Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators
HTML articles powered by AMS MathViewer
- by Juan Cavero, Steve Hofmann and José María Martell PDF
- Trans. Amer. Math. Soc. 371 (2019), 2797-2835 Request permission
Abstract:
Let $\Omega \subset \mathbb {R}^{n+1}$, $n\ge 2$, be a 1-sided chord-arc domain; that is, a domain which satisfies interior corkscrew and Harnack chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path connectedness), and whose boundary $\partial \Omega$ is $n$-dimensional Ahlfors regular. Consider $L_0$ and $L$ two real symmetric divergence form elliptic operators, and let $\omega _{L_0}$, $\omega _L$ be the associated elliptic measures. We show that if $\omega _{L_0}\in A_\infty (\sigma )$, where $\sigma =H^n{\left |_{ {\partial \Omega }}\right .}$, and $L$ is a perturbation of $L_0$ (in the sense that the discrepancy between $L_0$ and $L$ satisfies certain Carleson measure condition), then $\omega _L\in A_\infty (\sigma )$. Moreover, if $L$ is a sufficiently small perturbation of $L_0$, then one can preserve the reverse Hölder classes; that is, if for some $1<p<\infty$, one has $\omega _{L_0}\in RH_p(\sigma )$, then $\omega _{L}\in RH_p(\sigma )$. Equivalently, if the Dirichlet problem with data in $L^{p’}(\sigma )$ is solvable for $L_0$, then it is for $L$ also. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg; Fefferman, Kenig, and Pipher; and Milakis, Pipher, and Toro in more benign settings. As a consequence of our methods, we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $A_\infty (\sigma )$, then necessarily $\Omega$ is in fact a nontangentially accessible domain (and hence chord-arc), and therefore its boundary is uniformly rectifiable.References
- Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian, Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators, Acta Math. 187 (2001), no. 2, 161–190. MR 1879847, DOI 10.1007/BF02392615
- P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson measures, trees, extrapolation, and $T(b)$ theorems, Publ. Mat. 46 (2002), no. 2, 257–325. MR 1934198, DOI 10.5565/PUBLMAT_{4}6202_{0}1
- Jonas Azzam, Steve Hofmann, José María Martell, Kaj Nyström, and Tatiana Toro, A new characterization of chord-arc domains, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 967–981. MR 3626548, DOI 10.4171/JEMS/685
- J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), no. 3, 477–483. MR 874032, DOI 10.1007/BF01389238
- Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
- L. Carleson and J. Garnett. Interpolating sequences and separation properties. J. Analyse Math., 28:273–299, 1975.
- J. Cavero, S. Hofmann, J.M. Martell, and T. Toro. Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates. Work in progress (2017).
- Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
- Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119–1138. MR 859772, DOI 10.2307/2374598
- G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (1990), no. 3, 831–845. MR 1078740, DOI 10.1512/iumj.1990.39.39040
- G. David and S. Semmes, Singular integrals and rectifiable sets in $\textbf {R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517
- Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
- Luis Escauriaza, The $L^p$ Dirichlet problem for small perturbations of the Laplacian, Israel J. Math. 94 (1996), 353–366. MR 1394581, DOI 10.1007/BF02762711
- R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), no. 1, 65–124. MR 1114608, DOI 10.2307/2944333
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- Michael Grüter and Kjell-Ove Widman, The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), no. 3, 303–342. MR 657523, DOI 10.1007/BF01166225
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- Steve Hofmann, Phi Le, José María Martell, and Kaj Nyström, The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability, Anal. PDE 10 (2017), no. 3, 513–558. MR 3641879, DOI 10.2140/apde.2017.10.513
- Steve Hofmann and John L. Lewis, The Dirichlet problem for parabolic operators with singular drift terms, Mem. Amer. Math. Soc. 151 (2001), no. 719, viii+113. MR 1828387, DOI 10.1090/memo/0719
- Steve Hofmann and José María Martell, A note on $A_\infty$ estimates via extrapolation of Carleson measures, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 44, Austral. Nat. Univ., Canberra, 2010, pp. 143–166. MR 2655385
- Steve Hofmann and José María Martell, $A_\infty$ estimates via extrapolation of Carleson measures and applications to divergence form elliptic operators, Trans. Amer. Math. Soc. 364 (2012), no. 1, 65–101. MR 2833577, DOI 10.1090/S0002-9947-2011-05397-3
- Steve Hofmann and José María Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 3, 577–654 (English, with English and French summaries). MR 3239100, DOI 10.24033/asens.2223
- S. Hofmann and J.M. Martell, and T. Toro. General divergence form elliptic operators on domains with ADR boundaries, and on 1-sided NTA domains. Work in progress, 2014.
