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 & 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.  doi: 10.1090/S0002-9947-98-01943-6.  Google Scholar

[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.  Google Scholar

[3]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints,, J. Math. Pures Appl., 76 (1997), 265.  doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[4]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs. Oxford University Press, (2000).   Google Scholar

[5]

R. Argiolas, A two-phase variational problem with curvature,, Matematiche (Catania), 58 (2003), 131.   Google Scholar

[6]

I. Athanasopoulos, L. A. Caffarelli, C. Kenig and S. Salsa, An area-Dirichlet integral minimization problem,, Comm. Pure Appl. Math., 54 (2001), 479.   Google Scholar

[7]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics. CRC Press, (1992).   Google Scholar

[8]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer Verlag, (2001).   Google Scholar

[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.  doi: 10.1515/crll.2000.031.  Google Scholar

[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.  doi: 10.1090/S0002-9947-02-02892-1.  Google Scholar

[11]

D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free boundary problems I,, J. Analyse Math., 34 (1978), 86.  doi: 10.1007/BF02790009.  Google Scholar

[12]

B. Kirchheim and J. Kristensen, Differentiability of convex envelopes,, C. R. Acad. Sci. Paris S閞. I Math., 333 (2001), 725.  doi: 10.1016/S0764-4442(01)02117-6.  Google Scholar

[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.  doi: 10.1512/iumj.1990.39.39051.  Google Scholar

[14]

J. L. Lewis, Capacitary functions in convex rings,, Arch. Rational Mech. Anal., 66 (1977), 201.  doi: 10.1007/BF00250671.  Google Scholar

[15]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[16]

F. Mazzone, A single phase variational problem involving the area of level surfaces,, Comm. Part. Diff. Eq., 28 (2003), 991.  doi: 10.1081/PDE-120021183.  Google Scholar

[17]

I. Tamanini, Regularity results for almost minimal oriented hypersurface in $R^n$,, Quaderni del Dipartimento di Matematica, (1994).   Google Scholar

show all references

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.  doi: 10.1090/S0002-9947-98-01943-6.  Google Scholar

[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.  Google Scholar

[3]

O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints,, J. Math. Pures Appl., 76 (1997), 265.  doi: 10.1016/S0021-7824(97)89952-7.  Google Scholar

[4]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs. Oxford University Press, (2000).   Google Scholar

[5]

R. Argiolas, A two-phase variational problem with curvature,, Matematiche (Catania), 58 (2003), 131.   Google Scholar

[6]

I. Athanasopoulos, L. A. Caffarelli, C. Kenig and S. Salsa, An area-Dirichlet integral minimization problem,, Comm. Pure Appl. Math., 54 (2001), 479.   Google Scholar

[7]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics. CRC Press, (1992).   Google Scholar

[8]

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer Verlag, (2001).   Google Scholar

[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.  doi: 10.1515/crll.2000.031.  Google Scholar

[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.  doi: 10.1090/S0002-9947-02-02892-1.  Google Scholar

[11]

D. Kinderlehrer, L. Nirenberg and J. Spruck, Regularity in elliptic free boundary problems I,, J. Analyse Math., 34 (1978), 86.  doi: 10.1007/BF02790009.  Google Scholar

[12]

B. Kirchheim and J. Kristensen, Differentiability of convex envelopes,, C. R. Acad. Sci. Paris S閞. I Math., 333 (2001), 725.  doi: 10.1016/S0764-4442(01)02117-6.  Google Scholar

[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.  doi: 10.1512/iumj.1990.39.39051.  Google Scholar

[14]

J. L. Lewis, Capacitary functions in convex rings,, Arch. Rational Mech. Anal., 66 (1977), 201.  doi: 10.1007/BF00250671.  Google Scholar

[15]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[16]

F. Mazzone, A single phase variational problem involving the area of level surfaces,, Comm. Part. Diff. Eq., 28 (2003), 991.  doi: 10.1081/PDE-120021183.  Google Scholar

[17]

I. Tamanini, Regularity results for almost minimal oriented hypersurface in $R^n$,, Quaderni del Dipartimento di Matematica, (1994).   Google Scholar

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