### Mathematics in Engineering

2020, Issue 4: 698-708. doi: 10.3934/mine.2020032
Research article Special Issues

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

• 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.

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

### Related Papers:

• 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.

 [1] Andersson J, Lindgren E, Shahgholian H (2013) Optimal regularity for the no-sign obstacle problem. Commun Pure Appl Math 66: 245-262. doi: 10.1002/cpa.21434 [2] Caffarelli LA (1998) The obstacle problem revisited. J Fourier Anal Appl 4: 383-402. doi: 10.1007/BF02498216 [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. doi: 10.1007/s00205-014-0734-0 [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. doi: 10.1080/03605309908821478 [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.
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