Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces
About this Title
Ariel Barton, Department of Mathematical Sciences, 309 SCEN, University of Arkansas, Fayetteville, AR 72701 and Svitlana Mayboroda, School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church Street SE, Minneapolis, Minnesota 55455
Publication: Memoirs of the American Mathematical Society
Publication Year:
2016; Volume 243, Number 1149
ISBNs: 978-1-4704-1989-9 (print); 978-1-4704-3446-5 (online)
DOI: https://doi.org/10.1090/memo/1149
Published electronically: April 12, 2016
Keywords: Elliptic equation,
boundary-value problem,
Besov space,
weighted Sobolev space
MSC: Primary 35J25; Secondary 31B20, 35C15, 46E35
Table of Contents
Chapters
- 1. Introduction
- 2. Definitions
- 3. The Main Theorems
- 4. Interpolation, Function Spaces and Elliptic Equations
- 5. Boundedness of Integral Operators
- 6. Trace Theorems
- 7. Results for Lebesgue and Sobolev Spaces: Historic Account and some Extensions
- 8. The Green’s Formula Representation for a Solution
- 9. Invertibility of Layer Potentials and Well-Posedness of Boundary-Value Problems
- 10. Besov Spaces and Weighted Sobolev Spaces
Abstract
This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable $t$-independent coefficients in spaces of fractional smoothness, in Besov and weighted $L^p$ classes. We establish:
(1) Mapping properties for the double and single layer potentials, as well as the Newton potential;
(2) Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given $L^p$ space automatically assures their solvability in an extended range of Besov spaces;
(3) Well-posedness for the non-homogeneous boundary value problems.
In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients.
- Pascal Auscher and Andreas Axelsson, Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I, Invent. Math. 184 (2011), no. 1, 47–115. MR 2782252, DOI 10.1007/s00222-010-0285-4
- M. Angeles Alfonseca, Pascal Auscher, Andreas Axelsson, Steve Hofmann, and Seick Kim, Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^\infty$ coefficients, Adv. Math. 226 (2011), no. 5, 4533–4606. MR 2770458, DOI 10.1016/j.aim.2010.12.014
- Pascal Auscher, Andreas Axelsson, and Steve Hofmann, Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems, J. Funct. Anal. 255 (2008), no. 2, 374–448. MR 2419965, DOI 10.1016/j.jfa.2008.02.007
- Pascal Auscher, Andreas Axelsson, and Alan McIntosh, Solvability of elliptic systems with square integrable boundary data, Ark. Mat. 48 (2010), no. 2, 253–287. MR 2672609, DOI 10.1007/s11512-009-0108-2
- M. S. Agranovich, On the theory of Dirichlet and Neumann problems for linear strongly elliptic systems with Lipschitz domains, Funktsional. Anal. i Prilozhen. 41 (2007), no. 4, 1–21, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 41 (2007), no. 4, 247–263. MR 2411602, DOI 10.1007/s10688-007-0023-x
- M. S. Agranovich, Potential-type operators and conjugation problems for second-order strongly elliptic systems in domains with a Lipschitz boundary, Funktsional. Anal. i Prilozhen. 43 (2009), no. 3, 3–25 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 43 (2009), no. 3, 165–183. MR 2583636, DOI 10.1007/s10688-009-0025-y
- Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on ${\Bbb R}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633–654. MR 1933726, DOI 10.2307/3597201
- Andreas Axelsson, Stephen Keith, and Alan McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), no. 3, 455–497. MR 2207232, DOI 10.1007/s00222-005-0464-x
- Pascal Auscher and Mihalis Mourgoglou, Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 446–482. MR 3254931, DOI 10.1112/plms/pdu008
- Pascal Auscher, Alan McIntosh, and Mihalis Mourgoglou, On $L^2$ solvability of BVPs for elliptic systems, J. Fourier Anal. Appl. 19 (2013), no. 3, 478–494. MR 3048587, DOI 10.1007/s00041-013-9266-5
