Analysis and geometry on non-smooth domains
Esta nota esta basada en la charla de posesión como Miembro Correspondiente de la Academia Colombiana de Ciencias Exactas Fisicas y Naturales. En ella describo algunos de los resultados recientes en un area de análisis que esta enfocada en entender la relación entre las propiedades geométricas de un...
Saved in:
| Institution: | Academia Colombiana De Ciencias Exactas Fisicas Y Naturales ACCEFYN |
|---|---|
| Main Authors: | , |
| Format: | Artículo de revista |
| Language: | Español |
| Published: |
Academia Colombiana de Ciencias Exactas, Físicas y Naturales
2018-01-12
|
| Subjects: | |
| Online Access: | https://repositorio.accefyn.org.co/handle/001/1009 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| id |
oai:repositorio.accefyn.org.co:001-1009 |
|---|---|
| recordtype |
dspace |
| spelling |
Toro, Tatiana Academia Colombiana de Ciencias Exactas, Físicas y Naturales 2021-11-15T15:21:04Z 2021-11-15T15:21:04Z 2018-01-12 https://repositorio.accefyn.org.co/handle/001/1009 https://doi.org/10.18257/raccefyn.512 Esta nota esta basada en la charla de posesión como Miembro Correspondiente de la Academia Colombiana de Ciencias Exactas Fisicas y Naturales. En ella describo algunos de los resultados recientes en un area de análisis que esta enfocada en entender la relación entre las propiedades geométricas de un dominio y el comportamiento hacia la frontera de las soluciones de ecuaciones diferenciales parciales en este dominio. This paper is a summary of the talk given with the occasion of the author’s induction as Corresponding Member of the Academia Colombiana de Ciencias Exactas Fisicas y Naturales. We describe recent results in an area of analysis which focuses on the relationship between the geometric properties of a domain and the behavior near the boundary of the solutions to canonical PDEs in this domain. application/pdf spa Academia Colombiana de Ciencias Exactas, Físicas y Naturales Bogotá, Colombia Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International https://creativecommons.org/licenses/by-nc/4.0/ info:eu-repo/semantics/openAccess Atribución-NoComercial 4.0 Internacional (CC BY-NC 4.0) Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales Analysis and geometry on non-smooth domains Artículo de revista info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_6501 DataPaper http://purl.org/redcol/resource_type/ARTREF Harmonic measure Elliptic measure Uniform rectifiability A∞-weight Domain of Lipschitz Medida armónica Medida elíptica Rectificabilidad uniforme Peso A∞ Dominio de Lipschitz Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales 41 521 527 161 Estudiantes, Profesores, Comunidad científica colombiana H. Aikawa,Characterization of a uniform domain by theboundary Harnack principle, Harmonic analysis and itsapplications, Yokohama Publ., Yokohama, (2006), 1-17 H. Aikawa,Characterization of a uniform domain by theboundary Harnack principle, Harmonic analysis and itsapplications, Yokohama Publ., Yokohama, (2006), 1-17 M. Akman, M. Badger, S. Hofmann and J. M.Martell,Rectifiability and elliptic measures on 1-sidedNTA domains with Ahlfors-David regular boundaries, toappear, Trans. Amer. Math. Soc. J. Azzam, J. Garnett, M. Mourgoglou, andX. Tolsa,Uniform rectifiability, elliptic measure,square functions and -approximability, preprint 2016,arXiv:1612.02650 J. Azzam, S. Hofmann, J. M. Martell, S.Mayboroda, M. Mourgoglou, X. Tolsa and A. Volberg,Harmonic measure is rectifiable if it is absolutely contin-uous with respect to the co-dimension one Hausdorff mea-sure, C. R. Math. Acad. Sci. Paris 354 (2016), no. 4,351-355 J. Azzam, S. Hofmann, J. M. Martell, S.Mayboroda, M. Mourgoglou, X. Tolsa and A. VolbergRectifiability of harmonic measure,to appear in GAFA . Azzam, S. Hofmann, J. M. Martell, K. Nys-tröm and T. Toro,A new characterization of chord-arcdomains, to appear, J. European Math. Soc M. Akman, S. Hofmann, J. M. Martell and T.Toro, in preparation L. Caffarelli, E. Fabes, S. Mortola and S. Salsa,Boundary behavior of nonnegative solutions of elliptic op-erators in divergence form., Indiana Univ. Math. J.30(1981), no. 4, 621–640 L. Caffarelli, E. Fabes and C. Kenig,Completely sin-gular harmonic-elliptic measures, Indiana U. Math. J.30(1981), 917-924 . Cavera, S. Hofmann and J.M. Martell,Perturba-tions of elliptic operators in 1-sided chord-arc