Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalities

Jana Ježková

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 1, page 63-80
  • ISSN: 0010-2628

Abstract

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The local boundedness of weak solutions to variational inequalities (obstacle problem) with the linear growth condition is obtained. Consequently, an analogue of a theorem by Reshetnyak about a.eḋifferentiability of weak solutions to elliptic divergence type differential equations is proved for variational inequalities.

How to cite

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Ježková, Jana. "Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalities." Commentationes Mathematicae Universitatis Carolinae 35.1 (1994): 63-80. <http://eudml.org/doc/247628>.

@article{Ježková1994,
abstract = {The local boundedness of weak solutions to variational inequalities (obstacle problem) with the linear growth condition is obtained. Consequently, an analogue of a theorem by Reshetnyak about a.eḋifferentiability of weak solutions to elliptic divergence type differential equations is proved for variational inequalities.},
author = {Ježková, Jana},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasi-linear elliptic equations and inequalities; weak solution; local boundedness; pointwise differentiability; difference quotient; obstacle problem; local boundedness; pointwise differentiability; variational inequalities; linear growth},
language = {eng},
number = {1},
pages = {63-80},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalities},
url = {http://eudml.org/doc/247628},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Ježková, Jana
TI - Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalities
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 1
SP - 63
EP - 80
AB - The local boundedness of weak solutions to variational inequalities (obstacle problem) with the linear growth condition is obtained. Consequently, an analogue of a theorem by Reshetnyak about a.eḋifferentiability of weak solutions to elliptic divergence type differential equations is proved for variational inequalities.
LA - eng
KW - quasi-linear elliptic equations and inequalities; weak solution; local boundedness; pointwise differentiability; difference quotient; obstacle problem; local boundedness; pointwise differentiability; variational inequalities; linear growth
UR - http://eudml.org/doc/247628
ER -

References

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  1. Bojarski B., Pointwise differentiability of weak solutions of elliptic divergence type equations, Bull. Acad. Polon. Sci. 33 (1985), 1-6. (1985) Zbl0572.35011MR0798721
  2. Federer H., Geometric Measure Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1969. Zbl0874.49001MR0257325
  3. Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, New Jersey, 1983. Zbl0516.49003MR0717034
  4. Hajłasz P., Strzelecki P., A new proof of Reshetnyak's theorem concerning the pointwise differentiability of solution of quasilinear equations, Preprint, Institute of Mathematics, Warsaw University, PKIN IXp., 00-901 Warsaw. 
  5. Ladyzhenskaya O.A., Ural'tseva N.N., Linear and Quasilinear Elliptic Equations, 2nd ed., Nauka Press, Moscow, 1973, English translation Academic Press, New York, 1968. Zbl0177.37404MR0244627
  6. Moser J., A new proof of de Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. XIII (1960), 457-468. (1960) Zbl0111.09301MR0170091
  7. Moser J., On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. XIV (1961), 577-591. (1961) Zbl0111.09302MR0159138
  8. Reshetnyak Yu.G., Generalized derivatives and differentiability almost everywhere, Mat. Sb. 75 (117) (1968), 323-334 (in Russian) Math. USSR-Sb. 4 (1968), 293-302 (English translation). (1968) Zbl0176.12001MR0225159
  9. Reshetnyak Yu.G., O differentsiruemosti pochti vsyudu resheniĭ ellipticheskikh uravneniĭ, Sibirsk. Mat. Zh. XXVIII (1987), 193-195. (1987) MR0906049
  10. Serrin J., Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302. (1964) Zbl0128.09101MR0170096
  11. Stepanoff M.W., Sur les conditions de l'existence de la différentielle totale, Matematiceskij Sbornik, Rec. Math. Soc. Math. Moscou XXXII (1925), 511-527. (1925) 
  12. Ziemer W.P., Weakly Differentiable Functions, Springer-Verlag, Berlin-Heidelberg-New York, 1989. Zbl0692.46022MR1014685

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