# 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 (1998)

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

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topMikhail 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{MikhailBorsuk1998,

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 = {Mikhail 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 - Mikhail 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

top- [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.
- [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] C. Burch, The Dini condition and regularity of weak solutions of elliptic equations, J. Differential Equations 30 (1978), 308-323.
- [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] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.
- [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] G. Lieberman, The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data, Comm. Partial Differential Equations 11 (1986), 167-229.
- [8] E. Sperner, Schauder's existence theorem for α-Dini continuous data, Ark. Mat. 19 (1981), 193-216.

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