The Growth of Gradients of QC-mappings in n-dimensional Euclidean Space with Bounded Laplacian

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Authors: N. MUTAVDžIć

DOI: 10.46793/KgJMat2307.1029M


Here we review M. Mateljević’s article [?], with some novelities. We focus on mappings between smooth domains which have bounded Laplacian. As an application, if these mappings are quasiconformal, we obtain some results on the behavior of their partial derivatives on the boundary. In the last part of this article, we announce one new result of the author of [?], which has been recently presented on Belgrade Seminary of Complex Analysis.


PDE of the second order, Laplacian-Gradient Inequalities, Quasiconformal harmonic mappings, Boundary behavior of partial derivatives.


[1]   V. Božin and M. Mateljević, Quasiconformal and HQC mappings between Lyapunov Jordan domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XXI (2020), 107–132.

[2]   A. Gjokaj and D. Kalaj, QCH mappings between unit ball and domain with C1 boundary, Potential Anal. (2022), (to appear). arXiv:2005.05667

[3]   D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer-Verlag, Berlin, Second Edition, 1983.

[4]   E. Heinz, On certain nonlinear elliptic differential equations and univalent mappings, J. Anal. Math. 5 (1956), 197–272.

[5]   D. Kalaj, A priori estimate of gradient of a solution to certain differential inequality and quasiconformal mappings, J. Anal. Math. 119(1) (2013), 63–88.

[6]   D. Kalaj and M. Mateljević, Inner estimate and quasiconformal harmonic maps between smooth domains, J. Anal. Math. 100 (2006), 117–132.

[7]   D. Kalaj and E. Saksman, Quasiconformal maps with controlled Laplacian, J. Anal. Math. 137 (2019), 251–268.

[8]   D. Kalaj and A. Zlatičanin, Quasiconformal mappings with controlled Laplacian and Hölder continuity, Ann. Fenn. Math. 44 (2019), 797–803.