On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Magnanini, Rolando; Talenti, Giorgio

doi:10.1007/s00032-009-0103-x

Cited by 7 publications

(3 citation statements)

References 165 publications

(175 reference statements)

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“…In Theorem 3.6, we will also extend this result to a case in which the function p → f (|p|) is no longer differentiable at p = 0. Our interest on this kind of functionals (not considered in [Cr1]) is motivated by their relevance in the study of complex-valued solutions of the eikonal equation (see [MT1]- [MT4] and [CeM]).…”

Section: Introductionmentioning

confidence: 99%

Symmetry of minimizers with a level surface parallel to the boundary

Ciraolo¹,

Magnanini²,

Sakaguchi³

2015

J. Eur. Math. Soc.

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We consider the functionalwhere Ω is a bounded domain and f is a convex function. Under general assumptions on f , Crasta [Cr1] has shown that if IΩ admits a minimizer in W 1,1 0 (Ω) depending only on the distance from the boundary of Ω, then Ω must be a ball. With some restrictions on f , we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance).We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.

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Section: Introductionmentioning

confidence: 99%

Symmetry of minimizers with a level surface parallel to the boundary

Ciraolo¹,

Magnanini²,

Sakaguchi³

2015

J. Eur. Math. Soc.

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“…Evanescent wave tracking (EWT) is a theory of complex-valued eikonals that was proposed in [37] and was investigated from a mathematical point of view in [35] and [55,56,57,58,59]. Its basic ingredients are outlined below.…”

Section: Beyond Geometrical Opticsmentioning

confidence: 99%

Thoughts on the Busemann equation

Talenti

2011

Milan J. Math.

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“…Functionals of this kind occur in the study of complex-valued solutions of the eikonal equation (see [15]- [18]), as well as in the study of problems linked to traffic congestion (see [2]) and in variational problems which are relaxations of non-convex ones (see [5]). We have in mind the following two main examples of a function f :…”

Section: Introductionmentioning

confidence: 99%

A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals

Ciraolo

2013

Geometric Properties for Parabolic and Elliptic PDE's

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The study of geometric properties of partial differential equations has always attracted the interest of researchers and is now a broad and well-established research area, with contributions that often come from experts from disparate areas of mathematics, such as differential and convex geometry, functional analysis, calculus of variations, mathematical physics, to name a few.The interplay between partial differential equations and geometry has two main aspects: on the one hand, the former is classically a powerful tool for the investigation of important problems coming from differential geometry and, on the other hand, the latter gives useful and often decisive insights in the study of PDE's. Now that basic questions about PDE's, such as existence, uniqueness, stability and regularity of solutions for initial/boundary value problems, have been fairly understood, research on topological and/or geometric properties of their solutions have become more vigorous.Research on geometric aspects for parabolic and elliptic PDE's provides a vast variety of possibilities. Issues currently and actively studied comprehend among others: positivity of solutions; critical points: their structure, possible occurrence and evolution; spike-shaped solutions; symmetry and non-symmetry for ground states and overdetermined boundary value problems; stability of symmetric configurations; convexity, quasi-convexity or starshape of level sets; estimates on geometrically or physically relevant quantities such as surface area and curvature of level surfaces or torsional creep, eigenvalues and eigenfunctions; impact of curvature of the domain on the relevant solutions and their possible behavior for large or short times, and so on. Similarly wide is the assortment of mathematical tools and techniques, analytic and geometric, employed to analyze such issues: functional inequalities such as isoperimetric, Hardy or Brunn-Minkowski inequalities; Pohozaev-type identities; maximum principles; Harnack inequalities; asymptotics for solutions; moving-planes or sliding methods; Bernstein and Liouville-type theorems, viscosity-solutions techniques, et cetera.This volume aims to promote scientific collaboration in this very active area of research, by presenting recent results and informative surveys and by exploring new trends and techniques. It contains original papers and a few survey articles.

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On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Cited by 7 publications

References 165 publications

Symmetry of minimizers with a level surface parallel to the boundary

Symmetry of minimizers with a level surface parallel to the boundary

Thoughts on the Busemann equation

A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals

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