# 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

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topMarcus, 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 -

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