JournalsaihpcVol. 34, No. 2pp. 293–334

The obstacle problem with singular coefficients near Dirichlet data

  • Henrik Shahgholian

    Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
  • Karen Yeressian

    Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
The obstacle problem with singular coefficients near Dirichlet data cover
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Abstract

In this paper we study the behaviour of the free boundary close to its contact points with the fixed boundary B{x1=0}B \cap \{x_{1} = 0\} in the obstacle type problem

{div(x1au)=χ{u>0}inB+,u=0onB{x1=0}\left\{\begin{matrix} & \mathrm{div}(x_{1}^{a}\mathrm{∇}u) = \chi _{\{u > 0\}}\:\text{in}\:B^{ + }\text{,} \\ & u = 0\:\text{on}\:B \cap \{x_{1} = 0\} \\ \end{matrix}\right.

where a<1a < 1, B+=B{x1>0}B^{ + } = B \cap \{x_{1} > 0\}, B is the unit ball in Rn\mathbb{R}^{n} and n2n \geq 2 is an integer.

Let Γ=B+{u>0}\mathrm{\Gamma } = B^{ + } \cap \partial \{u > 0\} be the free boundary and assume that the origin is a contact point, i.e. 0 \in \mathrm{\Gamma }\limits^{‾}. 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 C1C^{1} function and there is a uniform modulus of continuity for the derivatives of this function.

Cite this article

Henrik Shahgholian, Karen Yeressian, The obstacle problem with singular coefficients near Dirichlet data. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 2, pp. 293–334

DOI 10.1016/J.ANIHPC.2015.12.003