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

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We prove an existence result for equations of the form { - D i ⁢ ( a i ⁢ j ⁢ ( x , u ) ⁢ D j ⁢ u ) = f   in ⁢ Ω u ∈ H 0 1 ⁢ ( Ω ) . where the coefficients a i ⁢ j ⁢ ( 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 i ⁢ j ⁢ ( x , s ) are supposed only Borel functions

How to cite

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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

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  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 T ⁢ f ⁢ ( u ) = f ⁢ ( x , u , D ⁢ u ) . 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

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