June  2018, 11(3): 465-476. doi: 10.3934/dcdss.2018025

(Non)local and (non)linear free boundary problems

1. 

Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50,20133 Milan, Italy

2. 

School of Mathematics and Statistics, University of Melbourne, 813 Swanston St, Parkville VIC 3010, Australia

3. 

Istituto di Matematica Applicata e Tecnologie Informatiche, Via Ferrata 1,27100 Pavia, Italy

* Corresponding author: Enrico Valdinoci

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: Supported by Australian Research Council grant N.E.W. (Nonlocal Equations at Work). Serena Dipierro is also supported by GNAMPA and Andrew Sisson fund 2017.

We discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin.

The main feature of these new free boundary problems is that they deeply take into account nonlinear energy superpositions and possibly nonlocal functionals.

The nonlocal parameter interpolates between volume and perimeter functionals, and so it can be seen as a fractional counterpart of classical free boundary problems, in which the bulk energy presents nonlocal aspects.

The nonlinear term in the energy superposition takes into account the possibility of modeling different regimes in terms of different energy levels and provides a lack of scale invariance, which in turn may cause a structural instability of minimizers that may vary from one scale to another.

Citation: Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.  doi: 10.1080/01630563.2014.901837.  Google Scholar

[2]

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[3]

H. W. AltL. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461.  doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar

[4]

I. AthanasopoulosL. A. CaffarelliC. Kenig and S. Salsa, An area-Dirichlet integral minimization problem, Comm. Pure Appl. Math., 54 (2001), 479-499.  doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2.  Google Scholar

[5]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 439-455.  Google Scholar

[6]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[7]

L. A. CaffarelliJ.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

[8]

L. CaffarelliX. Ros-Oton and J. Serra, Obstacle problems for integro-differential operators: regularity of solutions and free boundaries, Invent. Math., 208 (2017), 1155-1211.  doi: 10.1007/s00222-016-0703-3.  Google Scholar

[9]

L. A. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[10]

L. CaffarelliO. Savin and E. Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 901-924.  doi: 10.1016/j.anihpc.2014.04.004.  Google Scholar

[11]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6.  Google Scholar

[12]

J. Dávila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2002), 519-527.  doi: 10.1007/s005260100135.  Google Scholar

[13]

D. De Silva and J. M. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 335-367.  doi: 10.1016/j.anihpc.2011.11.003.  Google Scholar

[14]

D. De SilvaO. Savin and Y. Sire, A one-phase problem for the fractional Laplacian: Regularity of flat free boundaries, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 111-145.   Google Scholar

[15]

S. DipierroA. FigalliG. Palatucci and E. Valdinoci, Asymptotics of the $s$-perimeter as $s\searrow 0$, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.  doi: 10.3934/dcds.2013.33.2777.  Google Scholar

[16]

S. DipierroA. Karakhanyan and E. Valdinoci, New trends in free boundary problems, Adv. Nonlinear Stud., 17 (2017), 319-332.  doi: 10.1515/ans-2017-0002.  Google Scholar

[17]

S. DipierroA. Karakhanyan and E. Valdinoci, A class of unstable free boundary problems, Anal. PDE, 10 (2017), 1317-1359.  doi: 10.2140/apde.2017.10.1317.  Google Scholar

[18]

S. DipierroA. Karakhanyan and E. Valdinoci, A nonlinear free boundary problem with a self-driven Bernoulli condition, J. Funct. Anal., 273 (2017), 3549-3615.  doi: 10.1016/j.jfa.2017.07.014.  Google Scholar

[19]

S. DipierroO. Savin and E. Valdinoci, A nonlocal free boundary problem, SIAM J. Math. Anal., 47 (2015), 4559-4605.  doi: 10.1137/140999712.  Google Scholar

[20]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966.  doi: 10.4171/JEMS/684.  Google Scholar

[21]

S. DipierroO. Savin and E. Valdinoci, Boundary behavior of nonlocal minimal surfaces, J. Funct. Anal., 272 (2017), 1791-1851.  doi: 10.1016/j.jfa.2016.11.016.  Google Scholar

[22]

S. Dipierro, O. Savin and E. Valdinoci, Definition of Fractional Laplacian for Functions with Polynomial Growth Rev. Mat. Iberoam. Google Scholar

[23]

S. Dipierro and E. Valdinoci, On a fractional harmonic replacement, Discrete Contin. Dyn. Syst., 35 (2015), 3377-3392.  doi: 10.3934/dcds.2015.35.3377.  Google Scholar

[24]

S. Dipierro and E. Valdinoci, Nonlocal Minimal Surfaces: Interior Regularity, Quantitative Estimates and Boundary Stickiness Recent Dev. Nonlocal Theory, De Gruyter, Berlin. Google Scholar

[25]

S. Dipierro and E. Valdinoci, Continuity and Density Results for a One-Phase Nonlocal Free Boundary Problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1387-1428.  doi: 10.1016/j.anihpc.2016.11.001.  Google Scholar

[26]

A. Friedman, Free boundary problems in science and technology, Notices Amer. Math. Soc., 47 (2000), 854-861.   Google Scholar

[27]

N. Garofalo and A. Petrosyan, Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math., 177 (2009), 415-461.  doi: 10.1007/s00222-009-0188-4.  Google Scholar

[28]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation Department of Pure Mathematics, Australian National University, Canberra, 1977. ISBN 0-7081-1294-3. With notes by Graham H. Williams; Notes on Pure Mathematics, 10.  Google Scholar

[29]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.  Google Scholar

[30]

G. S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem, Comm. Partial Differential Equations, 23 (1998), 439-455.  doi: 10.1080/03605309808821352.  Google Scholar

show all references

References:
[1]