domains, inpreparation B. Dahlberg, On estimates for harmonic measure,Arch. Rat. Mech. Analysis65(1977), 272–288 B. Dahlberg,On estimates for harmonic measure, Arch.Rat. Mech. Analysis65(1977), 272–288 B. Dahlberg,On the absolute continuity of elliptic mea-sure.American Journal of Mathematics 108 (1986),1119-1138 G. David and D. Jerison,Lipschitz approximation to hy-persurfaces, harmonic measure, and singular integrals,Indiana Univ. Math. J.39(1990), no. 3, 831–845 G. David and S. Semmes, Singular integrals andrectifiable sets inRn: Au-dela des graphes lips-chitziens,Asterisque193(1991). G. David and S. Semmes,Analysis of and on Uni-formly Rectifiable Sets, Mathematical Monographs andSurveys38, AMS 1993 E. De Giorgi,Sulla differenziabilita eanaliticita delle es-tremali degli integrali multipli regolari, Mem. Acad. Sci.Torino3(1957), 25-43 M. Dindos, C. Kenig and J. Pipher,BMO solvabil-ity and the A∞condition for elliptic operators, J. Geom.Anal.21(2011), 78 -95. .L EscauriazaThe LpDirichlet problem for small pertur-bations of the Laplacian, Israel J. Math.94(1996), 353-366 L. Evans and R. Gariepy, Measure theory and fineproperties of functions. Revised edition. Textbooks inMathematics. CRC Press, Boca Raton, FL,(2015) R. Fefferman, C. Kenig, J. Pipher,The theory ofweights and the Dirichlet problem for elliptic equations.Ann. of Math. (2) 134 (1991), no. 1, 65–124 . Garnett, M. Mourgoglou, and X. Tolsa,Uniformrectifiability in terms of Carleson measure estimates and -approximability of bounded harmonic functions, preprint2016, arXiv:1611.00264. D. Gilbarg and N. Trudinger, Elliptic partial differ-ential equations of second order. Reprint of the 1998edition. Classics in Mathematics. Springer-Verlag,Berlin, 2001 S. Hofmann, C. Kenig, S. Mayboroda and J.Pipher,Square function/Non-tangential maximal func-tion estimates and the Dirichlet problem for non-symmetric elliptic operatorsto appear in Journal of theAMS S. Hofmann and P. Le,BMO solvability and absolutecontinuity of harmonic measure, arXiv:1607.00418 S. Hofmann and J.M. Martell,Uniform rectifiabil-ity and harmonic measure I: Uniform rectifiability impliesPoisson kernels in Lp, Ann. Sci. École Norm. Sup.47(2014), no. 3, 577–654 S. Hofmann and J.M. Martell,Uniform Rectifiabil-ity and harmonic measure IV: Ahlfors regularity plus Pois-son kernels in Lpimplies uniform rectifiability, preprint,arXiv:1505.06499. S. Hofmann, J.M. Martell, and S. Mayboroda,Uniform rectifiability, Carleson measure estimates,and approximation of harmonic functions,DukeMath. J.165 (2016), no. 12, 2331–2389. S. Hofmann, J. M. Martell, S. Mayboroda, X.Tolsa and A. VolbergAbsolute continuity between thesurface measure and harmonic measure implies rectifiabil-ity,, ArXiv: 1507.04409 . Hofmann, P. Le, J.M. Martell and K, Nyström,The weak-A∞property of harmonic and p-harmonic mea-sures, to appear in Anal. & PDE . Hofmann, P. Le, J.M. Martell and K, Nyström,The weak-A∞property of harmonic and p-harmonic mea-sures, to appear in Anal. & PDE S. Hofmann, J.M. Martell and T. Toro,A∞impliesNTA for a class of variable coefficient elliptic operators, toappear in J. of Diff. Eq., arXiv:1611.09561 S. Hofmann, J.M. Martell and I. Uriarte-Tuero,Uniform Rectifiability and Harmonic Measure II: Poissonkernels in Lpimply uniform rectfiability, Duke Math. J.163(2014), no. 8, 1601–1654 R. Hunt and R. Wheeden,Positive harmonic func-tions on Lipschitz domainsTrans. Amer. Math. Soc.147(1970), 507-527 D. Jerison and C. Kenig,Boundary behavior of harmonicfunctions in nontangentially accessible domains, Adv. inMath.46(1982), no. 1, 80–147. D. Jerison and C. Kenig,Boundary behavior of harmonicfunctions in nontangentially accessible domains, Adv. inMath.46(1982), no. 1, 80–147. C. Kenig, H. Koch, J. Pipher and T. Toro,A newapproach to absolute continuity of elliptic measure, withapplications to non-symmetric equations, Adv. Math.153(2000), no. 2, 231-298 C. Kenig, B. Kirchheim, J. Pipher and T. Toro,Square functions and the A∞property of elliptic measures,J. Geom. Anal.26(2016), no. 3, 2383–2410 C.E. Kenig and J. Pipher,The Dirichlet problem for el-liptic equations with drift terms, Publ. Mat.45, (2001),199–217 C. Kenig and T. Toro,Harmonic measure on locallyflat domainsDuke Math. J. 87 (1997), no. 3, 509–551 C. Kenig and T. Toro,Free boundary regularity forharmonic measures and Poisson kernels, Ann. of Math.