On the regularity of solutions of quasilinear Poisson equations
DOI:
https://doi.org/10.15407/dopovidi2018.10.009Keywords:
Dirichlet problem, logarithmic and Newtonian potentials, potential theory, quasiconformal mappings, quasilinear Poisson equation, Sobolev classesAbstract
We study the Dirichlet problem for quasilinear partial differential equations of the form Δu(z)=h(z)f(u(z)) in the unit disk D⊂C with continuous boundary data. Here, the function h:D→R belongs to the class Lp(D), p>1, and the continuous function f:R→R is assumed to have the nondecreasing |f| of |t| and such that f(t)/t→0 as t→∞. We prove the existence of a continuous solution u of the problem in the Sobolev class Wloc2,p(D). Moreover, we show that if p>2, then u∈Cloc1,α(D) with α=(p−2)/p .
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