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Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators

  • Nestor GuillenEmail author
  • Jun Kitagawa
  • Russell W. Schwab
Article
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Abstract

Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.

Keywords

Dirichlet-to-Neumann Integro-differential Nonlocal Elliptic equation Boundary process Fully nonlinear Levy measures Boundary operators 

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Notes

Acknowledgments

The authors all acknowledge partial support from the NSF leading to the completion of this work: N. Guillen DMS-1201413 and DMS-1700307; J. Kitagawa DMS-1700094; R. Schwab DMS-1665285. They would like to thank Rodrigo Bañuelos and Renming Song for helpful information on background results appearing in Section ??. They would also like to thank the anonymous referee for suggesting some helpful changes to the presentation of the results.

References

  1. 1.
    Barles, G: Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154(1), 191–224 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bass, R.F., Kassmann, M.: Hölder continuity of harmonic functions with respect to operators of variable order. Comm. Partial Differ. Equ. 30(7–9), 1249–1259 (2005)zbMATHGoogle Scholar
  3. 3.
    Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230(4–6), 1859–1894 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bony, J-M, Courrège, P., Priouret, P.: Sur la forme intégro-différentielle du générateur infinitésimal d’un semi-groupe de Feller sur une variété différentiable. C. R. Acad. Sci. Paris Sér. A-B 263, A207–A210 (1966)zbMATHGoogle Scholar
  6. 6.
    Caffarelli, LA., Cabré, X: Fully Nonlinear Elliptic Equations, Volume 43 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1995)Google Scholar
  7. 7.
    Caffarelli, L, Silvestre, L: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62(5), 597–638 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30(4), 621–640 (1981)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Caffarelli, L., Crandall, M.G., Kocan, M., Swięch, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49 (4), 365–397 (1996)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Caffarelli, L.A., Souganidis, P.E., Wang, L.: Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58(3), 319–361 (2005)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lara, HC: Regularity for fully non linear equations with non local drift. arXiv:1210.4242 [math.AP] (2012)
  12. 12.
    Lara, HC, Dávila, G: Regularity for solutions of non local parabolic equations. Calc. Var. PDE, published online (2012)Google Scholar
  13. 13.
    Lara, HC, Dávila, G: Regularity for solutions of nonlocal, nonsymmetric equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29(6), 833–859 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chang-Lara, HA, Dávila, G: Hölder estimates for non-local parabolic equations with critical drift. J. Differ. Equ. 260(5), 4237–4284 (2016)zbMATHGoogle Scholar
  15. 15.
    Cho, S: Two-sided global estimates of the Green’s function of parabolic equations. Potential Anal. 25(4), 387–398 (2006)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Courrege, P: Sur la forme intégro-différentielle des opérateurs de \( c^{\infty }_k\) dans c satisfaisant au principe du maximum. Séminaire Brelot-Choquet-Deny. Théorie du Potentiel 10 (1), 1–38 (1965)Google Scholar
  17. 17.
    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fabes, E., Garofalo, N., Marí n Malave, S., Salsa, S.: Fatou theorems for some nonlinear elliptic equations. Rev. Mat. Iberoamericana 4(2), 227–251 (1988)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Feldman, M: Regularity of Lipschitz free boundaries in two-phase problems for fully nonlinear elliptic equations. Ind. Univ. Math. J. 50(3), 1171–1200 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionzbMATHGoogle Scholar
  21. 21.
    Grüter, M, Widman, K-O: The green function for uniformly elliptic equations. Manuscripta Math. 37(3), 303–342 (1982)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Guillen, N, Schwab, RW: Aleksandrov–Bakelman–Pucci type estimates for integro-differential equations. Arch. Ration. Mech. Anal. 206(1), 111–157 (2012).  https://doi.org/10.1007/s00205-012-0529-0 MathSciNetzbMATHGoogle Scholar
  23. 23.
    Guillen, N, Schwab, RW: Min-max formulas for nonlocal elliptic operators. arXiv preprint arXiv:1606.08417 (2016)
  24. 24.
