Abstract
This paper considers the theory of higher-order divergence-form elliptic differential equations. In particular, we provide new generalizations of several well-known tools from the theory of second-order equations. These tools are the Caccioppoli inequality, Meyers’s reverse Hölder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17, 35–92 (1964)
Agranovich, M.S.: On the theory of Dirichlet and Neumann problems for linear strongly elliptic systems with Lipschitz domains. Funktsional. Anal. i Prilozhen. 41(4), 1–21, 96 (2007). doi:10.1007/s10688-007-0023-x. English translation: Funct. Anal. Appl. 41(4), 247–263 (2007)
Auscher, P., Hofmann, S., McIntosh, A., Tchamitchian, P.: The Kato square root problem for higher order elliptic operators and systems on \({\mathbb{R}}^n\). J. Evol. Equ. 1(4), 361–385 (2001). doi:10.1007/PL00001377. Dedicated to the memory of Tosio Kato
Auscher, P., McIntosh, A., Tchamitchian, P.: Heat kernels of second order complex elliptic operators and applications. J. Funct. Anal. 152(1), 22–73 (1998). doi:10.1006/jfan.1997.3156
Auscher, P., Qafsaoui, M.: Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form. J. Funct. Anal. 177(2), 310–364 (2000). doi:10.1006/jfan.2000.3643
Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1970/1971)
Barton, A., Hofmann, S., Mayboroda, S.: Square function estimates on layer potentials for higher-order elliptic equations. ArXiv e-prints (2015). arXiv:1508.04988 [math.AP]
Barton, A., Mayboroda, S.: Higher-order elliptic equations in non-smooth domains: a partial survey. In: Harmonic analysis, partial differential equations, complex analysis, banach spaces, andoperator theory. Celebrating Cora Sadosky’s life, vol. 1. AWM-Springer (2016) (To appear)
Campanato, S.: Sistemi ellittici in forma divergenza. Regolarità all’interno. Quaderni. [Publications]. Scuola Normale Superiore Pisa, Pisa (1980)
Cho, S., Dong, H., Kim, S.: Global estimates for Green’s matrix of second order parabolic systems with application to elliptic systems in two dimensional domains. Potential Anal. 36(2), 339–372 (2012). doi:10.1007/s11118-011-9234-0
Cohen, J., Gosselin, J.: Adjoint boundary value problems for the biharmonic equation on \(C^1\) domains in the plane. Ark. Mat. 23(2), 217–240 (1985). doi:10.1007/BF02384427
Dalla Riva, M.: A family of fundamental solutions of elliptic partial differential operators with real constant coefficients. Integral Equ. Oper. Theory 76(1), 1–23 (2013). doi:10.1007/s00020-013-2052-6
Dalla Riva, M., Morais, J., Musolino, P.: A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients. Math. Methods Appl. Sci. 36(12), 1569–1582 (2013). doi:10.1002/mma.2706
Dolzmann, G., Müller, S.: Estimates for Green’s matrices of elliptic systems by \(L^p\) theory. Manuscr. Math. 88(2), 261–273 (1995). doi:10.1007/BF02567822
Dong, H., Kim, S.: Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains. Trans. Am. Math. Soc. 361(6), 3303–3323 (2009). doi:10.1090/S0002-9947-09-04805-3
Duduchava, R.: The Green formula and layer potentials. Integral Equ. Oper. Theory 41(2), 127–178 (2001). doi:10.1007/BF01295303
Evans, L.C.: Partial differential equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)
Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
Frehse, J.: An irregular complex valued solution to a scalar uniformly elliptic equation. Calc. Var. Partial Differ. Equ. 33(3), 263–266 (2008). doi:10.1007/s00526-007-0131-8
Friedman, A.: On fundamental solutions of elliptic equations. Proc. Am. Math. Soc. 12, 533–537 (1961)
Fuchs, M.: The Green matrix for strongly elliptic systems of second order with continuous coefficients. Z. Anal. Anwendungen 5(6), 507–531 (1986)
Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, NJ (1983)
Grüter, M., Widman, K.O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37(3), 303–342 (1982). doi:10.1007/BF01166225
Hofmann, S., Kim, S.: The Green function estimates for strongly elliptic systems of second order. Manuscr. Math. 124(2), 139–172 (2007). doi:10.1007/s00229-007-0107-1
John, F.: Plane waves and spherical means applied to partial differential equations. Interscience Publishers, New York-London (1955)
Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147(1–2), 71–88 (1981). doi:10.1007/BF02392869
Kang, K., Kim, S.: Global pointwise estimates for Green’s matrix of second order elliptic systems. J. Differ. Equ. 249(11), 2643–2662 (2010). doi:10.1016/j.jde.2010.05.017
Kenig, C.E., Ni, W.M.: On the elliptic equation \(Lu-k+K\,{\rm exp}[2u]=0\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12(2), 191–224 (1985). http://www.numdam.org/item?id=ASNSP_1985_4_12_2_191_0
Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 3(17), 43–77 (1963)
Mayboroda, S., Maz’ya, V.: Boundedness of the Hessian of a biharmonic function in a convex domain. Commun. Partial Differ. Equ. 33(7–9), 1439–1454 (2008). doi:10.1080/03605300801891919
Mayboroda, S., Maz’ya, V.: Pointwise estimates for the polyharmonic Green function in general domains. In: Analysis, partial differential equations and applications, Oper. Theory Adv. Appl., vol. 193, pp. 143–158. Birkhäuser Verlag, Basel (2009)
Maz’ya, V.: The Wiener test for higher order elliptic equations. Duke Math. J. 115(3), 479–512 (2002). doi:10.1215/S0012-7094-02-11533-6
Maz’ya, V., Mitrea, M., Shaposhnikova, T.: The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients. J. Anal. Math. 110, 167–239 (2010). doi:10.1007/s11854-010-0005-4
Meyers, N.G.: An \(L^{p}\)-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 3(17), 189–206 (1963)
Mitrea, I., Mitrea, M.: Multi-layer potentials and boundary problems for higher-order elliptic systems in Lipschitz domains. Lecture Notes in Mathematics, Vol. 2063. Springer, Heidelberg (2013)
Morrey Jr., C.B.: Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer, New York Inc., New York (1966)
Moser, J.: On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14, 577–591 (1961)
Ortner, N., Wagner, P.: A survey on explicit representation formulae for fundamental solutions of linear partial differential operators. Acta Appl. Math. 47(1), 101–124 (1997). doi:10.1023/A:1005784017770
Pipher, J., Verchota, G.C.: Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators. Ann. Math. 142(1), 1–38 (1995). doi:10.2307/2118610
Rosén, A.: Layer potentials beyond singular integral operators. Publ. Mat. 57(2), 429–454 (2013). doi:10.5565/PUBLMAT_57213_08
Verchota, G.C.: Potentials for the Dirichlet problem in Lipschitz domains. In: Potential theory—ICPT 94 (Kouty, 1994), pp. 167–187. de Gruyter, Berlin (1996)
Verchota, G.C.: The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194(2), 217–279 (2005). doi:10.1007/BF02393222
Verchota, G.C.: Boundary coerciveness and the Neumann problem for 4th order linear partial differential operators. In: Around the research of Vladimir Maz’ya. II, Int. Math. Ser. (N.Y.), Vol. 12, pp. 365–378. Springer, New York (2010). doi:10.1007/978-1-4419-1343-2_17
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barton, A. Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients. manuscripta math. 151, 375–418 (2016). https://doi.org/10.1007/s00229-016-0839-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-016-0839-x