Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

Research output: Chapter in Book/Conference paperConference paper

Abstract

We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation aijuij = up, u ≥ 0, p ∈ [0, 1), with bounded discontinuous coefficients aij having small BMO norm. We consider the simplest discontinuity of the form x x|x| 2 at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when p = 0) cannot be smooth at the points of discontinuity of aij(x). To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

Original languageEnglish
Title of host publicationDiscrete and Continuous Dynamical Systems - Series A
Pages6073-6090
Number of pages18
Volume38
Edition12
DOIs
Publication statusPublished - 1 Dec 2018

Publication series

NameDiscrete and Continuous Dynamical Systems- Series A
PublisherSouthwest Missouri State University
ISSN (Print)1078-0947

Fingerprint

Discontinuous Coefficients
Invariance
Free Boundary
Irregular
Discontinuity
Scale Invariance
Second Order Elliptic Equations
Obstacle Problem
Homogeneity
Estimate
Blow-up
Norm
Form

Cite this

Dipierro, S., Karakhanyan, A., & Valdinoci, E. (2018). Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. In Discrete and Continuous Dynamical Systems - Series A (12 ed., Vol. 38, pp. 6073-6090). (Discrete and Continuous Dynamical Systems- Series A). https://doi.org/10.3934/dcds.2018262
Dipierro, Serena ; Karakhanyan, Aram ; Valdinoci, Enrico. / Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - Series A. Vol. 38 12. ed. 2018. pp. 6073-6090 (Discrete and Continuous Dynamical Systems- Series A).
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Dipierro, S, Karakhanyan, A & Valdinoci, E 2018, Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. in Discrete and Continuous Dynamical Systems - Series A. 12 edn, vol. 38, Discrete and Continuous Dynamical Systems- Series A, pp. 6073-6090. https://doi.org/10.3934/dcds.2018262

Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. / Dipierro, Serena; Karakhanyan, Aram; Valdinoci, Enrico.

Discrete and Continuous Dynamical Systems - Series A. Vol. 38 12. ed. 2018. p. 6073-6090 (Discrete and Continuous Dynamical Systems- Series A).

Research output: Chapter in Book/Conference paperConference paper

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T1 - Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients

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N2 - We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation aijuij = up, u ≥ 0, p ∈ [0, 1), with bounded discontinuous coefficients aij having small BMO norm. We consider the simplest discontinuity of the form x x|x|− 2 at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when p = 0) cannot be smooth at the points of discontinuity of aij(x). To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

AB - We provide an integral estimate for a non-divergence (non-variational) form second order elliptic equation aijuij = up, u ≥ 0, p ∈ [0, 1), with bounded discontinuous coefficients aij having small BMO norm. We consider the simplest discontinuity of the form x x|x|− 2 at the origin. As an application we show that the free boundary corresponding to the obstacle problem (i.e. when p = 0) cannot be smooth at the points of discontinuity of aij(x). To implement our construction, an integral estimate and a scale invariance will provide the homogeneity of the blow-up sequences, which then can be classified using ODE arguments.

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Dipierro S, Karakhanyan A, Valdinoci E. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. In Discrete and Continuous Dynamical Systems - Series A. 12 ed. Vol. 38. 2018. p. 6073-6090. (Discrete and Continuous Dynamical Systems- Series A). https://doi.org/10.3934/dcds.2018262