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

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.

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.

Reçu le :
Accepté le :
Publié le :
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)
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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. http://archive.numdam.org/articles/10.5802/aif.3278/

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