Dirichlet and Neumann boundary values of solutions to higher order elliptic equations
Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1627-1678.

We show that if u is a solution to a linear elliptic differential equation of order 2m2 in the half-space with t-independent coefficients, and if u satisfies certain area integral estimates, then the Dirichlet and Neumann boundary values of u exist and lie in a Lebesgue space L p ( n ) or Sobolev space W ˙ ±1 p ( n ). Even in the case where u is a solution to a second order equation, our results are new for certain values of p.

On montre que si u est une solution d’une équation aux dérivées partielles elliptique d’ordre 2m2 dans le demi-espace à coefficients indépendants de t, et u satisfait certaines conditions d’intégrales de surface, alors les données aux frontières de Dirichlet et de Neumann de u existent et appartiennent à un espace de Lebesgue L p ( n ) ou un espace de Sobolev W ˙ ±1 p ( n ). Même dans le cas où u est une solution d’une équation de second ordre, nos résultats sont nouveaux pour certaines valeurs de p.

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DOI: 10.5802/aif.3278
Classification: 35J67, 35J30, 31B10
Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values
Mot clés : Equation elliptique, équation différentielle d’ordre supérieur, données aux frontières de type Dirichlet, données aux frontières de type Neumann

Barton, Ariel 1; Hofmann, Steve 2; Mayboroda, Svitlana 3

1 Department of Mathematical Sciences 309 SCEN University of Arkansas Fayetteville, AR 72701 (USA)
2 202 Math Sciences Bldg. University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematics University of Minnesota Minneapolis, MN 55455 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Dirichlet and {Neumann} boundary values of solutions to higher order elliptic equations},
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Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana. Dirichlet and Neumann boundary values of solutions to higher order elliptic equations. Annales de l'Institut Fourier, Volume 69 (2019) no. 4, pp. 1627-1678. doi : 10.5802/aif.3278. https://aif.centre-mersenne.org/articles/10.5802/aif.3278/

[1] Adolfsson, Vilhelm; Pipher, Jill The inhomogeneous Dirichlet problem for Δ 2 in Lipschitz domains, J. Funct. Anal., Volume 159 (1998) no. 1, pp. 137-190 | DOI | MR | Zbl

[2] Agmon, Shmuel Multiple layer potentials and the Dirichlet problem for higher order elliptic equations in the plane. I, Commun. Pure Appl. Math., Volume 10 (1957), pp. 179-239 | DOI | MR | Zbl

[3] Agranovich, Mikhail S. On the theory of Dirichlet and Neumann problems for linear strongly elliptic systems with Lipschitz domains, Funkts. Anal. Prilozh., Volume 41 (2007) no. 4, pp. 1-21 English translation: Funct. Anal. Appl. 41 (2007), no. 4, p. 247-263 | DOI | MR | Zbl

[4] Agranovich, Mikhail S. Potential-type operators and conjugation problems for second-order strongly elliptic systems in domains with a Lipschitz boundary, Funkts. Anal. Prilozh., Volume 43 (2009) no. 3, pp. 3-25 | DOI | MR

[5] Alfonseca, M. Angeles; Auscher, Pascal; Axelsson, Andreas; Hofmann, Steve; Kim, Seick Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L coefficients, Adv. Math., Volume 226 (2011) no. 5, pp. 4533-4606 | DOI | MR | Zbl

[6] Auscher, Pascal; Axelsson, Andreas Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I, Invent. Math., Volume 184 (2011) no. 1, pp. 47-115 | DOI | MR | Zbl

[7] Auscher, Pascal; Axelsson, Andreas; Hofmann, Steve Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems, J. Funct. Anal., Volume 255 (2008) no. 2, pp. 374-448 | DOI | MR | Zbl

[8] Auscher, Pascal; Axelsson, Andreas; McIntosh, Alan Solvability of elliptic systems with square integrable boundary data, Ark. Mat., Volume 48 (2010) no. 2, pp. 253-287 | DOI | MR | Zbl

[9] Auscher, Pascal; Mourgoglou, Mihalis Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems, Proc. Lond. Math. Soc., Volume 109 (2014) no. 2, pp. 446-482 | DOI | MR | Zbl

[10] Auscher, Pascal; Qafsaoui, Mahmoud Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form, J. Funct. Anal., Volume 177 (2000) no. 2, pp. 310-364 | DOI | MR | Zbl

