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Applicable Analysis
An International Journal
Volume 100, 2021 - Issue 10
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Articles

The Dirichlet problem for nonlocal elliptic equations

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Pages 2093-2107 | Received 30 Jun 2018, Accepted 02 Oct 2019, Published online: 14 Oct 2019

References

  • Tian R, Tang Y. Stochastic entropy solutions for stochastic nonlinear transport equations. Entropy. 2018;20(6):395. doi: 10.3390/e20060395
  • Kang J, Tang Y. Value function regularity in option pricing problems under a pure jump model. Appl Math Optim. 2017;76:303–321.
  • Kang J, Tang Y. Asymptotical behavior of partial integral-differential equation on nonsymmetric layered stable processes. Asymptotic Anal. 2017;102:55–70.
  • Applebaum D. Lévy processes and Stochastic calculus. Vol. 93. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press; 2004.
  • Duan J. An introduction to Stochastic dynamics. Vol. 51. New York: Cambridge University Press; 2015.
  • Yanovsky VV, Chechkin AV, Schertzer D, et al. Lévy anomalous diffusion and fractional Fokker–Planck equation. Physica A. 2000;282(1–2):13–34.
  • Alibaud N, Andreianov B, Bendahmane M. Renormalized solutions of the fractional Laplace equation. C R Math. 2010;348(13):759–762.
  • Chen H, Véron L. Semilinear fractional elliptic equations with gradient nonlinearity involving measures. J Funct Anal. 2014;266(8):5467–5492.
  • Felsinger M, Kassmann M, Voigt P. The Dirichlet problem for nonlocal operators. Math Z. 2015;279(3–4):779–809.
  • Ros-Oton X, Serra J. Fractional Laplacian: Pohozaev identity and nonexistence results. C R Acad Sci Paris, Ser I. 2012;350(910):505–508.
  • Ros-Oton X, Serra J. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J Math Pures Appl. 2014;101(3):275–302.
  • Ros-Oton X. Nonlocal elliptic equations in bounded domains: a survey. Publ Mat. 2016;60(1):3–26.
  • Servadei R, Valdinoci E. Weak and viscosity solutions of the fractional Laplace equation. Publ Mat. 2014;58(1):133–154.
  • Du Q, Gunzburger M, Lehoucq RB, et al. Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 2012;54(4):667–696.
  • Gao T, Duan J, Li X, et al. Mean exit time and escape probability for dynamical systems driven by Lévy noises. SIAM J Sci Comput. 2014;36(3):887–906.
  • Ros-Oton X, Serra J. Regularity theory for general stable operators. J Differ Eq. 2016;260(12):8675–8715.
  • Karisen K, Petitta F, Ulusoy S. A duality approach to the fractional Laplacian with measure data. Publ Mat. 2011;55(1):151–161.
  • Chen H, Véron L. Semilinear fractional elliptic equations involving measures. J Differ Eq. 2014;257(5):1457–1486.
  • Lv G, Duan J, He J. Nonlocal elliptic equations involving measures. J Math Anal Appl. 2015;432(2):1106–1118.
  • Kuusi T, Mingione G, Sire Y. Nonlocal equations with measure data. Commun Math Phys. 2015;337(3):1317–1368.
  • Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136(5):521–573.
  • Wu Z, Yin J, Wang C. Elliptic and parabolic equations. Beijing: World Scientific Publishing; 2006.
  • Bogachev VI, Krylov NV, Röckner M. On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm Partial Differ Eq. 2001;26(11):2037–2080.
  • Bogachev VI, Krylov NV, Röckner M. Regularity of invariant measures: the case of non-constant diffusion part. J Funct Anal. 1996;138(1):223–242.
  • Bogachev VI, Krylov NV, Röckner M. Elliptic and parabolic equations for measures. Russ Math Surv. 2009;64(6):973–1078.
  • Barles G, Imbert C. Second-order elliptic integro-differential equations: viscosity solutions theory revisited. Ann I H Poincaré A N. 2008;25(3):567–585.
  • Hoh W, Jacob N. On the Dirichlet problem for pseudo differential operators generating Feller semigroups. J Funct Anal. 1996;137(137):19–48.
  • Lepeltier JP, Marchal B. Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. Ann Inst H Poincaré. 1976;12(1):43–103.
  • Bass RF, Chen ZQ. Systems of equations driven by stable processes. Probab Theory Relat Fields. 2006;134(2):175–214.
  • Priola E. Pathwise uniqueness for singular SDEs driven by stable processes. Osaka J Math. 2012;49(2):421–447.
  • Veretennikov AJ. On strong solutions and explicit formulas for solutions of stochastic integral equations. Math USSR Sbornik. 1980;39(3):434–452.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer-Verlag; 1983.
  • Kallenberg O. Foundations of modern probability. New York (NY): Springer; 1997.
  • Friedman A. Stochastic differential equations and applications. Vol. 2. New York (NY): Springer; 2000.

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