- Pascal Auscher, Alan McIntosh, and Philippe Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1998), no. 1, 22–73. MR 1600066, DOI 10.1006/jfan.1997.3156
- Pascal Auscher and Andreas Rosén, Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II, Anal. PDE 5 (2012), no. 5, 983–1061. MR 3022848, DOI 10.2140/apde.2012.5.983
- P. Auscher and P. Tchamitchian, Calcul fontionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 721–778 (French, with English and French summaries). MR 1340951
- Pascal Auscher and Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172 (English, with English and French summaries). MR 1651262
- Pascal Auscher, Regularity theorems and heat kernel for elliptic operators, J. London Math. Soc. (2) 54 (1996), no. 2, 284–296. MR 1405056, DOI 10.1112/jlms/54.2.284
- Andreas Axelsson, Non-unique solutions to boundary value problems for non-symmetric divergence form equations, Trans. Amer. Math. Soc. 362 (2010), no. 2, 661–672. MR 2551501, DOI 10.1090/S0002-9947-09-04673-X
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
- Ariel Barton, Elliptic partial differential equations with almost-real coefficients, Mem. Amer. Math. Soc. 223 (2013), no. 1051, vi+108. MR 3086390, DOI 10.1090/S0065-9266-2012-00677-0
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
- Ariel Barton and Svitlana Mayboroda, The Dirichlet problem for higher order equations in composition form, J. Funct. Anal. 265 (2013), no. 1, 49–107. MR 3049881, DOI 10.1016/j.jfa.2013.03.013
- Luis A. Caffarelli, Eugene B. Fabes, and Carlos E. Kenig, Completely singular elliptic-harmonic measures, Indiana Univ. Math. J. 30 (1981), no. 6, 917–924. MR 632860, DOI 10.1512/iumj.1981.30.30067
- J. A. Chao, J. E. Gilbert, and P. A. Tomas, Molecular decompositions in $H^{p}$-theory, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981, pp. 115–119. MR 639473
- 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
- Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909–996. MR 951745, DOI 10.1002/cpa.3160410705
- Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
- Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97–108. MR 735881, DOI 10.1007/BF02384374
- Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465. MR 890159, DOI 10.2307/1971407
- Martin Dindos, Carlos Kenig, and Jill Pipher, BMO solvability and the $A_\infty$ condition for elliptic operators, J. Geom. Anal. 21 (2011), no. 1, 78–95. MR 2755677, DOI 10.1007/s12220-010-9142-3
- B. E. J. Dahlberg, C. E. Kenig, J. Pipher, and G. C. Verchota, Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1425–1461 (English, with English and French summaries). MR 1600375
- Martin Dindos, Stefanie Petermichl, and Jill Pipher, The $L^p$ Dirichlet problem for second order elliptic operators and a $p$-adapted square function, J. Funct. Anal. 249 (2007), no. 2, 372–392. MR 2345337, DOI 10.1016/j.jfa.2006.11.012
- Martin Dindoš and David J. Rule, Elliptic equations in the plane satisfying a Carleson measure condition, Rev. Mat. Iberoam. 26 (2010), no. 3, 1013–1034. MR 2789374, DOI 10.4171/RMI/625
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- R. Fefferman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc. 2 (1989), no. 1, 127–135. MR 955604, DOI 10.1090/S0894-0347-1989-0955604-8
- Robert A. Fefferman, Large perturbations of elliptic operators and the solvability of the $L^p$ Dirichlet problem, J. Funct. Anal. 118 (1993), no. 2, 477–510. MR 1250271, DOI 10.1006/jfan.1993.1152
- Michael Frazier and Björn Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), no. 4, 777–799. MR 808825, DOI 10.1512/iumj.1985.34.34041
- Michael Frazier and Björn Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170. MR 1070037, DOI 10.1016/0022-1236(90)90137-A
- Eugene B. Fabes, David S. Jerison, and Carlos E. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. (2) 119 (1984), no. 1, 121–141. MR 736563, DOI 10.2307/2006966
- 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
