph by Pólya and Szegö [32, Section 1.29]; more recently, they have found application in different variational problems, see [12,13,15,16,17,18]. Clearly, asking that the solution of problem (3) is a web function is a more severe restriction than imposing just the constancy of its normal derivative along the boundary as in (1).…”

A new symmetry criterion based on the distance function and applications to PDEʼs

“…Often, symmetry questions arise in problems in which a crucial role is played by the distance function from the boundary of an open bounded domain Ω ⊂ R n , d Ω (x) := dist(x, ∂Ω). This happens for instance when studying PDE's related with mass transportation theory (see [7,13,14]), or minimization problems in the class of so-called web functions, namely functions which only depend on d Ω (see [19,21,22,23,24,25,37]). Symmetry questions in these frameworks, which will be described more precisely below, pushed us to set up a new roundedness criterion, which brings into play the distance function in a more intrinsic way than merely through the boundary curvatures.…”

“…Let us mention that the functions ϕ and λ already appeared in the literature in different contexts: concerning the function ϕ, it is a crucial tool in the proof of the isoperimetric inequality à la Gromov (see e.g. [3, §1.6.8]), and appears also in mathematical models for granular materials [14,16,26]; concerning the function λ, its regularity has been studied in [15,40,42], while some of its applications to variational problems can be found in [18,21,22,23].…”