A New Elliptic Measure on Lower Dimensional Sets
|
摘要
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.
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.
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.
引文
[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,ArXiv:1803.07975
[4]Bishop,C.,Jones,P.:Harmonic measure and arclength.Ann.of Math.(2),132,511-547(1990)
[5]Caffarelli,L.,Fabes,E.,Kenig,C.:Completely singular elliptic-harmonic measures.Indiana Univ.Math.J.,30(6),917-924(1981)
[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)
[7]Carbonaro,A.,Dragicevic,O.:Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients.Preprint,arXiv:1611.00653
[8]Cialdea,A.,Mazya,V.:Criterion for the Lp-dissipativity of second order differential operators with complex coefficients.J.Math.Pures Appl.,84(9),1067-1100(2005)
[9]Coifman,R.R.,McIntosh,A.,Meyer,Y.:Lintegrale de Cauchy definit un operateur borne sur L2pour les courbes lipschitziennes.Ann.of Math.(2),116(2),361-387(1982)
[10]Dahlberg,B.E.:Estimates of harmonic measure.Arch.Rational Mech.Anal.,65(3),275-288(1977)
[11]David,G.:Morceaux de graphes lipschitziens et integrales singulières sur une surface.(French)[Pieces of Lipschitz graphs and singular integrals on a surface].Rev.Mat.Iberoamericana,4(1),73-114(1988)
[12]David,G.:Operateurs d’integrale singulière sur les surfaces regulières.Ann.Sci.Ecole Norm.Sup.(4),21(2),225-258(1988)
[13]David,G.:Wavelets and singular integrals on curves and surfaces.Lecture Notes in Mathematics,1465,Springer-Verlag,Berlin,1991
[14]David,G.,Jerison,D.:Lipschitz approximation to hypersurfaces,harmonic measure,and singular integrals.Indiana Univ.Math.J.,39(3),831-845(1990)
[15]David,G.,Semmes,S.:Singular integrals and rectifiable sets in Rn:Beyond Lipschitz graphs.Asterisque,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.,ArXiv:1702.05503,to appear
[19]David,G.,Feneuil,J.,Mayboroda,S.:Dahlberg’s theorem in higher co-dimension.Preprint,arXiv: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)
[22]Dindos,M.,Pipher,J.:Regularity theory for solutions to second order elliptic operators with complex coefficients and the LpDirichlet problem.Preprint,arXiv: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)
[24]Dindos,M.,Petermichl,S.,Pipher,J.:The LpDirichlet problem for second order elliptic operators and a p-adapted square function.J.Funct.Anal.,249(2),372-392(2007)
[25]Dindos,M.,Petermichl,S.,Pipher,J.:BMO solvability and the A∞condition for second order parabolic operators.Ann.Inst.H.Poincare Anal.Non Lineaire,34(5),1155-1180(2017)
[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)
[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)
[28]Fabes,E.,Jerison,D.,Kenig,C.:The Wiener test for degenerate elliptic equations.Ann.Inst.Fourier(Grenoble),32(3),151-182(1982)
[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
[30]Feneuil,J.,Mayboroda,S.,Zihui,Z.:The Dirichlet problem with complex coefficients in higher codimension.In preparation
[31]Garnett,J.,Mourgoglou,M.,Tolsa,X.:Uniform rectifiability in terms of Carleson measure estimates andε-approximability of bounded harmonic functions.Duke Math.J.,167(8),1473-1524(2018)
[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
[33]Gruter,M.,Widman,K.O.:The Green function for uniformly elliptic equations.Manuscripta Math.,37(3),303-342(1982)
[34]Haj lasz,P.,Koskela,P.:Sobolev meets Poincare.C.R.Acad.Sci.Paris Ser.I Math.,320(10),1211-1215(1995)
[35]Haj lasz,P.,Koskela,P.:Sobolev met Poincare.Mem.Amer.Math.Soc.,145(688),(2000)
[36]Hofmann,S.,Kim,S.:The Green function estimates for strongly elliptic systems of second order.Manuscripta Math.,124(2),139-172(2007)
[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)
[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)
[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)
[43]Jerison,D.,Kenig,C.:Boundary behavior of harmonic functions in nontangentially accessible domains.Adv.in Math.,46(1),80-147(1982)
