Quasilinear elliptic equations with discontinuous coefficients

Lucio Boccardo; Giuseppe Buttazzo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1988)

  • Volume: 82, Issue: 1, page 21-28
  • ISSN: 1120-6330

Abstract

top
We prove an existence result for equations of the form \begin{cases}&-D_{i}(a_{ij}(x,u)D_{j}u)=f\quad\text{in}\,\Omega\\ &u\in H_{0}^{1}(\Omega).\end{cases} where the coefficients a_{ij}(x,s) satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable s. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients a_{ij}(x,s) are supposed only Borel functions

How to cite

top

Boccardo, Lucio, and Buttazzo, Giuseppe. "Quasilinear elliptic equations with discontinuous coefficients." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 82.1 (1988): 21-28. <http://eudml.org/doc/287485>.

@article{Boccardo1988,
abstract = {We prove an existence result for equations of the form $$\begin\{cases\} &- D\_\{i\} (a\_\{ij\}(x,u) D\_\{j\}u) = f \quad \text\{in\} \, \Omega\\ &u \in H\_\{0\}^\{1\}(\Omega). \end\{cases\}$$ where the coefficients $a_\{ij\}(x,s)$ satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable $s$. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients $a_\{ij\}(x,s)$ are supposed only Borel functions},
author = {Boccardo, Lucio, Buttazzo, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Quasilinear elliptic equations; Dirichlet problems; Semicontinuity; Calculus of variations; quasilinear elliptic equations; semicontinuity; calculus of variations},
language = {eng},
month = {3},
number = {1},
pages = {21-28},
publisher = {Accademia Nazionale dei Lincei},
title = {Quasilinear elliptic equations with discontinuous coefficients},
url = {http://eudml.org/doc/287485},
volume = {82},
year = {1988},
}

TY - JOUR
AU - Boccardo, Lucio
AU - Buttazzo, Giuseppe
TI - Quasilinear elliptic equations with discontinuous coefficients
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/3//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 1
SP - 21
EP - 28
AB - We prove an existence result for equations of the form $$\begin{cases} &- D_{i} (a_{ij}(x,u) D_{j}u) = f \quad \text{in} \, \Omega\\ &u \in H_{0}^{1}(\Omega). \end{cases}$$ where the coefficients $a_{ij}(x,s)$ satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable $s$. Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients $a_{ij}(x,s)$ are supposed only Borel functions
LA - eng
KW - Quasilinear elliptic equations; Dirichlet problems; Semicontinuity; Calculus of variations; quasilinear elliptic equations; semicontinuity; calculus of variations
UR - http://eudml.org/doc/287485
ER -

References

top
  1. AMBROSIO, L. - A few lower semicontinuity results for integral functionals. «Rend. Accad. Naz. Sci. XL Mem. Mat. Sci. Fis. Natur.», (to appear). Zbl0642.49007MR930856
  2. AMBROSIO, L., BUTTAZZO, G. and LEACI, A. - Continuous operators of the form Tf(u)=f(x,u,Du). Boll. Un. Mat. Ital. (to appear). Zbl0639.47051MR938993
  3. BOCCARDO, L., MURAT, F. (1982) - Remarques sur l'homogénéisation de certains problèmes quasi-linéaires. «Portugal. Math.», 41, 535-562. Zbl0524.35042MR766874
  4. DUGUNDJI, J. and GRANAS, A. (1982) - Fixed Point Theory. Polish Scientific Publishers, Warszawa. Zbl0483.47038
  5. GIACHETTI, D. (1983) - Controllo ottimale in problemi vincolati. «Boll. Un. Mat. Ital.», 2-B, 445-468. Zbl0522.49006MR716742
  6. LADYZHENSKAYA, O.A. and URALTSEVA, N.N. (1968) - Linear and Quasilinear Elliptic Equations. Academic Press, New York. Zbl0164.13002MR244627
  7. LERAY, J. and LIONS, J.L. (1965) - Quelques résultats de Visik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. «Bull. Soc. Math. France», 93, 97-107. Zbl0132.10502MR194733
  8. MARCUS, M. and MIZEL, V.J. (1972) - Absolute continuity on tracks and mappings of Sobolev spaces. «Arch. Rational Mech. Anal.», 45, 294-320. Zbl0236.46033MR338765
  9. STAMPACCHIA, G. (1964) - Formes bilinéaires coercitives sur les ensembles convexes. «C.R. Acad. Sci. Paris», 258, 4413-4416. Zbl0124.06401MR166591
  10. STAMPACCHIA, G. (1966) - Equations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathématiques Supérieures n° 16, Les Presses de l'Université de Montréal, Montréal. Zbl0151.15501MR251373

NotesEmbed ?

top

You must be logged in to post comments.