# American Institute of Mathematical Sciences

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 and 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. [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. [3] 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.  doi: 10.1090/S0002-9947-1984-0732100-6. [4] I. Athanasopoulos, L. A. Caffarelli, C. 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. [5] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 439-455. [6] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331. [7] L. A. Caffarelli, J.-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. [8] L. Caffarelli, X. 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. [9] L. A. Caffarelli, S. 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. [10] L. Caffarelli, O. 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. [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. [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. [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. [14] D. De Silva, O. 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. [15] S. Dipierro, A. Figalli, G. 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. [16] S. Dipierro, A. Karakhanyan and E. Valdinoci, New trends in free boundary problems, Adv. Nonlinear Stud., 17 (2017), 319-332.  doi: 10.1515/ans-2017-0002. [17] S. Dipierro, A. Karakhanyan and E. Valdinoci, A class of unstable free boundary problems, Anal. PDE, 10 (2017), 1317-1359.  doi: 10.2140/apde.2017.10.1317. [18] S. Dipierro, A. 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. [19] S. Dipierro, O. Savin and E. Valdinoci, A nonlocal free boundary problem, SIAM J. Math. Anal., 47 (2015), 4559-4605.  doi: 10.1137/140999712. [20] S. Dipierro, O. 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. [21] S. Dipierro, O. 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. [22] S. Dipierro, O. Savin and E. Valdinoci, Definition of Fractional Laplacian for Functions with Polynomial Growth Rev. Mat. Iberoam. [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. [24] S. Dipierro and E. Valdinoci, Nonlocal Minimal Surfaces: Interior Regularity, Quantitative Estimates and Boundary Stickiness Recent Dev. Nonlocal Theory, De Gruyter, Berlin. [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. [26] A. Friedman, Free boundary problems in science and technology, Notices Amer. Math. Soc., 47 (2000), 854-861. [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. [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. [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. [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.

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##### 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. [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. [3] 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.  doi: 10.1090/S0002-9947-1984-0732100-6. [4] I. Athanasopoulos, L. A. Caffarelli, C. 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. [5] J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, (2001), 439-455. [6] L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331. [7] L. A. Caffarelli, J.-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. [8] L. Caffarelli, X. 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. [9] L. A. Caffarelli, S. 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. [10] L. Caffarelli, O. 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. [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. [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. [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. [14] D. De Silva, O. 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. [15] S. Dipierro, A. Figalli, G. 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. [16] S. Dipierro, A. Karakhanyan and E. Valdinoci, New trends in free boundary problems, Adv. Nonlinear Stud., 17 (2017), 319-332.  doi: 10.1515/ans-2017-0002. [17] S. Dipierro, A. Karakhanyan and E. Valdinoci, A class of unstable free boundary problems, Anal. PDE, 10 (2017), 1317-1359.  doi: 10.2140/apde.2017.10.1317. [18] S. Dipierro, A. 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. [19] S. Dipierro, O. Savin and E. Valdinoci, A nonlocal free boundary problem, SIAM J. Math. Anal., 47 (2015), 4559-4605.  doi: 10.1137/140999712. [20] S. Dipierro, O. 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. [21] S. Dipierro, O. 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. [22] S. Dipierro, O. Savin and E. Valdinoci, Definition of Fractional Laplacian for Functions with Polynomial Growth Rev. Mat. Iberoam. [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. [24] S. Dipierro and E. Valdinoci, Nonlocal Minimal Surfaces: Interior Regularity, Quantitative Estimates and Boundary Stickiness Recent Dev. Nonlocal Theory, De Gruyter, Berlin. [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. [26] A. Friedman, Free boundary problems in science and technology, Notices Amer. Math. Soc., 47 (2000), 854-861. [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. [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. [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. [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.
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