[44]Jones,P.W.:Square functions,Cauchy integrals,analytic capacity,and harmonic measure.Harmonic Analysis and Partial Differential Equations.Lecture Notes in Math.1384,Springer-erlag,1989
[45]Jones,P.W.:Lipschitz and bi-lipschitz functions.Revista Matematica Iberoamericana,4(1),115-122(1988)
[46]Kenig,C.E.:Harmonic analysis techniques for second order elliptic boundary value problems.CBMSRegional Conference Series in Mathematics 83(AMS,Providence,RI,1994)
[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)
[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)
[49]Kenig,C.,Pipher,J.:The Dirichlet problem for elliptic equations with drift terms.Publ.Mat.,45(1),199-217(2001)
[50]Lewis,J.,Nystrom,K.:Quasi-linear PDEs and low-dimensional sets.JEMS,to appear
[51]Lewis,J.,Nystrom,K.,Vogel,A.:On the dimension of p-harmonic measure in space.J.Eur.Math.Soc.(JEMS),15(6),2197-2256(2013)
[52]Mattila,P.,Melnikov,M.,Verdera,J.:The Cauchy integral,analytic capacity,and uniform rectifiability.Ann.of Math.(2),144(1),127-136(1996)
[53]Mayboroda,S.,Zhao,A.:Square function estimates,BMO Dirichlet problem,and absolute continuity of harmonic measure on lower-dimensional sets.Preprint,arXiv:1802.09648
[54]Modica,L.,Mortola,S.:Construction of a singular elliptic-harmonic measure.Manuscripta Math.,33(1),81-98(1980/81)
[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)
[56]Okikiolu.K.:Characterization of subsets of rectifiable curves in Rn.J.of the London Math.Soc.,46,336-348(1992)
[57]Semmes,S.:Analysis vs.geometry on a class of rectifiable hypersurfaces in Rn.Indiana Univ.Math.J.,39(4),1005-1035(1990)
[58]Stampacchia,G.:Le problème de Dirichlet pour lesequations elliptiques du second ordrèa coefficients discontinus.Ann.Inst.Fourier,15,189-258(1965)
[59]Tolsa,X.:Uniform rectifiability,Calderon-Zygmund operators with odd kernel,and quasiorthogonality.Proc.Lond.Math.Soc.(3),98(2),393-426(2009)
[60]Zhao,Z.:BMO solvability and the A∞condition of the elliptic measure in uniform domains.J.Geom.Anal.,arXiv:1602.00717,to appear
[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,ArXiv:1803.07975
[4]Bishop,C.,Jones,P.:Harmonic measure and arclength.Ann.of Math.(2),132,511-547(1990)
[5]Caffarelli,L.,Fabes,E.,Kenig,C.:Completely singular elliptic-harmonic measures.Indiana Univ.Math.J.,30(6),917-924(1981)
[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)
[7]Carbonaro,A.,Dragicevic,O.:Convexity of power functions and bilinear embedding for divergence-form operators with complex coefficients.Preprint,arXiv:1611.00653
[8]Cialdea,A.,Mazya,V.:Criterion for the Lp-dissipativity of second order differential operators with complex coefficients.J.Math.Pures Appl.,84(9),1067-1100(2005)
[9]Coifman,R.R.,McIntosh,A.,Meyer,Y.:Lintegrale de Cauchy definit un operateur borne sur L2pour les courbes lipschitziennes.Ann.of Math.(2),116(2),361-387(1982)
[10]Dahlberg,B.E.:Estimates of harmonic measure.Arch.Rational Mech.Anal.,65(3),275-288(1977)
[11]David,G.:Morceaux de graphes lipschitziens et integrales singulières sur une surface.(French)[Pieces of Lipschitz graphs and singular integrals on a surface].Rev.Mat.Iberoamericana,4(1),73-114(1988)
[12]David,G.:Operateurs d’integrale singulière sur les surfaces regulières.Ann.Sci.Ecole Norm.Sup.(4),21(2),225-258(1988)
[13]David,G.:Wavelets and singular integrals on curves and surfaces.Lecture Notes in Mathematics,1465,Springer-Verlag,Berlin,1991
[14]David,G.,Jerison,D.:Lipschitz approximation to hypersurfaces,harmonic measure,and singular integrals.Indiana Univ.Math.J.,39(3),831-845(1990)
[15]David,G.,Semmes,S.:Singular integrals and rectifiable sets in Rn:Beyond Lipschitz graphs.Asterisque,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.,ArXiv:1702.05503,to appear
[19]David,G.,Feneuil,J.,Mayboroda,S.:Dahlberg’s theorem in higher co-dimension.Preprint,arXiv: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)
