The limit as 𝑝→∞ in free boundary problems with fractional 𝑝-Laplacians

da Silva, João; Rossi, Julio

doi:10.1090/tran/7559

The limit as $p\to \infty$ in free boundary problems with fractional $p$-Laplacians
HTML articles powered by AMS MathViewer

by João Vítor da Silva and Julio D. Rossi PDF

Trans. Amer. Math. Soc. 371 (2019), 2739-2769 Request permission

Abstract:

We study the $p$-fractional optimal design problem under volume constraint taking special care of the case when $p$ is large, obtaining in the limit a free boundary problem modeled by the Hölder infinity Laplacian operator. A necessary and sufficient condition is imposed in order to obtain the uniqueness of solutions to the limiting problem, and, under this condition, we find precisely the optimal configuration for the limit problem. We also prove the sharp regularity (locally $C^{0, s}$) for any limiting solution. Finally, we establish some geometric properties for solutions and their free boundaries.

References

N. Aguilera, H. W. Alt, and L. A. Caffarelli, An optimization problem with volume constraint, SIAM J. Control Optim. 24 (1986), no. 2, 191–198. MR 826512, DOI 10.1137/0324011
N. E. Aguilera, L. A. Caffarelli, and J. Spruck, An optimization problem in heat conduction, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 3, 355–387 (1988). MR 951225
Gunnar Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551–561 (1967). MR 217665, DOI 10.1007/BF02591928
Guy Barles and Cyril Imbert, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 3, 567–585. MR 2422079, DOI 10.1016/j.anihpc.2007.02.007
T. Bhattacharya, E. DiBenedetto, and J. Manfredi, Limits as $p\to \infty$ of $\Delta _pu_p=f$ and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino Special Issue (1989), 15–68 (1991). Some topics in nonlinear PDEs (Turin, 1989). MR 1155453
C. Bjorland, L. Caffarelli, and A. Figalli, Nonlocal tug-of-war and the infinity fractional Laplacian, Comm. Pure Appl. Math. 65 (2012), no. 3, 337–380. MR 2868849, DOI 10.1002/cpa.21379
Julián Fernández Bonder, Sandra Martínez, and Noemi Wolanski, An optimization problem with volume constraint for a degenerate quasilinear operator, J. Differential Equations 227 (2006), no. 1, 80–101. MR 2233955, DOI 10.1016/j.jde.2006.03.006
Jean Bourgain, Haïm Brezis, and Petru Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications, J. Anal. Math. 87 (2002), 77–101. Dedicated to the memory of Thomas H. Wolff. MR 1945278, DOI 10.1007/BF02868470
Luis A. Caffarelli, Jean-Michel Roquejoffre, and Yannick Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1151–1179. MR 2677613, DOI 10.4171/JEMS/226
Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68, American Mathematical Society, Providence, RI, 2005. MR 2145284, DOI 10.1090/gsm/068
Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260. MR 2354493, DOI 10.1080/03605300600987306
Antonin Chambolle, Erik Lindgren, and Régis Monneau, A Hölder infinity Laplacian, ESAIM Control Optim. Calc. Var. 18 (2012), no. 3, 799–835. MR 3041665, DOI 10.1051/cocv/2011182
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Raúl Ferreira and Mayte Pérez-Llanos, Limit problems for a fractional $p$-Laplacian as $p\to \infty$, NoDEA Nonlinear Differential Equations Appl. 23 (2016), no. 2, Art. 14, 28. MR 3478965, DOI 10.1007/s00030-016-0368-z
Erik Lindgren and Peter Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), no. 1-2, 795–826. MR 3148135, DOI 10.1007/s00526-013-0600-1
Juan Manfredi, Arshak Petrosyan, and Henrik Shahgholian, A free boundary problem for $\infty$-Laplace equation, Calc. Var. Partial Differential Equations 14 (2002), no. 3, 359–384. MR 1899452, DOI 10.1007/s005260100107
Krerley Oliveira and Eduardo V. Teixeira, An optimization problem with free boundary governed by a degenerate quasilinear operator, Differential Integral Equations 19 (2006), no. 9, 1061–1080. MR 2262097
Juan J. Manfredi, Julio D. Rossi, and Stephanie J. Somersille, An obstacle problem for tug-of-war games, Commun. Pure Appl. Anal. 14 (2015), no. 1, 217–228. MR 3299035, DOI 10.3934/cpaa.2015.14.217
Julio D. Rossi and Eduardo V. Teixeira, A limiting free boundary problem ruled by Aronsson’s equation, Trans. Amer. Math. Soc. 364 (2012), no. 2, 703–719. MR 2846349, DOI 10.1090/S0002-9947-2011-05322-5
J. D. Rossi, E. V. Teixeira, and J. M. Urbano, Optimal regularity at the free boundary for the infinity obstacle problem, Interfaces Free Bound. 17 (2015), no. 3, 381–398. MR 3421912, DOI 10.4171/IFB/347
Julio D. Rossi and Peiyong Wang, The limit as $p\to \infty$ in a two-phase free boundary problem for the $p$-Laplacian, Interfaces Free Bound. 18 (2016), no. 1, 115–135. MR 3511343, DOI 10.4171/IFB/359
Eduardo V. Teixeira, The nonlinear optimization problem in heat conduction, Calc. Var. Partial Differential Equations 24 (2005), no. 1, 21–46. MR 2157849, DOI 10.1007/s00526-004-0313-6
Eduardo V. Teixeira, Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint, Interfaces Free Bound. 9 (2007), no. 1, 133–148. MR 2317302, DOI 10.4171/IFB/159
Eduardo V. Teixeira, Optimal design problems in rough inhomogeneous media: existence theory, Amer. J. Math. 132 (2010), no. 6, 1445–1492. MR 2766175
Eduardo V. Teixeira, Regularity for the fully nonlinear dead-core problem, Math. Ann. 364 (2016), no. 3-4, 1121–1134. MR 3466862, DOI 10.1007/s00208-015-1247-3
Eduardo V. Teixeira and Rafayel Teymurazyan, Optimal design problems with fractional diffusions, J. Lond. Math. Soc. (2) 92 (2015), no. 2, 338–352. MR 3404027, DOI 10.1112/jlms/jdv034