- Eugene Fabes, Osvaldo Mendez, and Marius Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal. 159 (1998), no. 2, 323–368. MR 1658089, DOI 10.1006/jfan.1998.3316
- Jens Frehse, An irregular complex valued solution to a scalar uniformly elliptic equation, Calc. Var. Partial Differential Equations 33 (2008), no. 3, 263–266. MR 2429531, DOI 10.1007/s00526-007-0131-8
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- 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
- Ana Grau de la Herran, Generalized local TB theorem and applications, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–University of Missouri - Columbia. MR 3121947
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- Steve Hofmann and Seick Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math. 124 (2007), no. 2, 139–172. MR 2341783, DOI 10.1007/s00229-007-0107-1
- Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher, The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients, Math. Ann. 361 (2015), no. 3-4, 863–907. MR 3319551, DOI 10.1007/s00208-014-1087-6
- Steve Hofmann, Carlos Kenig, Svitlana Mayboroda, and Jill Pipher, Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators, J. Amer. Math. Soc. 28 (2015), no. 2, 483–529. MR 3300700, DOI 10.1090/S0894-0347-2014-00805-5
- Steve Hofmann, Svitlana Mayboroda, and Mihalis Mourgoglou, Layer potentials and boundary value problems for elliptic equations with complex $L^\infty$ coefficients satisfying the small Carleson measure norm condition, Adv. Math. 270 (2015), 480–564. MR 3286542, DOI 10.1016/j.aim.2014.11.009
- Steve Hofmann, Marius Mitrea, and Andrew J. Morris, The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^\infty$ coefficients, Proc. Lond. Math. Soc. (3) 111 (2015), no. 3, 681–716. MR 3396088, DOI 10.1112/plms/pdv035
- Björn Jawerth, Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand. 40 (1977), no. 1, 94–104. MR 454618, DOI 10.7146/math.scand.a-11678
- David S. Jerison and Carlos E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113 (1981), no. 2, 367–382. MR 607897, DOI 10.2307/2006988
- David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI 10.1006/jfan.1995.1067
- Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
- Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1282720
- Doyoon Kim, Trace theorems for Sobolev-Slobodeckij spaces with or without weights, J. Funct. Spaces Appl. 5 (2007), no. 3, 243–268. MR 2352844, DOI 10.1155/2007/471535
- C. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2000), no. 2, 231–298. MR 1770930, DOI 10.1006/aima.1999.1899
- Nigel Kalton and Marius Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 (1998), no. 10, 3903–3922. MR 1443193, DOI 10.1090/S0002-9947-98-02008-X
- Nigel Kalton, Svitlana Mayboroda, and Marius Mitrea, Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations, Interpolation theory and applications, Contemp. Math., vol. 445, Amer. Math. Soc., Providence, RI, 2007, pp. 121–177. MR 2381891, DOI 10.1090/conm/445/08598
- V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR 1788991
- V. A. Kondrat′ev and O. A. Oleĭnik, Boundary value problems for partial differential equations in nonsmooth domains, Uspekhi Mat. Nauk 38 (1983), no. 2(230), 3–76 (Russian). MR 695471
- Carlos E. Kenig and Jill Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), no. 3, 447–509. MR 1231834, DOI 10.1007/BF01244315
- Carlos E. Kenig and Jill Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients. II, Duke Math. J. 81 (1995), no. 1, 227–250 (1996). A celebration of John F. Nash, Jr. MR 1381976, DOI 10.1215/S0012-7094-95-08112-5
- Carlos E. Kenig and David J. Rule, The regularity and Neumann problem for non-symmetric elliptic operators, Trans. Amer. Math. Soc. 361 (2009), no. 1, 125–160. MR 2439401, DOI 10.1090/S0002-9947-08-04610-2
- G. Kyriazis, Decomposition systems for function spaces, Studia Math. 157 (2003), no. 2, 133–169. MR 1981430, DOI 10.4064/sm157-2-3
