The obstacle problem with singular coefficients near Dirichlet data

Shahgholian, Henrik; Yeressian, Karen

doi:10.1016/j.anihpc.2015.12.003

Shahgholian, Henrik ; Yeressian, Karen

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 293-334.

Résumé

In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary $B \cap {x_{1} = 0}$ in the obstacle type problem

{\begin{matrix} div (x_{1}^{a} \nabla u) = χ_{{u > 0}} in B^{+}, \\ u = 0 on B \cap {x_{1} = 0} \end{matrix}

where

a < 1

B^{+} = B \cap {x_{1} > 0}

, B is the unit ball in

R^{n}

and

n \geq 2

is an integer.

Let $Γ = B^{+} \cap \partial {u > 0}$ be the free boundary and assume that the origin is a contact point, i.e. $0 \in \overline{Γ}$ . We prove that the free boundary touches the fixed boundary uniformly tangentially at the origin, near to the origin it is the graph of a $C^{1}$ function and there is a uniform modulus of continuity for the derivatives of this function.

MR Zbl

DOI : 10.1016/j.anihpc.2015.12.003

Classification : 35R35, 35J60
Mots clés : Free boundary, Obstacle problem, Singular coefficient, Regularity of free boundaries

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     author = {Shahgholian, Henrik and Yeressian, Karen},
     title = {The obstacle problem with singular coefficients near {Dirichlet} data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {293--334},
     publisher = {Elsevier},
     volume = {34},
     number = {2},
     year = {2017},
     doi = {10.1016/j.anihpc.2015.12.003},
     mrnumber = {3610934},
     zbl = {1515.35369},
     language = {en},
     url = {http://www.numdam.org./articles/10.1016/j.anihpc.2015.12.003/}
}

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Shahgholian, Henrik; Yeressian, Karen. The obstacle problem with singular coefficients near Dirichlet data. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 293-334. doi : 10.1016/j.anihpc.2015.12.003. http://www.numdam.org./articles/10.1016/j.anihpc.2015.12.003/

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