    Guillen, N, Schwab, R.W.: Neumann homogenization via integro-differential operators. Discrete Contin. Dynam. Syst 36(7), 3677–3703 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Halmos, P.R.: Measure Theory. D. Van Nostrand Company, Inc., New York (1950)zbMATHGoogle Scholar
  26. 26.
    Hsu, P: On excursions of reflecting Brownian motion. Trans. Am. Math. Soc. 296(1), 239–264 (1986)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hu, B, Nicholls, D.P.: Analyticity of Dirichlet-Neumann operators on Hölder and Lipschitz domains. SIAM J. Math. Anal. 37(1), 302–320 (2005)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Hueber, H., Sieveking, M.: Uniform bounds for quotients of Green functions on C 1,1-domains. Ann. Inst. Fourier (Grenoble) 32(1), vi, 105–117 (1982)zbMATHGoogle Scholar
  29. 29.
    Ikeda, N, Watanabe, S: Stochastic differential equations and diffusion processes. Elsevier (1981)Google Scholar
  30. 30.
    Ishii, H., Lions, P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equ. 83(1), 26–78 (1990)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Jerison, DS, Kenig, CE: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80–147 (1982)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Jost, J: Riemannian Geometry and Geometric Analysis, 6th edn. Universitext. Springer, Heidelberg (2011)Google Scholar
  33. 33.
    Kassmann, M., Mimica, A.: Analysis of jump processes with nondegenerate jumping kernels. Stochastic Process. Appl. 123(2), 629–650 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Kassmann, M, Mimica, A: Intrinsic scaling properties for nonlocal operators. J. Eur. Math. Soc. (JEMS), to appearGoogle Scholar
  35. 35.
    Kassmann, M., Rang, M., Schwab, R.W.: Integro-differential equations with nonlinear directional dependence. Ind. Univ. Math. J. 63(5), 1467–1498 (2014)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Kenig, CE: Potential theory of non-divergence form elliptic equations. In: Dirichlet Forms, pp 89–128. Springer (1993)Google Scholar
  37. 37.
    Kim, S, Kim, Y-C, Lee, K-A: Regularity for fully nonlinear integro-differential operators with regularly varying kernels. Potential Anal. 44(4), 673–705 (2016)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Krylov, N.V., Safonov, M.V.: A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 161–175, 239 (1980)MathSciNetGoogle Scholar
  39. 39.
    Lieberman, G.M., Trudinger, N.S.: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Am. Math. Soc. 295(2), 509–546 (1986)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Lions, P.-L., Trudinger, N.S.: Linear oblique derivative problems for the uniformly elliptic Hamilton-Jacobi-Bellman equation. Math. Z. 191(1), 1–15 (1986)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17, 43–77 (1963)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Petersen, P: Riemannian Geometry, Volume 171 of Graduate Texts in Mathematics, 3rd edn. Springer, Cham (2016)Google Scholar
  43. 43.
    Sakai, T.: Riemannian Geometry, Volume 149 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1996). Translated from the 1992 Japanese original by the authorGoogle Scholar
  44. 44.
    Sato, K-I, Ueno, T: Multi-dimensional diffusion and the markov process on the boundary. J. Math. Kyoto Univ. 4(3), 529–605 (1965)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Schwab, R.W., Silvestre, L.: Regularity for parabolic integro-differential equations with very irregular kernels. Analysis and PDE, to appear (2015)Google Scholar
  46. 46.
    Silvestre, L: Hölder estimates for solutions of integro-differential equations like the fractional Laplace. Indiana Univ. Math. J. 55(3), 1155–1174 (2006)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Silvestre, L: On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion. Adv. Math. 226(2), 2020–2039 (2011)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Silvestre, L, Sirakov, B: Boundary regularity for viscosity solutions of fully nonlinear elliptic equations. Comm. Partial Differ. Equ. 39(9), 1694–1717 (2014)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Song, R, Vondracek, Z: Harnack inequality for some classes of markov processes. Math. Z. 246(1), 177–202 (2004)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Zhao, ZX: Uniform boundedness of conditional gauge and Schrödinger equations. Comm. Math. Phys. 93(1), 19–31 (1984)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Nestor Guillen
    • 1
    Email author
  • Jun Kitagawa
    • 2
  • Russell W. Schwab
    • 2
  1. 1.Department of MathematicsUniversity of Massachusetts, AmherstAmherstUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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