On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent
ESAIM: Control, Optimisation and Calculus of Variations (2010)
- Volume: 4, page 667-686
- ISSN: 1292-8119
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topDemengel, Françoise. "On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 667-686. <http://eudml.org/doc/197312>.
@article{Demengel2010,
abstract = {
Let Ω be a smooth bounded domain in $\{\bf R\}^n$,
n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star =
\{n\over n-1\}$. We are concerned here with the existence of solution in
$BV(\Omega)$, positive or not, to the problem:
$$
\left\\{
\begin\{array\}\{rl\}
-\{\rm div\}\ \sigma+a(x) sign\ u &= f|u|^\{1^\star-2\} u\cr
\sigma.\nabla u &= |\nabla u|\ \{\rm in\}\ \Omega\cr
u\ \{\rm is\ not \ identically\ zero\}, &-\sigma.n (u) = |u|\ \{\rm on \}\ \partial
\Omega.\end\{array\}\right.$$
This problem is closely related to the extremal functions for the problem of
the best constant of $W^\{1,1\}(\Omega)$ into $L^\{N\over N-1\}(\Omega)$.
},
author = {Demengel, Françoise},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {BV functions; best constant for Sobolev embeddings.; 1-Laplacian; critical Sobolev exponent; existence},
language = {eng},
month = {3},
pages = {667-686},
publisher = {EDP Sciences},
title = {On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent},
url = {http://eudml.org/doc/197312},
volume = {4},
year = {2010},
}
TY - JOUR
AU - Demengel, Françoise
TI - On Some Nonlinear Partial Differential Equations Involving the “1”-Laplacian and Critical Sobolev Exponent
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 667
EP - 686
AB -
Let Ω be a smooth bounded domain in ${\bf R}^n$,
n > 1, let a and f be continuous functions on $\bar\Omega$, $1^\star =
{n\over n-1}$. We are concerned here with the existence of solution in
$BV(\Omega)$, positive or not, to the problem:
$$
\left\{
\begin{array}{rl}
-{\rm div}\ \sigma+a(x) sign\ u &= f|u|^{1^\star-2} u\cr
\sigma.\nabla u &= |\nabla u|\ {\rm in}\ \Omega\cr
u\ {\rm is\ not \ identically\ zero}, &-\sigma.n (u) = |u|\ {\rm on }\ \partial
\Omega.\end{array}\right.$$
This problem is closely related to the extremal functions for the problem of
the best constant of $W^{1,1}(\Omega)$ into $L^{N\over N-1}(\Omega)$.
LA - eng
KW - BV functions; best constant for Sobolev embeddings.; 1-Laplacian; critical Sobolev exponent; existence
UR - http://eudml.org/doc/197312
ER -
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