N. Abatangelo and E. Valdinoci, A notion of nonlocal curvature, Numer. Funct. Anal. Optim., 35 (2014), 793-815.  doi: 10.1080/01630563.2014.901837.  Google Scholar

[2]

H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math., 325 (1981), 105-144.   Google Scholar

[3]

H. W. AltL. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431-461.  doi: 10.1090/S0002-9947-1984-0732100-6.  Google Scholar

[4]

I. AthanasopoulosL. A. CaffarelliC. Kenig and S. Salsa, An area-Dirichlet integral minimization problem, Comm. Pure Appl. Math., 54 (2001), 479-499.  doi: 10.1002/1097-0312(200104)54:4<479::AID-CPA3>3.0.CO;2-2.  Google Scholar

[5]

J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 439-455.  Google Scholar

[6]

L. CaffarelliJ.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[7]

L. A. CaffarelliJ.-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS), 12 (2010), 1151-1179.  doi: 10.4171/JEMS/226.  Google Scholar

[8]

L. CaffarelliX. Ros-Oton and J. Serra, Obstacle problems for integro-differential operators: regularity of solutions and free boundaries, Invent. Math., 208 (2017), 1155-1211.  doi: 10.1007/s00222-016-0703-3.  Google Scholar

[9]

L. A. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[10]

L. CaffarelliO. Savin and E. Valdinoci, Minimization of a fractional perimeter-Dirichlet integral functional, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 901-924.  doi: 10.1016/j.anihpc.2014.04.004.  Google Scholar

[11]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6.  Google Scholar

[12]

J. Dávila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2002), 519-527.  doi: 10.1007/s005260100135.  Google Scholar

[13]

D. De Silva and J. M. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 335-367.  doi: 10.1016/j.anihpc.2011.11.003.  Google Scholar

[14]

D. De SilvaO. Savin and Y. Sire, A one-phase problem for the fractional Laplacian: Regularity of flat free boundaries, Bull. Inst. Math. Acad. Sin. (N.S.), 9 (2014), 111-145.   Google Scholar

[15]

S. DipierroA. FigalliG. Palatucci and E. Valdinoci, Asymptotics of the $s$-perimeter as $s\searrow 0$, Discrete Contin. Dyn. Syst., 33 (2013), 2777-2790.  doi: 10.3934/dcds.2013.33.2777.  Google Scholar

[16]

S. DipierroA. Karakhanyan and E. Valdinoci, New trends in free boundary problems, Adv. Nonlinear Stud., 17 (2017), 319-332.  doi: 10.1515/ans-2017-0002.  Google Scholar

[17]

S. DipierroA. Karakhanyan and E. Valdinoci, A class of unstable free boundary problems, Anal. PDE, 10 (2017), 1317-1359.  doi: 10.2140/apde.2017.10.1317.  Google Scholar

[18]

S. DipierroA. Karakhanyan and E. Valdinoci, A nonlinear free boundary problem with a self-driven Bernoulli condition, J. Funct. Anal., 273 (2017), 3549-3615.  doi: 10.1016/j.jfa.2017.07.014.  Google Scholar

[19]

S. DipierroO. Savin and E. Valdinoci, A nonlocal free boundary problem, SIAM J. Math. Anal., 47 (2015), 4559-4605.  doi: 10.1137/140999712.  Google Scholar

[20]

S. DipierroO. Savin and E. Valdinoci, All functions are locally $s$-harmonic up to a small error, J. Eur. Math. Soc. (JEMS), 19 (2017), 957-966.  doi: 10.4171/JEMS/684.  Google Scholar

[21]

S. DipierroO. Savin and E. Valdinoci, Boundary behavior of nonlocal minimal surfaces, J. Funct. Anal., 272 (2017), 1791-1851.  doi: 10.1016/j.jfa.2016.11.016.  Google Scholar

[22]

S. Dipierro, O. Savin and E. Valdinoci, Definition of Fractional Laplacian for Functions with Polynomial Growth Rev. Mat. Iberoam. Google Scholar

[23]

S. Dipierro and E. Valdinoci, On a fractional harmonic replacement, Discrete Contin. Dyn. Syst., 35 (2015), 3377-3392.  doi: 10.3934/dcds.2015.35.3377.  Google Scholar

[24]

S. Dipierro and E. Valdinoci, Nonlocal Minimal Surfaces: Interior Regularity, Quantitative Estimates and Boundary Stickiness Recent Dev. Nonlocal Theory, De Gruyter, Berlin. Google Scholar

[25]

S. Dipierro and E. Valdinoci, Continuity and Density Results for a One-Phase Nonlocal Free Boundary Problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1387-1428.  doi: 10.1016/j.anihpc.2016.11.001.  Google Scholar

[26]

A. Friedman, Free boundary problems in science and technology, Notices Amer. Math. Soc., 47 (2000), 854-861.   Google Scholar

[27]

N. Garofalo and A. Petrosyan, Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem, Invent. Math., 177 (2009), 415-461.  doi: 10.1007/s00222-009-0188-4.  Google Scholar

[28]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation Department of Pure Mathematics, Australian National University, Canberra, 1977. ISBN 0-7081-1294-3. With notes by Graham H. Williams; Notes on Pure Mathematics, 10.  Google Scholar

[29]

V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.  doi: 10.1006/jfan.2002.3955.  Google Scholar

[30]

G. S. Weiss, Partial regularity for weak solutions of an elliptic free boundary problem, Comm. Partial Differential Equations, 23 (1998), 439-455.  doi: 10.1080/03605309808821352.  Google Scholar

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