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On free boundary problems in two dimensions

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Abstract

Let L be a linear elliptic operator in two dimensions with analytic coefficients and of second order, andu(x, y) a solution of Lu=0 in a simply connected domain ω with rectifiable boundary Γ. Suppose ψ(x, y) analytic on ω∪Γ and L ψ≠0 there.H is shown that ifu and ψ coincide with first derivatives on an open portion Γ0 of Γ, then Γ0 permits the representation λ=x (θ),y=y (θ) withx(θ),y(θ)analytic functions of a real parameter θ.

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References

  1. Lewy H., Stampacchia G.,On the regularity of the solutions of a Variational inequality. Comm. Pure Appl. Math.22 (1969), 133–188.

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  4. Kinderlehrer D., Nirenberg L.,Regularity in free boundary problems, Ann. Scuola Norm. Sup. Ser. IV4 (1977) treats similar problems inn≧2 dimensions, assuming Hölder continuity of the tangent manifold.

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Authors and Affiliations

  1. Departments of Mathematics, University of California, 94720, Berkeley, California

    Hans Lewy & Tang Zhiyuan

  2. Department of Mathematics, University of California, Berkeley, California

    Hans Lewy & Tang Zhiyuan

  3. Chengdu, China

    Hans Lewy & Tang Zhiyuan

Authors
  1. Hans Lewy
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  2. Tang Zhiyuan
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Lewy, H., Zhiyuan, T. On free boundary problems in two dimensions. Rend. Circ. Mat. Palermo 34, 325–336 (1985). https://doi.org/10.1007/BF02844529

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  • DOI: https://doi.org/10.1007/BF02844529

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