Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

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Abstract

In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.

Mathematics Subject Classification

35K86 35K87 

References

  1. 1.
    Bensoussan, A., Lions, J.-L.: Applications of variational inequalities in stochastic vontrol. North-Holland, Amsterdam, (1982)Google Scholar
  2. 2.
    Bögelein, V., Duzaar, F., Scheven, C.: The obstacle problem for parabolic minimizers, J. Evol. Equ. (2017).  https://doi.org/10.1007/s00028-017-0384-4
  3. 3.
    Chung, K.L., Walsh, J.B.: Meyer’s theorem on predictability. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 85–93 (1972)CrossRefMATHGoogle Scholar
  4. 4.
    Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28 (4), 741–808 (1999)MathSciNetMATHGoogle Scholar
  5. 5.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Publishing Co., Amsterdam (1978)MATHGoogle Scholar
  6. 6.
    Djehiche B., Hamadène S., Popier, A.: The finite horizon optimal multiple switching problem. SIAM J. Control Optim. 48 (4), 2751–2770 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    El Asri, B., Hamadène, S.: The finite horizon optimal multi-modes switching problem: The viscosity solution approach. Appl. Math. Optim. 60, 213–235 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Getoor, R.K., Sharpe, M.J.: Naturality, standardness, and weak duality for Markov processes. Z. Wahrsch. verw. Gebiete 67, 1–62 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hamadène, S., Jeanblanc, M.: On the starting and stopping problem: application in reversible investments Math. Oper. Res. 32, 182–192 (2007)MathSciNetMATHGoogle Scholar
  10. 10.
    Hamadène, S., Morlais, M.A.: Viscosity solutions of systems of PDEs with interconnected obstacles and multi-modes switching problem. arXiv:1104.2689v2
  11. 11.
    Hamadène, S., Zhang, J.: Switching problem and related system of reflected backward SDEs. Stochastic Process. Appl. 120, 403–426 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hu, Y., Tang, S.: Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Related Fields 147, 89–121 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Klimsiak, T.: Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form. Stochastic Process. Appl. 122, 134–169 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Klimsiak, T.: Cauchy problem for semilinear parabolic equation with time-dependent obstacles: a BSDEs approach. Potential Anal. 39, 99–140 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Klimsiak, T.: Semi-Dirichlet forms, Feynman-Kac functionals and the Cauchy problem for semilinear parabolic equations. J. Funct. Anal. 268, 1205–1240 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Klimsiak, T.: Reflected BSDEs on filtered probability spaces. Stochastic Process. Appl. 125, 4204–4241 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Klimsiak, T.: Reduced measures for semilinear elliptic equations involving Dirichlet operators. Calc. Var. Partial Differential Equations 55, no. 4, Paper No. 78, 27 pp. (2016)Google Scholar
  18. 18.
    Klimsiak, T.: Systems of quasi-variational inequalities related to the switching problem. arXiv:1609.02221v2 (2016)
  19. 19.
    Klimsiak, T., Rozkosz, A.: Dirichlet forms and semilinear elliptic equations with measure data. J. Funct. Anal. 265, 890–925 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Klimsiak, T., Rozkosz, A.: Obstacle problem for semilinear parabolic equations with measure data. J. Evol. Equ. 15, 457–491 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Klimsiak, T., Rozkosz, A.: Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form. NoDEA Nonlinear Differential Equations Appl. 22, 1911–1934 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lenglart, E.; Lépingle, D.; Pratelli, M. Présentation unifiée de certaines inégalités de la théorie des martingales. (French) With an appendix by Lenglart. Lecture Notes in Math. 784, Seminar on Probability, XIV (Paris, 1978/1979) pp. 26–52, Springer, Berlin, 1980Google Scholar
  23. 23.
    Lundström, N., Nyström, K., Olofsson, M.: Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness. Manuscripta Math. 145, 407–432 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lundström, N. Nyström, K. Olofsson, M.: Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type. Ann. Mat. Pura Appl. (4) 193, 1213–1247 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mignot, F., Puel, J.P.: Inéquations d’évolution paraboliques avec convexes dépendant du temps. Applications aux inéquations quasi-variationnelles d’évolution. Arch. Ration. Mech. Anal. 64, 59–91 (1977)CrossRefMATHGoogle Scholar
  26. 26.
    Oshima Y.: Semi-Dirichlet forms and Markov processes. Walter de Gruyter, Berlin (2013)CrossRefMATHGoogle Scholar
  27. 27.
    Oshima, Y.: Some properties of Markov processes associated with time dependent Dirichlet forms. Osaka J. Math. 29, 103–127 (1992)MathSciNetMATHGoogle Scholar
  28. 28.
    Peng, S.: Monotonic Limit Theorem of BSDE and Nonlinear Decomposition Theorem of Doob-Meyers Type. Probab. Theory Related Fields 113, 473–499 (1999)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Petitta, F., Ponce, A.C., Porretta, A.: Diffuse measures and nonlinear parabolic equations. J. Evol. Equ. 11, 861–905 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Pierre, M.: Problèmes d’évolution avec contraintes unilatérales et potentiel paraboliques. Comm. Partial Differential Equations 4, 1149–1197 (1979)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Pierre, M.: Représentant précis d’un potentiel parabolique. Séminaire de Théorie du Potentiel, Paris, No. 5, Lecture Notes in Math. 814, 186–228 (1980)CrossRefGoogle Scholar
  32. 32.
    Pierre, M.: Parabolic capacity and Sobolev spaces, Siam J. Math. Anal. 14, 522–533 (1983)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Stannat, W.: The theory of generalized Dirichlet forms and its applications in analysis and stochastics. Mem. Amer. Math. Soc. 142, no. 678, viii+101 pp. (1999)Google Scholar
  34. 34.
    M. Topolewski, Reflected BSDEs with general filtration and two completely separated barriers. arXiv:1611.06745 (2016)

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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