The 𝐿^{𝑝} Dirichlet boundary problem for second order elliptic Systems with rough coefficients

Dindoš, Martin; Hwang, Sukjung; Mitrea, Marius

doi:10.1090/tran/8306

The $L^p$ Dirichlet boundary problem for second order elliptic Systems with rough coefficients
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by Martin Dindoš, Sukjung Hwang and Marius Mitrea PDF

Trans. Amer. Math. Soc. 374 (2021), 3659-3701 Request permission

Abstract:

Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in an interval of the form $\big (2-\varepsilon ,\frac {2(n-1)}{n-2}+\varepsilon \big )$ for some small $\varepsilon >0$). The main novel aspect of our result is that the coefficients of the operator do not have to be constant, or have very high regularity; instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A significant example of a system to which our result may be applied is the Lamé system for isotropic inhomogeneous materials. We show that our result applies to isotropic materials with Poisson ratio $\nu <0.396$.

Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi–Nash–Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving $L^p$ solvability results for $p$ near $2$.

References

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 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
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
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
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 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
Russell M. Brown and Irina Mitrea, The mixed problem for the Lamé system in a class of Lipschitz domains, J. Differential Equations 246 (2009), no. 7, 2577–2589. MR 2503013, DOI 10.1016/j.jde.2009.01.008
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
B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 3, 109–135 (English, with French summary). MR 865663, DOI 10.5802/aif.1062
B. E. J. Dahlberg, C. E. Kenig, and G. C. Verchota, Boundary value problems for the systems of elastostatics in Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 795–818. MR 975122, DOI 10.1215/S0012-7094-88-05735-3
M. Dindoš, The $L^p$ Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system, available at arXiv:2006.13015.
Martin Dindoš and Sukjung Hwang, The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition, Rev. Mat. Iberoam. 34 (2018), no. 2, 767–810. MR 3809457, DOI 10.4171/RMI/1003
Martin Dindoš and Marius Mitrea, The stationary Navier-Stokes system in nonsmooth manifolds: the Poisson problem in Lipschitz and $C^1$ domains, Arch. Ration. Mech. Anal. 174 (2004), no. 1, 1–47. MR 2092995, DOI 10.1007/s00205-004-0320-y
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 Jill Pipher, Regularity theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem, Adv. Math. 341 (2019), 255–298. MR 3872848, DOI 10.1016/j.aim.2018.07.035
Martin Dindoš, Jill Pipher, and David Rule, Boundary value problems for second-order elliptic operators satisfying a Carleson condition, Comm. Pure Appl. Math. 70 (2017), no. 7, 1316–1365. MR 3666568, DOI 10.1002/cpa.21649
Eugene Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Potential theory—surveys and problems (Prague, 1987) Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 55–80. MR 973881, DOI 10.1007/BFb0103344
E. B. Fabes, C. E. Kenig, and G. C. Verchota, The Dirichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 57 (1988), no. 3, 769–793. MR 975121, DOI 10.1215/S0012-7094-88-05734-1
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
J. Feneuil, S. Mayboroda, and Z. Zhao, Dirichlet problem in domains with lower dimensional boundaries, available at arXiv:1810.06805., DOI 10.4171/rmi/1208
Wen Jie Gao, Layer potentials and boundary value problems for elliptic systems in Lipschitz domains, J. Funct. Anal. 95 (1991), no. 2, 377–399. MR 1092132, DOI 10.1016/0022-1236(91)90035-4
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 and José María Martell, $L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), no. 2, 497–515. MR 2006497, DOI 10.5565/PUBLMAT_{4}7203_{1}2
Carlos E. Kenig, Elliptic boundary value problems on Lipschitz domains, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 131–183. MR 864372
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
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
José María Martell, Dorina Mitrea, Irina Mitrea, and Marius Mitrea, The Dirichlet problem for elliptic systems with data in Köthe function spaces, Rev. Mat. Iberoam. 32 (2016), no. 3, 913–970. MR 3556056, DOI 10.4171/RMI/903
P. Mott and C. Roland, Limits to Poisson’s ratio in isotropic materials – general result for arbitrary deformation, Physica Scripta 87 (2012).
Zhongwei Shen, Necessary and sufficient conditions for the solvability of the $L^p$ Dirichlet problem on Lipschitz domains, Math. Ann. 336 (2006), no. 3, 697–725. MR 2249765, DOI 10.1007/s00208-006-0022-x
Zhongwei Shen, The $L^p$ Dirichlet problem for elliptic systems on Lipschitz domains, Math. Res. Lett. 13 (2006), no. 1, 143–159. MR 2200052, DOI 10.4310/MRL.2006.v13.n1.a11
Zhongwei Shen, Extrapolation for the $L^p$ Dirichlet problem in Lipschitz domains, Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 1074–1084. MR 3952704, DOI 10.1007/s10114-019-8199-6
Gunther Uhlmann and Jenn-Nan Wang, Complex spherical waves for the elasticity system and probing of inclusions, SIAM J. Math. Anal. 38 (2007), no. 6, 1967–1980. MR 2299437, DOI 10.1137/060651434
B. Yan, Existence and regularity theory for nonlinear elliptic systems and multiple integrals in the calculus of variations, Lecture notes, MSU, https://users.math.msu.edu/users/yan/full-notes.pdf.