Geodesic
When does the free boundary enter into corner points of the fixed boundary?
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 35, Tome 310 (2004), pp. 213-225 Cet article a éte moissonné depuis la source Math-Net.Ru

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Our prime goal in this note is to lay the ground for studying free boundaries close to the corner points of a fixed, Lipschitz boundary. Our study is restricted to 2-space dimensions, and to the obstacle problem. Our main result states that the free boundary can not enter into a corner $x^0$ of the fixed boundary, if the (interior) angle is less than $\pi$, provided the boundary datum is zero close to the point $x^0$. For larger angles and other boundary datum the free boundary may enter into corners, as discussed in the text. 
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H. Shahgholian. When does the free boundary enter into corner points of the fixed boundary?. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 35, Tome 310 (2004), pp. 213-225. http://geodesic.mathdoc.fr/item/ZNSL_2004_310_a10/

[1] D. Apushkinskaya, H. Shahgholian, N. Uraltseva, “Boundary estimates for solutions to the parabolic free boundary problem”, Zap. Nauchn. Semin. POMI, 271, 2000, 39–55 | MR | Zbl

[2] I. Blank, H. Shahgholian, “Boundary regularity and compactness for overdetermined problems”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2:4 (2003), 787–802 | MR | Zbl

[3] L. A. Caffarelli, “Compactness methods in free boundary problems”, Comm. P.D.E., 5 (1980), 427–448 | DOI | MR | Zbl

[4] A. Friedman, Variational Principles and Free Boundary Problems, Wiley, New York, 1982 | MR

[5] B. Gustafsson, H. Shahgholian, “Existence and geometric properties of solutions of a free boundary problem in potential theory”, J. Reine Angew. Math., 473 (1996), 137–179 | MR | Zbl

[6] L. Karp, H. Shahgholian, “Regularity of a free boundary at the infinity point”, Comm. Partial Differential Equations, 25:11–12 (2000), 2055–2086 | DOI | MR | Zbl

[7] J. F. Rodrigues, Obstacle problems in mathematical physics, North-Holland Mathematics Studies, 134, North-Holland Publishing Co., Amsterdam, 1987 | MR | Zbl

[8] H. Shahgholian, N. Uraltseva, “Regularity properties of a free boundary near contact points with the fixed boundary”, Duke Math. J., 116:1 (2003), 1–34 | DOI | MR | Zbl

[9] G. S. Weiss, “A homogeneity improvement approach to the obstacle problem”, Inv. Math., 138 (1999), 23–50 | DOI | MR | Zbl

[10] G. S. Weiss, “An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary”, Interfaces Free Bound., 3:2 (2001), 121–128 | DOI | MR | Zbl