On the interior Bernoulli free boundary problem for the fractional Laplacian on an interval

• Kulczycki, Tadeusz ^[1] ; Wszoła, Jacek ^[1]
  1. [1] Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb. Wyspiańskiego 27, 50-370, Wrocław, Poland
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 1, 2025, págs. 11-34
• Idioma: inglés
• DOI: 10.1007/s13348-023-00417-5
• Enlaces
  • Texto completo
• Resumen

  • We study the structure of solutions of the interior Bernoulli free boundary problem for (-\Delta )^{\alpha /2} on an interval D with parameter \lambda > 0. In particular, we show that there exists a constant \lambda _{\alpha ,D} > 0 (called the Bernoulli constant) such that the problem has no solution for \lambda \in (0,\lambda _{\alpha ,D}) , at least one solution for \lambda = \lambda _{\alpha ,D} and at least two solutions for \lambda > \lambda _{\alpha ,D}. We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for \alpha = 1 that there exists solutions of the interior Bernoulli free boundary problem for (-\Delta )^{\alpha /2} on an interval which are not minimizers of the corresponding variational problem.

• Referencias bibliográficas
  • Acker, A.: Uniqueness and monotonicity of solutions for the interior Bernoulli free boundary problem in the convex n-dimensional case’. Nonlinear...
  • Alt, H.W., Caffarelli, L.A.: Existence and regularity for a minimum problem with a free boundary. J. Reine Angew. Math. 325, 105–144 (1981)
  • Bianchini, C., Salani, P.: Concavity properties for elliptic free boundary problems. Nonlinear Anal. 71(10), 4461–4470 (2009)
  • Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Stat. 20, 293–335 (2000)
  • Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondracek, Z.: Potential Analysis of Stable Processes and its Extensions,...
  • Bogdan, K., Kulczycki, T., Kwaśnicki, M.: Estimates and structure of a-harmonic functions. Probab. Theory Relat. Fields 140, 345–381 (2008)
  • Caffarelli, L.A., Mellet, A., Sire, Y.: Traveling waves for a boundary reaction–diffusion equation. Adv. Math. 230(2), 433–457 (2012)
  • Caffarelli, L.A., Roquejoffre, J.-M., Sire, Y.: Variational problems with free boundaries for the fractional Laplacian. J. Eur. Math. Soc....
  • Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
  • Cardaliaguet, P., Tahraoui, R.: Some uniqueness results for the Bernoulli interior free-boundary problems in convex domains. Electron. J....
  • Chen, Z.-Q.: Multidimensional symmetric stable processes. Korean J. Comput. Appl. Math. 6(2), 227–266 (1999)
  • De Silva, D., Roquejoffre, J.M.: Regularity in a one-phase free boundary problem for the fractional Laplacian. Ann. Inst. H. Poincare Anal....
  • De Silva, D., Savin, O.: C regularity of certain thin free boundaries. Indiana Univ. Math. J. 64(5), 1575–1608 (2015)
  • De Silva, D., Savin, O.: Regularity of Lipschitz free boundaries for the thin one-phase problem. J. Eur. Math. Soc. 17(6), 1293–1326 (2015)
  • De Silva, D., Savin, O., Sire, Y.: A one-phase problem for the fractional Laplacian: regularity of flat free boundaries. Bull. Inst. Math....
  • Engelstein, M., Kauranen, A., Prats, M., Sakellaris, G., Sire, Y.: Minimizers for the thin one-phase free boundary problem. Commun. Pure Appl....
  • Fernández-Real, X., Ros-Oton, X.: Stable cones in the thin one-phase problem. Am. J. Math. (in press) (2022)
  • Flucher, M., Rumpf, M.: Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486, 165–204...
  • Henrot, A., Shahgholian, H.: Convexity of free boundaries with Bernoulli type boundary condition. Nonlinear Anal. 28(5), 815–823 (1997)
  • Henrot, A., Shahgholian, H.: Existence of classical solution to a free boundary problem for the p-Laplace operator: (II) the interior convex...
  • Jarohs, S., Kulczycki, T., Salani, P.: On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian....
  • Kulczycki, T.: Gradient estimates of q-harmonic functions of fractional Schrödinger operator. Potential Anal. 39(1), 69–98 (2013)