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A New Elliptic Measure on Lower Dimensional Sets

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Abstract

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n − 1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.

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References

  1. Azzam, J.: Semi-uniform domains and a characterization of the A ∞ property for harmonic measure, preprint

  2. Azzam, J., Hofmann, S., Martell, J. M., et al.: A new characterization of chord-arc domains. JEMS, to appear

  3. Azzam, J., Mourgoglou, M., Tolsa, X.: A geometric characterization of the weak-A ∞ condition for harmonic measure. Preprint, Ar**v:1803.07975

  4. Bishop, C., Jones, P.: Harmonic measure and arclength. Ann. of Math. (2), 132, 511–547 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caffarelli, L., Fabes, E., Kenig, C.: Completely singular elliptic-harmonic measures. Indiana Univ. Math. J., 30(6), 917–924 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli, L., Fabes, E., Mortola, S., et al.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J., 30(4), 621–640 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carbonaro, A., Dragičevič, O.: Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients. Preprint, ar**v:1611.00653

  8. Cialdea, A., Maz’ya, V.: Criterion for the L p-dissipativity of second order differential operators with complex coefficients. J. Math. Pures Appl., 84(9), 1067–1100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Coifman, R. R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes. Ann. of Math. (2), 116(2), 361–387 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dahlberg, B. E.: Estimates of harmonic measure. Arch. Rational Mech. Anal., 65(3), 275–288 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. David, G.: Morceaux de graphes lipschitziens et intégrales singulières sur une surface. (French) [Pieces of Lipschitz graphs and singular integrals on a surface]. Rev. Mat. Iberoamericana, 4(1), 73–114 (1988)

    Article  MathSciNet  Google Scholar 

  12. David, G.: Opérateurs d’intégrale singulière sur les surfaces régulières. Ann. Sci. Ecole Norm. Sup. (4), 21(2), 225–258 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. David, G.: Wavelets and singular integrals on curves and surfaces. Lecture Notes in Mathematics, 1465, Springer-Verlag, Berlin, (1991)

    Google Scholar 

  14. David, G., Jerison, D.: Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals. Indiana Univ. Math. J., 39(3), 831–845 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. David, G., Semmes, S.: Singular integrals and rectifiable sets in Rn Beyond Lipschitz graphs. Astérisque, 193 (1991)

  16. David, G., Semmes, C.: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, 38, American Mathematical Society, Providence, RI, (1993)

  17. David, G., Engelstein, M., Mayboroda, S.: Square functions estimates in co-dimensions larger than 1, in preparation

  18. David, G., Feneuil, J., Mayboroda, S.: Elliptic theory for sets with higher co-dimensional boundaries. Mem. Amer. Math. Soc, Ar**v:1702.05503, to appear

  19. David, G., Feneuil, J., Mayboroda, S.: Dahlberg’s theorem in higher co-dimension. Preprint, ar**v: 1704.00667

  20. David, G., Feneuil, J., Mayboroda, S.: Elliptic theory in domains with boundaries of mixed dimension. In preparation

  21. David, G., Toro, T.: Reifenberg Parameterizations for Sets with Holes. Mem. Amer. Math. Soc, 215(1012), (2012)

    Google Scholar 

  22. Dindoš, M., Pipher, J.: Regularity theory for solutions to second order elliptic operators with complex coefficients and the L p Dirichlet problem. Preprint, ar**v:1612.01568

  23. Dindos, M., Kenig, C., Pipher, J.: BMO solvability and the A ∞ condition for elliptic operators. J. Geom. Anal., 21(1), 78–95 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dindoš, M., Petermichl, S., Pipher, J.: The L p Dirichlet problem for second order elliptic operators and a p-adapted square function. J. Funct. Anal., 249(2), 372–392 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dindoš, M., Petermichl, S., Pipher, J.: BMO solvability and the A ∞ condition for second order parabolic operators. Ann. Inst. H. Poincaré Anal. Non Linéaire, 34(5), 1155–1180 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dong, H., Kim, S.: Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains. Trans. Amer. Math. Soc, 361(6), 3303–3323 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7(1), 77–116 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fabes, E., Jerison, D., Kenig, C.: The Wiener test for degenerate elliptic equations. Ann. Inst. Fourier (Grenoble), 32(3), 151–182 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fabes, E., Jerison, D., Kenig, C.: Boundary behavior of solutions to degenerate elliptic equations. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981). Wadsworth Math. Ser., Wadsworth, Belmont, CA, (1983, 577–589)

    Google Scholar 

  30. Feneuil, J., Mayboroda, S., Zihui, Z.: The Dirichlet problem with complex coefficients in higher co-dimension. In preparation

  31. Garnett, J., Mourgoglou, M., Tolsa, X.: Uniform rectifiability in terms of Carleson measure estimates and ε-approximability of bounded harmonic functions. Duke Math. J., 167(8), 1473–1524 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition

    Google Scholar 

  33. Grüter, M., Widman, K. O.: The Green function for uniformly elliptic equations. Manuscripta Math., 37(3), 303–342 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hajlasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math., 320(10), 1211–1215 (1995)

    MathSciNet  MATH  Google Scholar 

  35. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc., 145(688), (2000)

    Google Scholar 

  36. Hofmann, S., Kim, S.: The Green function estimates for strongly elliptic systems of second order. Manuscripta Math., 124(2), 139–172 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Hofmann, S., Kenig, C. E., Mayboroda, S., et al.: Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators. J. Amer. Math. Soc, 28(2), 483–529 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hofmann, S., Martell, J. M.: On quantitative absolute continuity of harmonic measure and big piece approximation by chord-arc domains, preprint

  39. Hofmann, S., Martell, J. M., Mayboroda, S.: Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions. Duke Math. J., 165(12), 2331–2389 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hofmann, S., Martell, J. M., Mayboroda, S.: Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains, preprint

  41. Hofmann, S., Martell, J. M., Mayboroda, S., et al.: Uniform rectifiability and elliptic operators with small Carleson norm, preprint

  42. Jerison, D., Kenig, C.: The Dirichlet problem in nonsmooth domains. Ann. of Math. (2), 113(2), 367–382 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. in Math., 46(1), 80–147 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  44. Jones, P. W.: Square functions, Cauchy integrals, analytic capacity, and harmonic measure. Harmonic Analysis and Partial Differential Equations. Lecture Notes in Math. 1384, Springererlag, (1989)

    Chapter  Google Scholar 

  45. Jones, P. W.: Lipschitz and bilipschitz functions. Revista Matematica Iberoamericana, 4(1), 115–122 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kenig, C. E.: Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics 83 (AMS, Providence, RI, (1994)

    MATH  Google Scholar 

  47. Kenig, C., Koch, H., Pipher, J., et al.: A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math., 153(2), 231–298 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kenig, C., Kirchheim, B., Pipher, J., et al.: Square Functions and the A ∞ Property of Elliptic Measures. J. Geom. Anal., 26(3), 2383–2410 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kenig, C., Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat., 45(1), 199–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  50. Lewis, J., Nyström, K.: Quasi-linear PDEs and low-dimensional sets. JEMS, to appear

  51. Lewis, J., Nyström, K., Vogel, A.: On the dimension of p-harmonic measure in space. J. Eur. Math. Soc. (JEMS), 15(6), 2197–2256 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  52. Mattila, P., Melnikov, M., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. (2), 144(1), 127–136 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mayboroda, S., Zhao, A.: Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets. Preprint, ar**v:1802.09648

  54. Modica, L., Mortola, S.: Construction of a singular elliptic-harmonic measure. Manuscripta Math., 33(1), 81–98 (1980/81)

    Article  MathSciNet  MATH  Google Scholar 

  55. Nazarov, F., Tolsa, X., Volberg, A.: On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1. Acta Math., 213(2), 237–321 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  56. Okikiolu. K.: Characterization of subsets of rectifiable curves in Rn. J. of the London Math. Soc., 46, 336–348 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  57. Semmes, S.: Analysis vs. geometry on a class of rectifiable hypersurfaces in Rn. Indiana Univ. Math. J., 39(4), 1005–1035 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  58. Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, 15, 189–258 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tolsa, X.: Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality. Proc. Lond. Math. Soc. (3), 98(2), 393–426 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zhao, Z.: BMO solvability and the A ∞ condition of the elliptic measure in uniform domains}. J. Geom. Anal., ar**v:1602.00717, to appear

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Authors and Affiliations

  1. Université Paris-Sud, Laboratoire de Mathématiques, UMR8628, Orsay, F-91405, France

    Guy David

  2. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Joseph Feneuil

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Svitlana Mayboroda

Authors
  1. Guy David
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  2. Joseph Feneuil
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  3. Svitlana Mayboroda
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Correspondence to Svitlana Mayboroda.

Additional information

Dedicated to Carlos Kenig on his 65th birthday with our gratitude for his mathematics and kindness

The first author was partially supported by the ANR, programme blanc GEOMETRYA ANR-12-BS01-0014, the European Community Marie Curie grant MANET 607643 and H2020 grant GHAIA 777822, and the Simons Collaborations in MPS grant 601941, GD. The third author was supported by the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089, the NSF RAISE-TAQ grant DMS 1839077, the Simons Fellowship, and the Simons Foundation grant 563916, SM.

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David, G., Feneuil, J. & Mayboroda, S. A New Elliptic Measure on Lower Dimensional Sets. Acta. Math. Sin.-English Ser. 35, 876–902 (2019). https://doi.org/10.1007/s10114-019-9001-5

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