- P. I. Lizorkin, Boundary properties of functions from “weight” classes, Soviet Math. Dokl. 1 (1960), 589–593. MR 0123103
- Svitlana Mayboroda, The Poisson problem on Lipschitz domains, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–University of Missouri - Columbia. MR 2709638
- Svitlana Mayboroda, The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients, Adv. Math. 225 (2010), no. 4, 1786–1819. MR 2680190, DOI 10.1016/j.aim.2010.04.019
- Norman G. Meyers, An $L^{p}$e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR 159110
- Norman G. Meyers, Mean oscillation over cubes and Hölder continuity, Proc. Amer. Math. Soc. 15 (1964), 717–721. MR 168712, DOI 10.1090/S0002-9939-1964-0168712-3
- Dorina Mitrea, A generalization of Dahlberg’s theorem concerning the regularity of harmonic Green potentials, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3771–3793. MR 2386245, DOI 10.1090/S0002-9947-08-04384-5
- Osvaldo Mendez and Marius Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl. 6 (2000), no. 5, 503–531. MR 1781091, DOI 10.1007/BF02511543
- Svitlana Mayboroda and Marius Mitrea, Sharp estimates for Green potentials on non-smooth domains, Math. Res. Lett. 11 (2004), no. 4, 481–492. MR 2092902, DOI 10.4310/MRL.2004.v11.n4.a7
- Dorina Mitrea and Irina Mitrea, On the regularity of Green functions in Lipschitz domains, Comm. Partial Differential Equations 36 (2011), no. 2, 304–327. MR 2763343, DOI 10.1080/03605302.2010.489629
- V. Maz’ya, M. Mitrea, and T. Shaposhnikova, The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients, J. Anal. Math. 110 (2010), 167–239. MR 2753293, DOI 10.1007/s11854-010-0005-4
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
- J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841
- S. M. Nikol′skiĭ, Priblizhenie funktsiĭ mnogikh peremennykh i teoremy vlozheniya, “Nauka”, Moscow, 1977 (Russian). Second edition, revised and supplemented. MR 506247
- S. M. Nikol′skiĭ, P. I. Lizorkin, and N. V. Miroshin, Weighted function spaces and their applications to the investigation of boundary value problems for degenerate elliptic equations, Izv. Vyssh. Uchebn. Zaved. Mat. 8 (1988), 4–30 (Russian); English transl., Soviet Math. (Iz. VUZ) 32 (1988), no. 8, 1–40. MR 971868
- Jaak Peetre, Interpolation functors and Banach couples, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 373–378. MR 0425636
- Andreas Rosén, Layer potentials beyond singular integral operators, Publ. Mat. 57 (2013), no. 2, 429–454. MR 3114777, DOI 10.5565/PUBLMAT_{5}7213_{0}8
- Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319
- David J. Rule, Non-symmetric elliptic operators on bounded Lipschitz domains in the plane, Electron. J. Differential Equations (2007), No. 144, 8. MR 2366037
- V. V. Shan′kov, The averaging operator with variable radius, and the inverse trace theorem, Sibirsk. Mat. Zh. 26 (1985), no. 6, 141–152, 191 (Russian). MR 816512
- I. Ja. Šneĭberg, Spectral properties of linear operators in interpolation families of Banach spaces, Mat. Issled. 9 (1974), no. 2(32), 214–229, 254–255 (Russian). MR 0634681
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Robert S. Strichartz, $H^p$ Sobolev spaces, Colloq. Math. 60/61 (1990), no. 1, 129–139. MR 1096364, DOI 10.4064/cm-60-61-1-129-139
- Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903
- Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540
- Mitchell H. Taibleson and Guido Weiss, The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 67–149. MR 604370
- S. V. Uspenskiĭ, Imbedding theorems for classes with weights, Trudy Mat. Inst. Steklov. 60 (1961), 282–303 (Russian). MR 0136980
- Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/0022-1236(84)90066-1
- Daniel Z. Zanger, The inhomogeneous Neumann problem in Lipschitz domains, Comm. Partial Differential Equations 25 (2000), no. 9-10, 1771–1808. MR 1778780, DOI 10.1080/03605302.2000.10824220