[11] Auscher, Pascal; Stahlhut, Sebastian Functional calculus for first order systems of Dirac type and boundary value problems, Mém. Soc. Math. Fr., Nouv. Sér. (2016) no. 144, vii+164 pages | MR | Zbl

[12] Barton, Ariel Extrapolation of well posedness for higher order elliptic systems with rough coefficients (http://arxiv.org/abs/1708.05079) | Zbl

[13] Barton, Ariel Elliptic partial differential equations with almost-real coefficients, Mem. Am. Math. Soc., Volume 223 (2013) no. 1051, vi+108 pages | DOI | MR | Zbl

[14] Barton, Ariel Gradient estimates and the fundamental solution for higher-order elliptic systems with rough coefficients, Manuscr. Math., Volume 151 (2016) no. 3-4, pp. 375-418 | DOI | MR | Zbl

[15] Barton, Ariel Trace and extension theorems relating Besov spaces to weighted averaged Sobolev spaces, Math. Inequal. Appl., Volume 21 (2018) no. 3, pp. 817-870 | MR | Zbl

[16] Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana Square function estimates on layer potentials for higher-order elliptic equations, Math. Nachr., Volume 290 (2017) no. 16, pp. 2459-2511 | DOI | MR | Zbl

[17] Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana The Neumann problem for higher order elliptic equations with symmetric coefficients, Math. Ann., Volume 371 (2018) no. 1-2, pp. 297-336 | DOI | MR | Zbl

[18] Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana Nontangential estimates and the Neumann problem for higher order elliptic equations (2018) (https://arxiv.org/abs/1808.07137) | Zbl

[19] Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana Bounds on layer potentials with rough inputs for higher order elliptic equations, Proc. Lond. Math. Soc., Volume 119 (2019) no. 3, pp. 613-653 | DOI | MR | Zbl

[20] Barton, Ariel; Mayboroda, Svitlana The Dirichlet problem for higher order equations in composition form, J. Funct. Anal., Volume 265 (2013) no. 1, pp. 49-107 | DOI | MR | Zbl

[21] Barton, Ariel; Mayboroda, Svitlana Higher-order elliptic equations in non-smooth domains: a partial survey, Harmonic analysis, partial differential equations, complex analysis, Banach spaces, and operator theory. Vol. 1 (Association for Women in Mathematics Series), Volume 4, Springer, 2016, pp. 55-121 | MR | Zbl

[22] Barton, Ariel; Mayboroda, Svitlana Layer potentials and boundary-value problems for second order elliptic operators with data in Besov spaces, Mem. Am. Math. Soc., Volume 243 (2016) no. 1149, v+110 pages | DOI | MR | Zbl

[23] Benedek, Agnes; Calderón, Alberto P.; Panzone, Rafael Convolution operators on Banach space valued functions, Proc. Natl. Acad. Sci. USA, Volume 48 (1962), pp. 356-365 | DOI | MR | Zbl

[24] Brewster, Kevin; Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius Extending Sobolev functions with partially vanishing traces from locally (ε,δ)-domains and applications to mixed boundary problems, J. Funct. Anal., Volume 266 (2014) no. 7, pp. 4314-4421 | DOI | MR | Zbl

[25] Brewster, Kevin; Mitrea, Marius Boundary value problems in weighted Sobolev spaces on Lipschitz manifolds, Mem. Differ. Equ. Math. Phys., Volume 60 (2013), pp. 15-55 | MR | Zbl

[26] Caffarelli, Luis A.; Fabes, Eugene B.; Kenig, Carlos E. Completely singular elliptic-harmonic measures, Indiana Univ. Math. J., Volume 30 (1981) no. 6, pp. 917-924 | DOI | MR | Zbl

[27] Caffarelli, Luis A.; Fabes, Eugene B.; Mortola, Stefano; Salsa, Sandro Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., Volume 30 (1981) no. 4, pp. 621-640 | DOI | MR | Zbl

[28] Calderón, Alberto P. On the behaviour of harmonic functions at the boundary, Trans. Am. Math. Soc., Volume 68 (1950), pp. 47-54 | DOI | MR | Zbl

[29] Campanato, Sergio Sistemi ellittici in forma divergenza. Regolarità all’interno, Pubblicazioni della Classe di Scienze: Quaderni, Scuola Normale Superiore Pisa, 1980, 187 pages | MR | Zbl

