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

Received : 2017-03-22
Accepted : 2018-06-12
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3278
Classification:  35J67,  35J30,  31B10
Keywords: Elliptic equation, higher order differential equation, Dirichlet boundary values, Neumann boundary values
@article{AIF_2019__69_4_1627_0,
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {4},
     year = {2019},
     pages = {1627-1678},
     doi = {10.5802/aif.3278},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_4_1627_0}
}
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/item/AIF_2019__69_4_1627_0/

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