Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains

Michail Borsuk

Annales Polonici Mathematici (1998)

  • Volume: 69, Issue: 2, page 129-154
  • ISSN: 0066-2216

Abstract

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We consider generalized solutions to the Dirichlet problem for linear elliptic second order equations in a domain bounded by a Dini-Lyapunov surface and containing a conical point. For such solutions we derive Dini estimates for the first order generalized derivatives.

How to cite

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Michail Borsuk. "Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains." Annales Polonici Mathematici 69.2 (1998): 129-154. <http://eudml.org/doc/270148>.

@article{MichailBorsuk1998,
abstract = {We consider generalized solutions to the Dirichlet problem for linear elliptic second order equations in a domain bounded by a Dini-Lyapunov surface and containing a conical point. For such solutions we derive Dini estimates for the first order generalized derivatives.},
author = {Michail Borsuk},
journal = {Annales Polonici Mathematici},
keywords = {elliptic equations; nonsmooth domains; Dini continuous; smoothness of generalized solutions; Dini continuity},
language = {eng},
number = {2},
pages = {129-154},
title = {Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains},
url = {http://eudml.org/doc/270148},
volume = {69},
year = {1998},
}

TY - JOUR
AU - Michail Borsuk
TI - Dini continuity of the first derivatives of generalized solutions to the Dirichlet problem for linear elliptic second order equations in nonsmooth domains
JO - Annales Polonici Mathematici
PY - 1998
VL - 69
IS - 2
SP - 129
EP - 154
AB - We consider generalized solutions to the Dirichlet problem for linear elliptic second order equations in a domain bounded by a Dini-Lyapunov surface and containing a conical point. For such solutions we derive Dini estimates for the first order generalized derivatives.
LA - eng
KW - elliptic equations; nonsmooth domains; Dini continuous; smoothness of generalized solutions; Dini continuity
UR - http://eudml.org/doc/270148
ER -

References

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  1. [1] A. Azzam and V. Kondrat'ev, Schauder-type estimates of solutions of second order elliptic systems in divergence form in non-regular domains, Comm. Partial Differential Equations 16 (1991), 1857-1878. Zbl0776.35011
  2. [2] M. Borsuk, Best-possible estimates of solutions of the Dirichlet problem for linear elliptic nondivergence equations of second order in a neighbourhood of a conical point of the boundary, Math. USSR-Sb. 74 (1993), 185-201. 
  3. [3] C. Burch, The Dini condition and regularity of weak solutions of elliptic equations, J. Differential Equations 30 (1978), 308-323. Zbl0362.35021
  4. [4] S. Eĭdel'man and M. Matiĭchuk, The Cauchy problem for parabolic systems with coefficients having low smoothness, Ukrain. Mat. Zh. 22 (1970), 22-36 (in Russian). 
  5. [5] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983. Zbl0562.35001
  6. [6] V. Kondrat'ev, I. Kopachek and O. Oleĭnik, On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second-order elliptic equation, Mat. Sb. 131 (1986), 113-125 (in Russian); English transl.: Math. USSR-Sb. 59 (1988). 
  7. [7] G. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), 167-229. Zbl0589.35036
  8. [8] E. Sperner, Schauder's existence theorem for α-Dini continuous data, Ark. Mat. 19 (1981), 193-216. Zbl0506.35028

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