A strongly degenerate quasilinear equation : the elliptic case

Fuensanta Andreu[1]; Vicent Caselles[2]; José Mazón[1]

  • [1] Universitat de Valencia Dept. de Análisis Matemático
  • [2] Universitat Pompeu-Fabra Dept. de Tecnologia

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

  • Volume: 3, Issue: 3, page 555-587
  • ISSN: 0391-173X

Abstract

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We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation u - div 𝐚 ( u , D u ) = v , where v ∈ L 1 , 𝐚 ( z , ξ ) = ∇ ξ f ( z , ξ ) , and f is a convex function of ξ with linear growth as ∥ ξ ∥ → ∞ , satisfying other additional assumptions. In particular, this class includes the case where f ( z , ξ ) = ϕ ( z ) ψ ( ξ ) , ϕ > 0 , ψ being a convex function with linear growth as ∥ ξ ∥ → ∞ . In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in L 1 .

How to cite

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Andreu, Fuensanta, Caselles, Vicent, and Mazón, José. "A strongly degenerate quasilinear equation : the elliptic case." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.3 (2004): 555-587. <http://eudml.org/doc/84540>.

@article{Andreu2004,
abstract = {We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u - \mathrm \{div\} \, \mathbf \{a\}(u,Du) = v$, where $v\!\in \! L^1$, $\mathbf \{a\}(z,\xi ) = \nabla _\xi f(z,\xi )$, and $f$ is a convex function of $\xi $ with linear growth as $\Vert \xi \Vert \rightarrow \infty $, satisfying other additional assumptions. In particular, this class includes the case where $f(z,\xi ) = \varphi (z)\psi (\xi )$, $\varphi &gt; 0$, $\psi $ being a convex function with linear growth as $\Vert \xi \Vert \rightarrow \infty $. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in $L^1$.},
affiliation = {Universitat de Valencia Dept. de Análisis Matemático; Universitat Pompeu-Fabra Dept. de Tecnologia; Universitat de Valencia Dept. de Análisis Matemático},
author = {Andreu, Fuensanta, Caselles, Vicent, Mazón, José},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {Neumann problem; entropy solution},
language = {eng},
number = {3},
pages = {555-587},
publisher = {Scuola Normale Superiore, Pisa},
title = {A strongly degenerate quasilinear equation : the elliptic case},
url = {http://eudml.org/doc/84540},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Andreu, Fuensanta
AU - Caselles, Vicent
AU - Mazón, José
TI - A strongly degenerate quasilinear equation : the elliptic case
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 3
SP - 555
EP - 587
AB - We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u - \mathrm {div} \, \mathbf {a}(u,Du) = v$, where $v\!\in \! L^1$, $\mathbf {a}(z,\xi ) = \nabla _\xi f(z,\xi )$, and $f$ is a convex function of $\xi $ with linear growth as $\Vert \xi \Vert \rightarrow \infty $, satisfying other additional assumptions. In particular, this class includes the case where $f(z,\xi ) = \varphi (z)\psi (\xi )$, $\varphi &gt; 0$, $\psi $ being a convex function with linear growth as $\Vert \xi \Vert \rightarrow \infty $. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in $L^1$.
LA - eng
KW - Neumann problem; entropy solution
UR - http://eudml.org/doc/84540
ER -

References

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