Artículo

Blanc, P.; da Silva, J.V.; Rossi, J.D."A limiting free boundary problem with gradient constraint and Tug-of-War games" (2019) Annali di Matematica Pura ed Applicata
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Abstract:

In this manuscript we deal with regularity issues and the asymptotic behaviour (as p→ ∞) of solutions for elliptic free boundary problems of p- Laplacian type (2 ≤ p< ∞): -Δpu(x)+λ0(x)χ{u>0}(x)=0inΩ⊂RN,with a prescribed Dirichlet boundary data, where λ> 0 is a bounded function and Ω is a regular domain. First, we prove the convergence as p→ ∞ of any family of solutions (up)p≥2, as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, {max{-Δ∞u∞,-|∇u∞|+χ{u∞>0}}=0inΩ∩{u∞≥0}u∞=Fon∂Ω.Next, we obtain uniqueness for solutions to this limit problem. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game. © 2019, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

Registro:

Documento: Artículo
Título:A limiting free boundary problem with gradient constraint and Tug-of-War games
Autor:Blanc, P.; da Silva, J.V.; Rossi, J.D.
Filiación:FCEyN, Department of Mathematics, Universidad de Buenos Aires, Ciudad Universitaria-Pabellón, Buenos Aires, C1428EGA, Argentina
Palabras clave:Existence/uniqueness of solutions; Free boundary problems; Lipschitz regularity estimates; Tug-of-War games; ∞-Laplace operator
Año:2019
DOI: http://dx.doi.org/10.1007/s10231-019-00825-0
Handle:http://hdl.handle.net/20.500.12110/paper_03733114_v_n_p_Blanc
Título revista:Annali di Matematica Pura ed Applicata
Título revista abreviado:Ann. Mat. Pura Appl.
ISSN:03733114
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03733114_v_n_p_Blanc

Referencias:

  • Alt, H.W., Phillips, D., A free boundary problem for semilinear elliptic equations (1986) J. Reine Angew. Math., 368, pp. 63-107
  • Aronsson, G., Crandall, M.G., Juutinen, P., A tour of the theory of absolutely minimizing functions (2004) Bull. Am. Math. Soc. (N.S.), 41 (4), pp. 439-505
  • Blanc, P., Pinasco, J.P., Rossi, J.D., Maximal operators for the p- Laplacian family (2017) Pac. J. Math., 287 (2), pp. 257-295
  • Crandall, M.G., Evans, L.C., Gariepy, R.F., Optimal Lipschitz extensions and the infinity Laplacian (2001) Calc. Var. Part. Differ. Equ., 13 (2), pp. 123-139
  • Crandall, M.G., Gunnarsson, G., Wang, P.Y., Uniqueness of ∞- harmonic functions and the eikonal equation (2007) Comm. Part. Differ. Equ., 32 (10-12), pp. 1587-1615
  • Crandall, M.G., Ishii, H., Lions, P.L., User’s guide to viscosity solutions of second order partial differential equations (1992) Bull. Am. Math. Soc. (N.S.), 27 (1), pp. 1-67
  • da Silva, J.V., Rossi, J.D., Salort, A., Maximal solutions for the ∞ - eigenvalue problem Adv. Calc. Var., , https://doi.org/10.1515/acv-2017-0024, to appear
  • Díaz, J.I., (1985) Nonlinear Partial Differential Equations and Free Boundaries Vol. I. Elliptic Equations, 106, pp. vii+323. , Research Notes in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 0-273-08572-7
  • Evans, L.C., Smart, C.K., Everywhere differentiability of infinity harmonic functions (2011) Calc. Var. Part. Differ. Equ., 42 (1-2), pp. 289-299
  • Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient (1993) Arch. Ration. Mech. Anal., 123 (1), pp. 51-74
  • Juutinen, P., Lindqvist, P., Manfredi, J.J., The ∞ -eigenvalue problem (1999) Arch. Ration. Mech. Anal., 148 (2), pp. 89-105
  • Juutinen, P., Parviainen, M., Rossi, J.D., Discontinuous gradient constraints and the infinity Laplacian (2016) Int. Math. Res. Not. IMRN, 8, pp. 2451-2492
  • Karp, L., Kilpeläinen, T., Petrosyan, A., Shahgholian, H., On the porosity of free boundaries in degenerate variational inequalities (2000) J. Differ. Equ., 164 (1), pp. 110-117
  • Koskela, P., Rohde, S., Hausdorff dimension and mean porosity (1997) Math. Ann., 309 (4), pp. 593-609
  • Lee, K.-A., Shahgholian, H., Hausdorff measure and stability for the p -obstacle problem (2 < p< ∞) (2003) J. Differ. Equ., 195 (1), pp. 14-24
  • Lindqvist, P., Lukkari, T., A curious equation involving the ∞ -Laplacian (2010) Adv. Calc. Var., 3 (4), pp. 409-421
  • Manfredi, J.J., Parviainen, M., Rossi, J.D., An asymptotic mean value characterization for p- harmonic functions (2010) Proc. Am. Math. Soc., 138 (3), pp. 881-889
  • Manfredi, J.J., Parviainen, M., Rossi, J.D., On the definition and properties of p -harmonious functions (2012) Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2), pp. 215-241
  • Manfredi, J.J., Regularity for minima of functionals with p -growth (1988) J. Differ. Equ., 76 (2), pp. 203-212
  • Peres, Y., Schramm, O., Sheffield, S., Wilson, D., Tug-of-war and the infinity Laplacian (2009) J. Am. Math. Soc., 22 (1), pp. 167-210
  • Rossi, J.D., Teixeira, E.V., A limiting free boundary problem ruled by Aronsson’s equation (2012) Trans. Am. Math. Soc., 364 (2), pp. 703-719
  • Rossi, J.D., Teixeira, E.V., Urbano, J.M., Optimal regularity at the free boundary for the infinity obstacle problem (2015) Interfaces Free Bound, 17 (3), pp. 381-398
  • Rossi, J.D., Wang, P., The limit as p→ ∞ in a two-phase free boundary problem for the p -Laplacian (2016) Interfaces Free Bound., 18 (1), pp. 115-135

Citas:

---------- APA ----------
Blanc, P., da Silva, J.V. & Rossi, J.D. (2019) . A limiting free boundary problem with gradient constraint and Tug-of-War games. Annali di Matematica Pura ed Applicata.
http://dx.doi.org/10.1007/s10231-019-00825-0
---------- CHICAGO ----------
Blanc, P., da Silva, J.V., Rossi, J.D. "A limiting free boundary problem with gradient constraint and Tug-of-War games" . Annali di Matematica Pura ed Applicata (2019).
http://dx.doi.org/10.1007/s10231-019-00825-0
---------- MLA ----------
Blanc, P., da Silva, J.V., Rossi, J.D. "A limiting free boundary problem with gradient constraint and Tug-of-War games" . Annali di Matematica Pura ed Applicata, 2019.
http://dx.doi.org/10.1007/s10231-019-00825-0
---------- VANCOUVER ----------
Blanc, P., da Silva, J.V., Rossi, J.D. A limiting free boundary problem with gradient constraint and Tug-of-War games. Ann. Mat. Pura Appl. 2019.
http://dx.doi.org/10.1007/s10231-019-00825-0