# American Institute of Mathematical Sciences

May  2013, 12(3): 1431-1443. doi: 10.3934/cpaa.2013.12.1431

## Convexity of the free boundary for an exterior free boundary problem involving the perimeter

 1 Xi'an Jiaotong-Liverpool University, Mathematical Sciences, 111 Ren'ai Road, Suzhou 215123, Jiangsu Prov., China 2 Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden

Received  November 2011 Revised  March 2012 Published  September 2012

We prove that if the given compact set $K$ is convex then a minimizer of the functional \begin{eqnarray*} I(v)=\int_{B_R} |\nabla v|^p dx+ Per(\{v>0\}), 1 < p < \infty, \end{eqnarray*} over the set $\{v\in W^{1,p}_0 (B_R)| v\equiv 1 \ \text{on} \ K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.
Citation: Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431
##### References:
 [1] A. Acker, On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions, Trans. Amer. Math. Soc., 350 (1998), 2981-3020. doi: 10.1090/S0002-9947-98-01943-6. [2] F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc., 4 (1976). doi: 10.1090/S0002-9904-1975-13681-0. [3] O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7. [4] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs. Oxford University Press, New York, 2000. [5] R. Argiolas, A two-phase variational problem with curvature, Matematiche (Catania), 58 (2003), 131-148. [6] I. Athanasopoulos, L. A. Caffarelli, C. Kenig and S. Salsa, An area-Dirichlet integral minimization problem, Comm. Pure Appl. Math., 54 (2001), 479-499. [7] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [8] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer Verlag, Berlin, 2001. [9] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator. I. The exterior convex case, J. Reine Angew. Math., 521 (2000), 85-97. doi: 10.1515/crll.2000.031. [10] A. Henrot and H. Shahgholian, The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition, Trans. Amer. Math. Soc., 354 (2002), 2399-2416. doi: 10.1090/S0002-9947-02-02892-1. [11] D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free boundary problems I, J. Analyse Math., 34 (1978), 86-119. doi: 10.1007/BF02790009. [12] B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris S閞. I Math., 333 (2001), 725-728. doi: 10.1016/S0764-4442(01)02117-6. [13] P. Laurence and E. Stredulinsky, Existence of regular solutions with convex levels for semilinear elliptic equations with nonmonotone $L^1$ nonlinearities. Part I, Indiana Univ. Math. J., 39 (1990), 1081-1114. doi: 10.1512/iumj.1990.39.39051. [14] J. L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224. doi: 10.1007/BF00250671. [15] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3. [16] F. Mazzone, A single phase variational problem involving the area of level surfaces, Comm. Part. Diff. Eq., 28 (2003), 991-1004. doi: 10.1081/PDE-120021183. [17] I. Tamanini, Regularity results for almost minimal oriented hypersurface in $R^n$, Quaderni del Dipartimento di Matematica, Universitá di Lecce 1, (1994).

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##### References:
 [1] A. Acker, On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions, Trans. Amer. Math. Soc., 350 (1998), 2981-3020. doi: 10.1090/S0002-9947-98-01943-6. [2] F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc., 4 (1976). doi: 10.1090/S0002-9904-1975-13681-0. [3] O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7. [4] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs. Oxford University Press, New York, 2000. [5] R. Argiolas, A two-phase variational problem with curvature, Matematiche (Catania), 58 (2003), 131-148. [6] I. Athanasopoulos, L. A. Caffarelli, C. Kenig and S. Salsa, An area-Dirichlet integral minimization problem, Comm. Pure Appl. Math., 54 (2001), 479-499. [7] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. [8] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer Verlag, Berlin, 2001. [9] A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the $p$-Laplace operator. I. The exterior convex case, J. Reine Angew. Math., 521 (2000), 85-97. doi: 10.1515/crll.2000.031. [10] A. Henrot and H. Shahgholian, The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition, Trans. Amer. Math. Soc., 354 (2002), 2399-2416. doi: 10.1090/S0002-9947-02-02892-1. [11] D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free boundary problems I, J. Analyse Math., 34 (1978), 86-119. doi: 10.1007/BF02790009. [12] B. Kirchheim and J. Kristensen, Differentiability of convex envelopes, C. R. Acad. Sci. Paris S閞. I Math., 333 (2001), 725-728. doi: 10.1016/S0764-4442(01)02117-6. [13] P. Laurence and E. Stredulinsky, Existence of regular solutions with convex levels for semilinear elliptic equations with nonmonotone $L^1$ nonlinearities. Part I, Indiana Univ. Math. J., 39 (1990), 1081-1114. doi: 10.1512/iumj.1990.39.39051. [14] J. L. Lewis, Capacitary functions in convex rings, Arch. Rational Mech. Anal., 66 (1977), 201-224. doi: 10.1007/BF00250671. [15] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3. [16] F. Mazzone, A single phase variational problem involving the area of level surfaces, Comm. Part. Diff. Eq., 28 (2003), 991-1004. doi: 10.1081/PDE-120021183. [17] I. Tamanini, Regularity results for almost minimal oriented hypersurface in $R^n$, Quaderni del Dipartimento di Matematica, Universitá di Lecce 1, (1994).
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