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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Azzam, J.: Semi-uniform domains and a characterization of the A ∞ property for harmonic measure, preprint
Azzam, J., Hofmann, S., Martell, J. M., et al.: A new characterization of chord-arc domains. JEMS, to appear
Azzam, J., Mourgoglou, M., Tolsa, X.: A geometric characterization of the weak-A ∞ condition for harmonic measure. Preprint, ArXiv:1803.07975
Bishop, C., Jones, P.: Harmonic measure and arclength. Ann. of Math. (2), 132, 511–547 (1990)
Caffarelli, L., Fabes, E., Kenig, C.: Completely singular elliptic-harmonic measures. Indiana Univ. Math. J., 30(6), 917–924 (1981)
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)
Carbonaro, A., Dragičevič, O.: Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients. Preprint, arXiv:1611.00653
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)
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)
Dahlberg, B. E.: Estimates of harmonic measure. Arch. Rational Mech. Anal., 65(3), 275–288 (1977)
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)
David, G.: Opérateurs d’intégrale singulière sur les surfaces régulières. Ann. Sci. Ecole Norm. Sup. (4), 21(2), 225–258 (1988)
David, G.: Wavelets and singular integrals on curves and surfaces. Lecture Notes in Mathematics, 1465, Springer-Verlag, Berlin, (1991)
David, G., Jerison, D.: Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals. Indiana Univ. Math. J., 39(3), 831–845 (1990)
David, G., Semmes, S.: Singular integrals and rectifiable sets in Rn Beyond Lipschitz graphs. Astérisque, 193 (1991)
David, G., Semmes, C.: Analysis of and on uniformly rectifiable sets. Mathematical Surveys and Monographs, 38, American Mathematical Society, Providence, RI, (1993)
David, G., Engelstein, M., Mayboroda, S.: Square functions estimates in co-dimensions larger than 1, in preparation
David, G., Feneuil, J., Mayboroda, S.: Elliptic theory for sets with higher co-dimensional boundaries. Mem. Amer. Math. Soc, ArXiv:1702.05503, to appear
David, G., Feneuil, J., Mayboroda, S.: Dahlberg’s theorem in higher co-dimension. Preprint, arXiv: 1704.00667
David, G., Feneuil, J., Mayboroda, S.: Elliptic theory in domains with boundaries of mixed dimension. In preparation
David, G., Toro, T.: Reifenberg Parameterizations for Sets with Holes. Mem. Amer. Math. Soc, 215(1012), (2012)
Dindoš, M., Pipher, J.: Regularity theory for solutions to second order elliptic operators with complex coefficients and the L p Dirichlet problem. Preprint, arXiv:1612.01568
Dindos, M., Kenig, C., Pipher, J.: BMO solvability and the A ∞ condition for elliptic operators. J. Geom. Anal., 21(1), 78–95 (2011)
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)
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)
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)
Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7(1), 77–116 (1982)
Fabes, E., Jerison, D., Kenig, C.: The Wiener test for degenerate elliptic equations. Ann. Inst. Fourier (Grenoble), 32(3), 151–182 (1982)
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)
Feneuil, J., Mayboroda, S., Zihui, Z.: The Dirichlet problem with complex coefficients in higher co-dimension. In preparation
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)
Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition
Grüter, M., Widman, K. O.: The Green function for uniformly elliptic equations. Manuscripta Math., 37(3), 303–342 (1982)
Hajlasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math., 320(10), 1211–1215 (1995)
Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc., 145(688), (2000)
Hofmann, S., Kim, S.: The Green function estimates for strongly elliptic systems of second order. Manuscripta Math., 124(2), 139–172 (2007)
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)
Hofmann, S., Martell, J. M.: On quantitative absolute continuity of harmonic measure and big piece approximation by chord-arc domains, preprint
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)
Hofmann, S., Martell, J. M., Mayboroda, S.: Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains, preprint
Hofmann, S., Martell, J. M., Mayboroda, S., et al.: Uniform rectifiability and elliptic operators with small Carleson norm, preprint
Jerison, D., Kenig, C.: The Dirichlet problem in nonsmooth domains. Ann. of Math. (2), 113(2), 367–382 (1981)
Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in nontangentially accessible domains. Adv. in Math., 46(1), 80–147 (1982)
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)
Jones, P. W.: Lipschitz and bilipschitz functions. Revista Matematica Iberoamericana, 4(1), 115–122 (1988)
Kenig, C. E.: Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series in Mathematics 83 (AMS, Providence, RI, (1994)
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)
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)
Kenig, C., Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat., 45(1), 199–217 (2001)
Lewis, J., Nyström, K.: Quasi-linear PDEs and low-dimensional sets. JEMS, to appear
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)
Mattila, P., Melnikov, M., Verdera, J.: The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. (2), 144(1), 127–136 (1996)
Mayboroda, S., Zhao, A.: Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets. Preprint, arXiv:1802.09648
Modica, L., Mortola, S.: Construction of a singular elliptic-harmonic measure. Manuscripta Math., 33(1), 81–98 (1980/81)
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)
Okikiolu. K.: Characterization of subsets of rectifiable curves in Rn. J. of the London Math. Soc., 46, 336–348 (1992)
Semmes, S.: Analysis vs. geometry on a class of rectifiable hypersurfaces in Rn. Indiana Univ. Math. J., 39(4), 1005–1035 (1990)
Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, 15, 189–258 (1965)
Tolsa, X.: Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality. Proc. Lond. Math. Soc. (3), 98(2), 393–426 (2009)
Zhao, Z.: BMO solvability and the A ∞ condition of the elliptic measure in uniform domains}. J. Geom. Anal., arXiv:1602.00717, to appear
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-019-9001-5
Keywords
- Elliptic measure in higher codimension
- degenerate elliptic operators
- absolute continuity
- Dahlberg’s theorem
- Dirichlet solvability.