On the regularity of solutions of quasilinear Poisson equations

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“…Basic facts from the potential theory. For the sake of completeness, we repeat the fun damental results concerning the potential theory in a plane, given in [8], and strengthen some of them.…”

“…The following theorem on the Newtonian potential strengthens Theorem 2 from [8]. It is important to obtain solutions of a higher regularity to the Poisson equations (8), as well as to the corresponding semilinear equations.…”

“…In series of our recent papers (see, e.g., [7,8]), we have proposed another application of the theory of quasiconformal mappings to the the study of semilinear partial differential equations of the form…”

“…In this section, we study the solvability problem for quasilinear Poisson equations of the form ( ). Schauder approach allows us to reduce the problem to the study of the corresponding linear Poisson equation from the previous section.where M is the constant from estimate (19) in[8]. Thus, by Theorem 1 from[8] in combination with the Arzela-Ascoli theorem (see, e.g., Theorem IV.6.7 in[14]), the operators ( ; ) → R by the norm ⋅ on the domain1 D .…”

On the quasilinear Poisson equations in the complex plane

“…The following theorem on the Newtonian potential strengthens Theorem 2 from [8]. It is important to obtain solutions of a higher regularity to the Poisson equations (8), as well as to the corresponding semilinear equations.…”