[30] Carleson, Lennart On the existence of boundary values for harmonic functions in several variables, Ark. Mat., Volume 4 (1962), pp. 393-399 | DOI | MR | Zbl

[31] Cohen, Jonathan; Gosselin, John The Dirichlet problem for the biharmonic equation in a C 1 domain in the plane, Indiana Univ. Math. J., Volume 32 (1983) no. 5, pp. 635-685 | DOI | MR | Zbl

[32] Cohen, Jonathan; Gosselin, John Adjoint boundary value problems for the biharmonic equation on C 1 domains in the plane, Ark. Mat., Volume 23 (1985) no. 2, pp. 217-240 | DOI | MR | Zbl

[33] Dahlberg, Björn E. J. Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain, Stud. Math., Volume 67 (1980) no. 3, pp. 297-314 | MR | Zbl

[34] Dahlberg, Björn E. J.; Jerison, David S.; Kenig, Carlos E. Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat., Volume 22 (1984) no. 1, pp. 97-108 | DOI | MR | Zbl

[35] Dahlberg, Björn E. J.; Kenig, Carlos E.; Pipher, Jill; Verchota, Gregory C. Area integral estimates for higher order elliptic equations and systems, Ann. Inst. Fourier, Volume 47 (1997) no. 5, pp. 1425-1461 | DOI | MR | Zbl

[36] Dindoš, Martin; Petermichl, Stefanie; Pipher, Jill The L p Dirichlet problem for second order elliptic operators and a p-adapted square function, J. Funct. Anal., Volume 249 (2007) no. 2, pp. 372-392 | DOI | MR | Zbl

[37] Dindoš, Martin; Pipher, Jill; Rule, David J. Boundary value problems for second-order elliptic operators satisfying a Carleson condition, Commun. Pure Appl. Math., Volume 70 (2017) no. 7, pp. 1316-1365 | DOI | MR | Zbl

[38] Fabes, Eugene B.; Mendez, Osvaldo; Mitrea, Marius Boundary layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains, J. Funct. Anal., Volume 159 (1998) no. 2, pp. 323-368 | DOI | MR | Zbl

[39] Fatou, Pierre Séries trigonométriques et séries de Taylor, Acta Math., Volume 30 (1906) no. 1, pp. 335-400 | DOI | MR | Zbl

[40] Hofmann, Steve; Kenig, Carlos E.; Mayboroda, Svitlana; Pipher, Jill The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients, Math. Ann., Volume 361 (2015) no. 3-4, pp. 863-907 | DOI | MR | Zbl

[41] Hofmann, Steve; Kenig, Carlos E.; Mayboroda, Svitlana; Pipher, Jill Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators, J. Am. Math. Soc., Volume 28 (2015) no. 2, pp. 483-529 | DOI | MR | Zbl

[42] Hofmann, Steve; Mayboroda, Svitlana; Mourgoglou, Mihalis Layer potentials and boundary value problems for elliptic equations with complex L coefficients satisfying the small Carleson measure norm condition, Adv. Math., Volume 270 (2015), pp. 480-564 | DOI | MR | Zbl

[43] Hofmann, Steve; Mitrea, Marius; Morris, Andrew J. The method of layer potentials in L p and endpoint spaces for elliptic operators with L coefficients, Proc. Lond. Math. Soc., Volume 111 (2015) no. 3, pp. 681-716 | DOI | MR | Zbl

[44] Hunt, Richard A.; Wheeden, Richard L. On the boundary values of harmonic functions, Trans. Am. Math. Soc., Volume 132 (1968), pp. 307-322 | DOI | MR | Zbl

[45] Hunt, Richard A.; Wheeden, Richard L. Positive harmonic functions on Lipschitz domains, Trans. Am. Math. Soc., Volume 147 (1970), pp. 507-527 | DOI | MR | Zbl

[46] Jawerth, Björn Some observations on Besov and Lizorkin-Triebel spaces, Math. Scand., Volume 40 (1977) no. 1, pp. 94-104 | DOI | MR | Zbl

[47] Jonsson, Alf; Wallin, Hans Function spaces on subsets of R n , Math. Rep., Chur, Volume 2 (1984) no. 1, xiv+221 pages | MR | Zbl

