Dirichlet and Neumann boundary values of solutions to higher order elliptic equations

Barton, Ariel; Hofmann, Steve; Mayboroda, Svitlana

doi:10.5802/aif.3278

Dirichlet and Neumann boundary values of solutions to higher order elliptic equations
[Conditions aux limites de type Dirichlet ou Neumann non-homogènes pour les solutions d’équations elliptiques d’ordre supérieur]

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)

Annales de l'Institut Fourier, Tome 69 (2019) no. 4, pp. 1627-1678.

Résumé
Abstract

On montre que si $u$ est une solution d’une équation aux dérivées partielles elliptique d’ordre $2 m \geq 2$ 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 $L^{p} (ℝ^{n})$ ou un espace de Sobolev ${\dot{W}}_{\pm 1}^{p} (ℝ^{n})$ . Même dans le cas où $u$ est une solution d’une équation de second ordre, nos résultats sont nouveaux pour certaines valeurs de $p$ .

We show that if $u$ is a solution to a linear elliptic differential equation of order $2 m \geq 2$ 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 $L^{p} (ℝ^{n})$ or Sobolev space ${\dot{W}}_{\pm 1}^{p} (ℝ^{n})$ . Even in the case where $u$ is a solution to a second order equation, our results are new for certain values of $p$ .

Reçu le : 2017-03-22
Accepté le : 2018-06-12
Publié le : 2019-09-16

Zbl

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

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Barton, Ariel and Hofmann, Steve and Mayboroda, Svitlana},
     title = {Dirichlet and {Neumann} boundary values of solutions to higher order elliptic equations},
     journal = {Annales de l'Institut Fourier},
     pages = {1627--1678},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     doi = {10.5802/aif.3278},
     zbl = {1394.35159},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3278/}
}

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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, Tome 69 (2019) no. 4, pp. 1627-1678. doi : 10.5802/aif.3278. https://aif.centre-mersenne.org/articles/10.5802/aif.3278/

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