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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

This contribution is part of the Special Issue: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday
Guest Editor: Gianmaria Verzini
Link: www.aimspress.com/mine/article/5753/special-articles

Special Issues: Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday

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.
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References

1. Andersson J, Lindgren E, Shahgholian H (2013) Optimal regularity for the no-sign obstacle problem. Commun Pure Appl Math 66: 245-262.    

2. Caffarelli LA (1998) The obstacle problem revisited. J Fourier Anal Appl 4: 383-402.    

3. Figalli A, Shahgholian H (2014) A general class of free boundary problems for fully nonlinear elliptic equations. Arch Ration Mech Anal 213: 269-286.    

4. Gilbarg D, Trudinger NS (2001) Elliptic Partial Differential Equations of Second Order, Springer.

5. Krylov NV (1999) Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space. Commun Part Diff Eq 24: 1611-1653.    

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) C1,1 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.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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