[48] Kenig, Carlos E.; Koch, Herbert; Pipher, Jill; Toro, Tatiana A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math., Volume 153 (2000) no. 2, pp. 231-298 | DOI | MR | Zbl

[49] Kenig, Carlos E.; Pipher, Jill The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math., Volume 113 (1993) no. 3, pp. 447-509 | DOI | MR | Zbl

[50] Kenig, Carlos E.; Pipher, Jill The Dirichlet problem for elliptic equations with drift terms, Publ. Mat., Barc., Volume 45 (2001) no. 1, pp. 199-217 | DOI | MR | Zbl

[51] Kenig, Carlos E.; Rule, David J. The regularity and Neumann problem for non-symmetric elliptic operators, Trans. Am. Math. Soc., Volume 361 (2009) no. 1, pp. 125-160 | DOI | MR | Zbl

[52] Kilty, Joel; Shen, Zhongwei A bilinear estimate for biharmonic functions in Lipschitz domains, Math. Ann., Volume 349 (2011) no. 2, pp. 367-394 | DOI | MR | Zbl

[53] Kim, Doyoon Trace theorems for Sobolev–Slobodeckij spaces with or without weights, J. Funct. Spaces Appl., Volume 5 (2007) no. 3, pp. 243-268 | DOI | MR | Zbl

[54] Lizorkin, Pjotr I. Boundary properties of functions from “weight” classes, Sov. Math., Dokl., Volume 1 (1960), pp. 589-593 | MR | Zbl

[55] Maz’ya, Vladimir; Mitrea, Marius; Shaposhnikova, Tatyana The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients, J. Anal. Math., Volume 110 (2010), pp. 167-239 | DOI | MR | Zbl

[56] Mitrea, Irina; Mitrea, Marius Boundary value problems and integral operators for the bi-Laplacian in non-smooth domains, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., Volume 24 (2013) no. 3, pp. 329-383 | DOI | MR | Zbl

[57] Mitrea, Irina; Mitrea, Marius Multi-layer potentials and boundary problems for higher-order elliptic systems in Lipschitz domains, Lecture Notes in Mathematics, 2063, Springer, 2013, x+424 pages | DOI | MR | Zbl

[58] Mitrea, Irina; Mitrea, Marius; Wright, M. Optimal estimates for the inhomogeneous problem for the bi-Laplacian in three-dimensional Lipschitz domains, J. Math. Sci., New York, Volume 172 (2011) no. 1, pp. 24-134 (translation of Probl. Mat. Anal. 51, p. 21-114) | DOI | MR | Zbl

[59] Modica, Luciano; Mortola, Stefano Construction of a singular elliptic-harmonic measure, Manuscr. Math., Volume 33 (1980) no. 1, pp. 81-98 | DOI | MR | Zbl

[60] Pipher, Jill; Verchota, Gregory C. Area integral estimates for the biharmonic operator in Lipschitz domains, Trans. Am. Math. Soc., Volume 327 (1991) no. 2, pp. 903-917 | DOI | MR | Zbl

[61] Pipher, Jill; Verchota, Gregory C. Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators, Ann. Math., Volume 142 (1995) no. 1, pp. 1-38 | DOI | MR | Zbl

[62] Privaloff, Ivan Sur une généralisation du théorème de Fatou, Mat. Sb., Volume 31 (1923) no. 2, pp. 232-235 | Zbl

[63] Shen, Zhongwei Necessary and sufficient conditions for the solvability of the L p Dirichlet problem on Lipschitz domains, Math. Ann., Volume 336 (2006) no. 3, pp. 697-725 | DOI | MR | Zbl

[64] Verchota, Gregory C. The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J., Volume 39 (1990) no. 3, pp. 671-702 | DOI | MR | Zbl

[65] Verchota, Gregory C. Potentials for the Dirichlet problem in Lipschitz domains, Potential theory—ICPT 94 (Kouty, 1994), Walter de Gruyter, 1996, pp. 167-187 | MR | Zbl

[66] Verchota, Gregory C. The biharmonic Neumann problem in Lipschitz domains, Acta Math., Volume 194 (2005) no. 2, pp. 217-279 | DOI | MR | Zbl

[67] Zanger, Daniel Z. The inhomogeneous Neumann problem in Lipschitz domains, Commun. Partial Differ. Equations, Volume 25 (2000) no. 9-10, pp. 1771-1808 | DOI | MR | Zbl

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