Remarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type

Morteza Fotouhi; Andreas Minne; Henrik Shahgholian; Georg S. Weiss; Morteza Fotouhi; Andreas Minne; Henrik Shahgholian; Georg S. Weiss

doi:10.3934/mine.2020032

Mathematics in Engineering

2020, Volume 2, Issue 4: 698-708. doi: 10.3934/mine.2020032

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Remarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type

1.
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
2.
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden
3.
Department of Mathematics, University of Duisburg-Essen, Essen, Germany

Received: 09 January 2020 Accepted: 12 May 2020 Published: 18 June 2020

The purpose of this note is to present a "new" approach to the decay rate of the solutions to the no-sign obstacle problem from the free boundary, based on Weiss-monotonicity formula. In presenting the approach we have chosen to treat a problem which is not touched earlier in the existing literature. Although earlier techniques may still work for this problem, we believe this approach gives a shorter proof, and may have wider applications.
- no-sign obstacle problem,
- singular elliptic equation,
- regularity of solutions
Citation: Morteza Fotouhi, Andreas Minne, Henrik Shahgholian, Georg S. Weiss. Remarks on the decay/growth rate of solutions to elliptic free boundary problems of obstacle type[J]. Mathematics in Engineering, 2020, 2(4): 698-708. doi: 10.3934/mine.2020032

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Abstract

The purpose of this note is to present a "new" approach to the decay rate of the solutions to the no-sign obstacle problem from the free boundary, based on Weiss-monotonicity formula. In presenting the approach we have chosen to treat a problem which is not touched earlier in the existing literature. Although earlier techniques may still work for this problem, we believe this approach gives a shorter proof, and may have wider applications.

References

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[6]	Petrosyan A, Shahgholian H, Uraltseva N (2012) Regularity of Free Boundaries in Obstacle-Type Problems, Providence: American Mathematical Society.
[7]	Rákosník J (1989) On embeddings and traces in Sobolev spaces with weights of power type, In: Approximation and Function Spaces, Warsaw: Banach Center Publ., 331-339.
[8]	Shahgholian H (2003) C^1,1 regularity in semilinear elliptic problems, Commun Pure Appl Math 56: 278-281.
[9]	Shahgholian H, Yeressian K (2017) The obstacle problem with singular coefficients near Dirichlet data, Ann Inst H Poincaré Anal Non Linéaire 34: 293-334.
[10]	Weiss GS (2001) An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interface Free Bound 3: 121-128.

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