# ANNALES DE L'INSTITUT FOURIER

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 $2m\ge 2$ 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 ${L}^{p}\left({ℝ}^{n}\right)$ or Sobolev space ${\stackrel{˙}{W}}_{±1}^{p}\left({ℝ}^{n}\right)$. 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 $2m\ge 2$ 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 ${L}^{p}\left({ℝ}^{n}\right)$ ou un espace de Sobolev ${\stackrel{˙}{W}}_{±1}^{p}\left({ℝ}^{n}\right)$. Même dans le cas où $u$ est une solution d’une équation de second ordre, nos résultats sont nouveaux pour certaines valeurs de $p$.

Accepted:
Published online:
DOI: 10.5802/aif.3278
Classification: 35J67, 35J30, 31B10
Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values
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)
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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/

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