- Steve Hofmann, José María Martell, and Tatiana Toro, $A_\infty$ implies NTA for a class of variable coefficient elliptic operators, J. Differential Equations 263 (2017), no. 10, 6147–6188. MR 3693172, DOI 10.1016/j.jde.2017.06.028
- Steve Hofmann, José María Martell, and Ignacio Uriarte-Tuero, Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^p$ imply uniform rectifiability, Duke Math. J. 163 (2014), no. 8, 1601–1654. MR 3210969, DOI 10.1215/00127094-2713809
- Steve Hofmann, Dorina Mitrea, Marius Mitrea, and Andrew J. Morris, $L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc. 245 (2017), no. 1159, v+108. MR 3589162, DOI 10.1090/memo/1159
- David S. Jerison and Carlos E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80–147. MR 676988, DOI 10.1016/0001-8708(82)90055-X
- Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
- C. Kenig, B. Kirchheim, J. Pipher, and T. Toro, Square functions and the $A_\infty$ property of elliptic measures, J. Geom. Anal. 26 (2016), no. 3, 2383–2410. MR 3511480, DOI 10.1007/s12220-015-9630-6
- Carlos E. Kenig and Jill Pipher, The Dirichlet problem for elliptic equations with drift terms, Publ. Mat. 45 (2001), no. 1, 199–217. MR 1829584, DOI 10.5565/PUBLMAT_{4}5101_{0}9
- John L. Lewis and Margaret A. M. Murray, Regularity properties of commutators and layer potentials associated to the heat equation, Trans. Amer. Math. Soc. 328 (1991), no. 2, 815–842. MR 1020043, DOI 10.1090/S0002-9947-1991-1020043-6
- Emmanouil Milakis, Jill Pipher, and Tatiana Toro, Harmonic analysis on chord arc domains, J. Geom. Anal. 23 (2013), no. 4, 2091–2157. MR 3107693, DOI 10.1007/s12220-012-9322-4
- Emmanouil Milakis, Jill Pipher, and Tatiana Toro, Perturbations of elliptic operators in chord arc domains, Harmonic analysis and partial differential equations, Contemp. Math., vol. 612, Amer. Math. Soc., Providence, RI, 2014, pp. 143–161. MR 3204862, DOI 10.1090/conm/612/12229
- Stephen Semmes, Analysis vs. geometry on a class of rectifiable hypersurfaces in $\textbf {R}^n$, Indiana Univ. Math. J. 39 (1990), no. 4, 1005–1035. MR 1087183, DOI 10.1512/iumj.1990.39.39048
- Zihui Zhao, BMO solvability and $A_{\infty }$ condition of the elliptic measures in uniform domains, J. Geom. Anal. 28 (2018), no. 2, 866–908. MR 3790485, DOI 10.1007/s12220-017-9845-9
Additional Information
- Juan Cavero
- Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, E-28049 Madrid, Spain
- MR Author ID: 1303184
- Email: juan.cavero@icmat.es
- Steve Hofmann
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 251819
- ORCID: 0000-0003-1110-6970
- Email: hofmanns@missouri.edu
- José María Martell
- Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, 13-15, E-28049 Madrid, Spain
- MR Author ID: 671782
- ORCID: 0000-0001-6788-4769
- Email: chema.martell@icmat.es
- Received by editor(s): August 22, 2017
- Received by editor(s) in revised form: January 9, 2018
- Published electronically: October 2, 2018
- Additional Notes: The first author was partially supported by la Caixa–Severo Ochoa international PhD Programme. The first and third authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554). They also acknowledge that the research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT
The second author was supported by NSF grant DMS-1664047. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 2797-2835
- MSC (2010): Primary 31B05, 35J08, 35J25; Secondary 42B99, 42B25, 42B37
- DOI: https://doi.org/10.1090/tran/7536
- MathSciNet review: 3896098