Boundary trace of positive solutions of nonlinear elliptic inequalities

Moshe Marcus[1]; Laurent Véron[2]

  • [1] Department of Mathematics, Israel Institute of Technology Technion, Haifa 32000, Israel
  • [2] Université François Rabelais Tours 37200, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 3, page 481-533
  • ISSN: 0391-173X

Abstract

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We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of - Δ u + g ( x , u ) ≥ 0 in a smooth domain Ω under very general assumptions on g . This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity is very degenerate near the boundary, for example if g ( x , u ) ≈ exp ( - ρ ∂ Ω - 1 ( x ) ) u q , we exhibit a new full boundary blow-up phenomenon.

How to cite

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Marcus, Moshe, and Véron, Laurent. "Boundary trace of positive solutions of nonlinear elliptic inequalities." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.3 (2004): 481-533. <http://eudml.org/doc/84538>.

@article{Marcus2004,
abstract = {We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ in a smooth domain $\Omega $ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity is very degenerate near the boundary, for example if $g(x,u)\approx \exp (-\rho ^\{-1\}_\{\partial \Omega \}(x))u^q$, we exhibit a new full boundary blow-up phenomenon.},
affiliation = {Department of Mathematics, Israel Institute of Technology Technion, Haifa 32000, Israel; Université François Rabelais Tours 37200, France},
author = {Marcus, Moshe, Véron, Laurent},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {481-533},
publisher = {Scuola Normale Superiore, Pisa},
title = {Boundary trace of positive solutions of nonlinear elliptic inequalities},
url = {http://eudml.org/doc/84538},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Marcus, Moshe
AU - Véron, Laurent
TI - Boundary trace of positive solutions of nonlinear elliptic inequalities
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 3
SP - 481
EP - 533
AB - We develop a new method for proving the existence of a boundary trace, in the class of Borel measures, of nonnegative solutions of $-\Delta u+g(x,u)\ge 0$ in a smooth domain $\Omega $ under very general assumptions on $g$. This new definition which extends the previous notions of boundary trace is based upon a sweeping technique by solutions of Dirichlet problems with measure boundary data. We also prove a boundary pointwise blow-up estimate of any solution of such inequalities in terms of the Poisson kernel. If the nonlinearity is very degenerate near the boundary, for example if $g(x,u)\approx \exp (-\rho ^{-1}_{\partial \Omega }(x))u^q$, we exhibit a new full boundary blow-up phenomenon.
LA - eng
UR - http://eudml.org/doc/84538
ER -

References

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