A limiting free boundary problem ruled by Aronsson's equation

Authors:
Julio D. Rossi and Eduardo V. Teixeira

Journal:
Trans. Amer. Math. Soc. **364** (2012), 703-719

MSC (2010):
Primary 35R35, 35J70, 62K05, 49L25

Published electronically:
September 13, 2011

MathSciNet review:
2846349

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the behavior of a -Dirichlet optimal design problem with volume constraint for large. As the limit of goes to infinity, we find a limiting free boundary problem governed by the infinity-Laplacian operator. We find a necessary and sufficient condition for uniqueness of the limiting problem and, under such a condition, we determine precisely the optimal configuration for the limiting problem. Finally, we establish convergence results for the free boundaries.

**1.**A. Acker and R. Meyer,*A free boundary problem for the 𝑝-Laplacian: uniqueness, convexity, and successive approximation of solutions*, Electron. J. Differential Equations (1995), No. 08, approx. 20 pp. (electronic). MR**1334863****2.**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**, 10.1137/0324011**3.**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. MR**618549****4.**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****5.**Gunnar Aronsson, Michael G. Crandall, and Petri Juutinen,*A tour of the theory of absolutely minimizing functions*, Bull. Amer. Math. Soc. (N.S.)**41**(2004), no. 4, 439–505. MR**2083637**, 10.1090/S0273-0979-04-01035-3**6.**T. Bhattacharya, E. DiBenedetto, and J. Manfredi,*Limits as 𝑝→∞ of Δ_{𝑝}𝑢_{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****7.**Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions,*User’s guide to viscosity solutions of second order partial differential equations*, Bull. Amer. Math. Soc. (N.S.)**27**(1992), no. 1, 1–67. MR**1118699**, 10.1090/S0273-0979-1992-00266-5**8.**Donatella Danielli and Arshak Petrosyan,*A minimum problem with free boundary for a degenerate quasilinear operator*, Calc. Var. Partial Differential Equations**23**(2005), no. 1, 97–124. MR**2133664**, 10.1007/s00526-004-0294-5**9.**L. C. Evans and W. Gangbo,*Differential equations methods for the Monge-Kantorovich mass transfer problem*, Mem. Amer. Math. Soc.**137**(1999), no. 653, viii+66. MR**1464149**, 10.1090/memo/0653**10.**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**, 10.1016/j.jde.2006.03.006**11.**J. García-Azorero, J. J. Manfredi, I. Peral, and J. D. Rossi,*The Neumann problem for the ∞-Laplacian and the Monge-Kantorovich mass transfer problem*, Nonlinear Anal.**66**(2007), no. 2, 349–366. MR**2279530**, 10.1016/j.na.2005.11.030**12.**Antoine Henrot and Henrik Shahgholian,*Convexity of free boundaries with Bernoulli type boundary condition*, Nonlinear Anal.**28**(1997), no. 5, 815–823. MR**1422187**, 10.1016/0362-546X(95)00192-X**13.**Antoine Henrot and Henrik 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. MR**1752296**, 10.1515/crll.2000.031**14.**Antoine Henrot and Henrik Shahgholian,*Existence of classical solutions to a free boundary problem for the 𝑝-Laplace operator. II. The interior convex case*, Indiana Univ. Math. J.**49**(2000), no. 1, 311–323. MR**1777029**, 10.1512/iumj.2000.49.1711**15.**Pekka Koskela, Juan J. Manfredi, and Enrique Villamor,*Regularity theory and traces of 𝒜-harmonic functions*, Trans. Amer. Math. Soc.**348**(1996), no. 2, 755–766. MR**1311911**, 10.1090/S0002-9947-96-01430-4**16.**John L. Lewis and Andrew L. Vogel,*Uniqueness in a free boundary problem*, Comm. Partial Differential Equations**31**(2006), no. 10-12, 1591–1614. MR**2273966**, 10.1080/03605300500455909**17.**C. Lederman,*A free boundary problem with a volume penalization*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**23**(1996), no. 2, 249–300. MR**1433424****18.**Juan Manfredi, Arshak Petrosyan, and Henrik Shahgholian,*A free boundary problem for ∞-Laplace equation*, Calc. Var. Partial Differential Equations**14**(2002), no. 3, 359–384. MR**1899452**, 10.1007/s005260100107**19.**Sandra Martínez,*An optimization problem with volume constraint in Orlicz spaces*, J. Math. Anal. Appl.**340**(2008), no. 2, 1407–1421. MR**2390940**, 10.1016/j.jmaa.2007.09.061**20.**Sandra Martínez and Noemi Wolanski,*A minimum problem with free boundary in Orlicz spaces*, Adv. Math.**218**(2008), no. 6, 1914–1971. MR**2431665**, 10.1016/j.aim.2008.03.028**21.**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****22.**Eduardo V. Teixeira,*The nonlinear optimization problem in heat conduction*, Calc. Var. Partial Differential Equations**24**(2005), no. 1, 21–46. MR**2157849**, 10.1007/s00526-004-0313-6**23.**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**, 10.4171/IFB/159**24.**Eduardo V. Teixeira,*A variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary*, Ann. Inst. H. Poincaré Anal. Non Linéaire**25**(2008), no. 4, 633–658 (English, with English and French summaries). MR**2436786**, 10.1016/j.anihpc.2007.02.006**25.**E. V. Teixeira,*Optimal design problems in rough inhomogeneous media. Existence theory.*Preprint. arXiv:0710.2936.**26.**E. V. Teixeira,*Optimal design problems in rough inhomogeneous media. Free boundary regularity theory.*In preparation.

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Additional Information

**Julio D. Rossi**

Affiliation:
Departamento de Análisis Matemático, Universidad de Alicante, Alicante, Spain

Email:
jrossi@dm.uba.ar

**Eduardo V. Teixeira**

Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici - Bloco 914, Fortaleza, CE - Brazil 60.455-760

Email:
eteixeira@ufc.br

DOI:
https://doi.org/10.1090/S0002-9947-2011-05322-5

Keywords:
Optimal design,
free boundary problems,
infinite Laplacian

Received by editor(s):
March 24, 2009

Received by editor(s) in revised form:
February 9, 2010

Published electronically:
September 13, 2011

Article copyright:
© Copyright 2011
American Mathematical Society