## Duke Mathematical Journal

### Regularity properties of a free boundary near contact points with the fixed boundary

#### Abstract

In the upper half of the unit ball $B\sp + =\{ |x|<1,x\sb 1>0\}$, let $u$ and $\Omega$ (a domain in $\mathbf {R}\sp n\sb + =\{x\in \mathbf {R}\sp n : x\sb 1>0\}$) solve the following overdetermined problem:

where $B$ is the unit ball with center at the origin, $\chi\sb \Omega$ denotes the characteristic function of $\Delta,\Pi=\{ x\sb 1=0\} ,n\geq 2$, and the equation is satisfied in the sense of distributions. We show (among other things) that if the origin is a contact point of the free boundary, that is, if $u(0)=|\nabla u(0)|=0$, then $\partial\Delta\cap B\sb {r\sb 0}$ is the graph of a $C\sp 1$-function over $\Pi\cap B\sb (r\sb 0)$. The $C\sp 1$-norm depends on the dimension and $\sup\sb {B\sp +}|u|$. The result is extended to general subdomains of the unit ball with $C\sp 3$-boundary.

#### Article information

Source
Duke Math. J. Volume 116, Number 1 (2003), 1-34.

Dates
First available in Project Euclid: 26 May 2004

http://projecteuclid.org/euclid.dmj/1085598234

Mathematical Reviews number (MathSciNet)
MR1950478

Digital Object Identifier
doi:10.1215/S0012-7094-03-11611-7

Zentralblatt MATH identifier
1050.35157

#### Citation

Shahgholian, Henrik; Uraltseva, Nina. Regularity properties of a free boundary near contact points with the fixed boundary. Duke Math. J. 116 (2003), no. 1, 1--34. doi:10.1215/S0012-7094-03-11611-7. http://projecteuclid.org/euclid.dmj/1085598234.

#### References

• H. W. Alt, L. A. Caffarelli, and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431--461.
• D. E. Apushkinskaya and N. N. Uraltseva, On the behavior of the free boundary near the boundary of the domain (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), 5--19.; English translation in J. Math. Sci. (New York) 87 (1997), 3267--3276.
• I. Athanasopoulos and L. A. Caffarelli, A theorem of real analysis and its application to free boundary problems, Comm. Pure Appl. Math. 38 (1985), 499--502.
• I. Athanasopoulos, L. A. Caffarelli, and S. Salsa, The free boundary in an inverse conductivity problem, J. Reine Angew. Math. 534 (2001), 1--31.
• H. Berestycki, L. A. Caffarelli, and L. Nirenberg, Uniform estimates for regularization of free boundary problems'' in Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math. 122, Dekker, New York, 1990, 567--619.
• H. Berestycki and B. Larrouturou, A semi-linear elliptic equation in a strip arising in a two-dimensional flame propagation model, J. Reine Angew. Math. 396 (1989), 14--40.
• H. Berestycki, B. Larrouturou, and L. Nirenberg, A nonlinear elliptic problem describing the propagation of a curved premixed flame'' in Mathematical Modeling in Combustion and Related Topics (Lyon, 1987), NATO Adv. Sci. Inst. Ser. E Appl. Sci. 140, Nijhoff, Dordrecht, 1988, 11--28.
• L. A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (1977), 155--184.
• --. --. --. --., Compactness methods in free boundary problems, Comm. Partial Differential Equations 5 (1980), 427--448.
• --------, The obstacle problem revisited, J. Fourier Anal. Appl. 4 (1998) 383--402.
• L. A. Caffarelli, D. Jerison, and C. E. Kenig, Ann. of Math. (2) 155 (2002), 369--404. \CMP1 906 591
• L. A. Caffarelli, L. Karp, and H. Shahgholian, Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math. (2) 151 (2000), 269--292.
• L. Caffarelli and J. Salazar, Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves, Trans. Amer. Math. Soc. 354 (2002), 3095--3115. \CMP1 897 393
• L. A. Caffarelli and H. Shahgholian, The structure of the singular set of a free boundary in potential theory, 2000, unpublished.
• J. R. Cannon and C. D. Hill, On the movement of a chemical reaction interface, Indiana Univ. Math. J. 20 (1970/1971), 429--454.
• C. M. Elliott, R. Schätzle, and B. E. E. Stoth, Viscosity solutions of a degenerate parabolic-elliptic system arising in the mean-field theory of superconductivity, Arch. Ration. Mech. Anal. 145 (1998), 99--127.
• A. Friedman, Variational Principles and Free-Boundary Problems, Pure Appl. Math., Wiley, New York, 1982.
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2d ed., Grundlehren Math. Wiss. 224, Springer, Berlin, 1983.
• A. Gurevich, Boundary regularity for free boundary problems, Comm. Pure Appl. Math. 52 (1999), 363--403.
• L. Karp and H. Shahgholian, Regularity of a free boundary problem, J. Geom. Anal. 9 (1999), 653--669.
• D. Kinderlehrer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 373--391.
• D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math. 88, Academic Press, New York, 1980.
• O. A. Ladyzhenskaya and N. N. Uraltseva, Estimates on the boundary of the domain of first derivatives of functions satisfying an elliptic or a parabolic inequality (in Russian), Trudy Mat. Inst. Steklov. 179 (1988), 102--125.; English translation in Proc. Steklov Inst. Math. 1989, no. 2, 109--135.
• N. N. Uraltseva, $C^1$ regularity of the boundary of a noncoincident set in a problem with an obstacle (in Russian), Algebra i Analiz 8, no. 2 (1996), 205--221.; English translation in St. Petersburg Math. J. 8, no. 2 (1997), 341--353.
• --. --. --. --., On the properties of a free boundary in a neighborhood of the points of contact with the known boundary (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 (1997), 303--312.; English translation in J. Math. Sci. (New York) 101 (2000), 3570--3576.
• --. --. --. --., On the contact of a free boundary with a given boundary (in Russian), Mat. Sb. 191, no. 2 (2000), 165--173.