Existence of entropy solutions for degenerate quasilinear elliptic equations in L 1

Albo Carlos Cavalheiro

Communications in Mathematics (2014)

  • Volume: 22, Issue: 1, page 57-69
  • ISSN: 1804-1388

Abstract

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In this article, we prove the existence of entropy solutions for the Dirichlet problem ( P ) - div [ ω ( x ) 𝒜 ( x , u , ∇ u ) ] = f ( x ) - div ( G ) , in Ω u ( x ) = 0 , on ∂ Ω where Ω is a bounded open set of N , N ≥ 2 , f ∈ L 1 ( Ω ) and G / ω ∈ [ L p ' ( Ω , ω ) ] N .

How to cite

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Cavalheiro, Albo Carlos. "Existence of entropy solutions for degenerate quasilinear elliptic equations in $L^1$." Communications in Mathematics 22.1 (2014): 57-69. <http://eudml.org/doc/261950>.

@article{Cavalheiro2014,
abstract = {In this article, we prove the existence of entropy solutions for the Dirichlet problem \[ (P)\{\left\lbrace \begin\{array\}\{ll\} -\mathrm \{div\} [\{\omega \}(x)\{\mathcal \{A\}\} (x,u,\{\nabla \}u)]=f(x)-\mathrm \{div\} (G),&\text\{in \}\Omega \\ u(x) = 0,&\text\{on \}\{\partial \Omega \} \end\{array\}\right.\} \] where $\Omega $ is a bounded open set of $^N$, $N\ge 2$, $f \in L^1(\Omega )$ and $G/\{\omega \} \in [L^\{p^\{\prime \}\}(\Omega , \omega )]^N$.},
author = {Cavalheiro, Albo Carlos},
journal = {Communications in Mathematics},
keywords = {degenerate elliptic equations; entropy solutions; weighted Sobolev spaces; degenerate elliptic equations; entropy solutions; weighted Sobolev spaces},
language = {eng},
number = {1},
pages = {57-69},
publisher = {University of Ostrava},
title = {Existence of entropy solutions for degenerate quasilinear elliptic equations in $L^1$},
url = {http://eudml.org/doc/261950},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Cavalheiro, Albo Carlos
TI - Existence of entropy solutions for degenerate quasilinear elliptic equations in $L^1$
JO - Communications in Mathematics
PY - 2014
PB - University of Ostrava
VL - 22
IS - 1
SP - 57
EP - 69
AB - In this article, we prove the existence of entropy solutions for the Dirichlet problem \[ (P){\left\lbrace \begin{array}{ll} -\mathrm {div} [{\omega }(x){\mathcal {A}} (x,u,{\nabla }u)]=f(x)-\mathrm {div} (G),&\text{in }\Omega \\ u(x) = 0,&\text{on }{\partial \Omega } \end{array}\right.} \] where $\Omega $ is a bounded open set of $^N$, $N\ge 2$, $f \in L^1(\Omega )$ and $G/{\omega } \in [L^{p^{\prime }}(\Omega , \omega )]^N$.
LA - eng
KW - degenerate elliptic equations; entropy solutions; weighted Sobolev spaces; degenerate elliptic equations; entropy solutions; weighted Sobolev spaces
UR - http://eudml.org/doc/261950
ER -

References

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  10. Kufner, A., John, O., Fučík, S., Function Spaces, 1977, Noordhoof International Publishing, Leyden, (1977) MR0482102
  11. Muckenhoupt, B., 10.1090/S0002-9947-1972-0293384-6, Trans. Am. Math. Soc., 165, 1972, 207-226, (1972) Zbl0236.26016MR0293384DOI10.1090/S0002-9947-1972-0293384-6
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