(2)150(1999), no. 2, 369-454 C. Kenig and T. Toro,Poisson kernel characteriza-tion of Reifenberg flat chord arc domains, Ann. Sci. ÉcoleNorm. Sup. (4)36(2003), 323-401 W. Littman, G. Stampacchia and H.F. Weinberger,Regular points for elliptic equations with discontinuous coefficientsAnn. Scuola Norm. Sup. Pisa (3)17(1963)43-77O. Martio,Capacity and measure densities, Ann. Acad.Sci. Fenn. Ser. A I Math.4(1979), 109-118 E. Milakis, J. Pipher and T. Toro,Harmonic Anal-ysis on Chord Arc Domains, J. Geom. Anal.23(2013),2091-2157. E. Milakis, J. Pipher and T. Toro,Perturbation ofelliptic operators in chord arc domains, ContemporaryMathematics (AMS)612(2014), 143 -161 E. Milakis and T. Toro,Divergence form operators inReifenberg flat domains, Mathematische Zeitschrift264(2010), 15-41 L. Modica and S. Mortola,Construction of a sin-gular elliptic-harmonic measure, Manuscripta Math.33(1980), 81-98 L. Modica, S. Mortola and S. Salsa,A nonvaria-tional second order elliptic operator with singular ellipticmeasureProc. of Amer. Math. Soc.84(1982), 225-230 J. Moser,On Harnack’s theorem for elliptic differentialequations, Comm. Pure Appl. Math.14(1961) 577-591 . Nash,Continuity of solutions of parabolic and ellipticequations, Amer. J. of Math80(1958), 931-954 . Semmes,A criterion for the boundedness of singularintegrals on on hypersurfaces, Trans. Amer. Math. Soc.311(1989), 501–513 N. Wiener,The Dirichlet problem, J. Math. Phys.3(1924), 127-146. Z. Zhao,BMO solvability and the A∞condition ofthe elliptic measure in uniform domains, preprint,arXiv:1602.00717 T. Toro & Z. Zhao,A∞implies rectifiability for ellipticoperators with V MO coefficients, in preparation http://purl.org/coar/access_right/c_abf2 http://purl.org/coar/version/c_970fb48d4fbd8a85 |
| institution |
Academia Colombiana De Ciencias Exactas Fisicas Y Naturales ACCEFYN |
| collection |
d_repositorio.accefyn.org.co-DSPACE |
| title |
Analysis and geometry on non-smooth domains |
| spellingShingle |
Analysis and geometry on non-smooth domains Toro, Tatiana Toro, Tatiana Academia Colombiana de Ciencias Exactas, Físicas y Naturales Harmonic measure Elliptic measure Uniform rectifiability A∞-weight Domain of Lipschitz Medida armónica Medida elíptica Rectificabilidad uniforme Peso A∞ Dominio de Lipschitz |
| title_short |
Analysis and geometry on non-smooth domains |
| title_full |
Analysis and geometry on non-smooth domains |
| title_fullStr |
Analysis and geometry on non-smooth domains |
| title_full_unstemmed |
Analysis and geometry on non-smooth domains |
| title_sort |
analysis and geometry on non-smooth domains |
| author |
Toro, Tatiana Toro, Tatiana Academia Colombiana de Ciencias Exactas, Físicas y Naturales |
| author_facet |
Toro, Tatiana Toro, Tatiana Academia Colombiana de Ciencias Exactas, Físicas y Naturales |
| building |
Repositorio digital |
| topic |
Harmonic measure Elliptic measure Uniform rectifiability A∞-weight Domain of Lipschitz Medida armónica Medida elíptica Rectificabilidad uniforme Peso A∞ Dominio de Lipschitz |
| topic_facet |
Harmonic measure Elliptic measure Uniform rectifiability A∞-weight Domain of Lipschitz Medida armónica Medida elíptica Rectificabilidad uniforme Peso A∞ Dominio de Lipschitz |
| publishDate |
2018-01-12 |
| language |
Español |
| publisher |
Academia Colombiana de Ciencias Exactas, Físicas y Naturales |
| format |
Artículo de revista |
| description |
Esta nota esta basada en la charla de posesión como Miembro Correspondiente de la Academia Colombiana de Ciencias Exactas Fisicas y Naturales. En ella describo algunos de los resultados recientes en un area de análisis que esta enfocada en entender la relación entre las propiedades geométricas de un dominio y el comportamiento hacia la frontera de las soluciones de ecuaciones diferenciales parciales en este dominio.
This paper is a summary of the talk given with the occasion of the author’s induction as Corresponding Member of the Academia Colombiana de Ciencias Exactas Fisicas y Naturales. We describe recent results in an area of analysis which focuses on the relationship between the geometric properties of a domain and the behavior near the boundary of the solutions to canonical PDEs in this domain.
|
| url |
https://repositorio.accefyn.org.co/handle/001/1009 |
| url_str_mv |
https://repositorio.accefyn.org.co/handle/001/1009 |
| _version_ |
1716733468761653248 |
| score |
11.246474 |