[22]Dindos,M.,Pipher,J.:Regularity theory for solutions to second order elliptic operators with complex coefficients and the LpDirichlet problem.Preprint,arXiv: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)
[24]Dindos,M.,Petermichl,S.,Pipher,J.:The LpDirichlet problem for second order elliptic operators and a p-adapted square function.J.Funct.Anal.,249(2),372-392(2007)
[25]Dindos,M.,Petermichl,S.,Pipher,J.:BMO solvability and the A∞condition for second order parabolic operators.Ann.Inst.H.Poincare Anal.Non Lineaire,34(5),1155-1180(2017)
[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)
[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)
[28]Fabes,E.,Jerison,D.,Kenig,C.:The Wiener test for degenerate elliptic equations.Ann.Inst.Fourier(Grenoble),32(3),151-182(1982)
[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
[30]Feneuil,J.,Mayboroda,S.,Zihui,Z.:The Dirichlet problem with complex coefficients in higher codimension.In preparation
[31]Garnett,J.,Mourgoglou,M.,Tolsa,X.:Uniform rectifiability in terms of Carleson measure estimates andε-approximability of bounded harmonic functions.Duke Math.J.,167(8),1473-1524(2018)
[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
[33]Gruter,M.,Widman,K.O.:The Green function for uniformly elliptic equations.Manuscripta Math.,37(3),303-342(1982)
[34]Haj lasz,P.,Koskela,P.:Sobolev meets Poincare.C.R.Acad.Sci.Paris Ser.I Math.,320(10),1211-1215(1995)
[35]Haj lasz,P.,Koskela,P.:Sobolev met Poincare.Mem.Amer.Math.Soc.,145(688),(2000)
[36]Hofmann,S.,Kim,S.:The Green function estimates for strongly elliptic systems of second order.Manuscripta Math.,124(2),139-172(2007)
[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)
[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)
[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)
[43]Jerison,D.,Kenig,C.:Boundary behavior of harmonic functions in nontangentially accessible domains.Adv.in Math.,46(1),80-147(1982)
[44]Jones,P.W.:Square functions,Cauchy integrals,analytic capacity,and harmonic measure.Harmonic Analysis and Partial Differential Equations.Lecture Notes in Math.1384,Springer-erlag,1989
[45]Jones,P.W.:Lipschitz and bi-lipschitz functions.Revista Matematica Iberoamericana,4(1),115-122(1988)
[46]Kenig,C.E.:Harmonic analysis techniques for second order elliptic boundary value problems.CBMSRegional Conference Series in Mathematics 83(AMS,Providence,RI,1994)
[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)
[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)
[49]Kenig,C.,Pipher,J.:The Dirichlet problem for elliptic equations with drift terms.Publ.Mat.,45(1),199-217(2001)
[50]Lewis,J.,Nystrom,K.:Quasi-linear PDEs and low-dimensional sets.JEMS,to appear
[51]Lewis,J.,Nystrom,K.,Vogel,A.:On the dimension of p-harmonic measure in space.J.Eur.Math.Soc.(JEMS),15(6),2197-2256(2013)
[52]Mattila,P.,Melnikov,M.,Verdera,J.:The Cauchy integral,analytic capacity,and uniform rectifiability.Ann.of Math.(2),144(1),127-136(1996)
[53]Mayboroda,S.,Zhao,A.:Square function estimates,BMO Dirichlet problem,and absolute continuity of harmonic measure on lower-dimensional sets.Preprint,arXiv:1802.09648
[54]Modica,L.,Mortola,S.:Construction of a singular elliptic-harmonic measure.Manuscripta Math.,33(1),81-98(1980/81)
[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)
[56]Okikiolu.K.:Characterization of subsets of rectifiable curves in Rn.J.of the London Math.Soc.,46,336-348(1992)
[57]Semmes,S.:Analysis vs.geometry on a class of rectifiable hypersurfaces in Rn.Indiana Univ.Math.J.,39(4),1005-1035(1990)
[58]Stampacchia,G.:Le problème de Dirichlet pour lesequations elliptiques du second ordrèa coefficients discontinus.Ann.Inst.Fourier,15,189-258(1965)
[59]Tolsa,X.:Uniform rectifiability,Calderon-Zygmund operators with odd kernel,and quasiorthogonality.Proc.Lond.Math.Soc.(3),98(2),393-426(2009)
[60]Zhao,Z.:BMO solvability and the A∞condition of the elliptic measure in uniform domains.J.Geom.Anal.,arXiv:1602.00717,to appear
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |