Semi-linear equations and quasiconformal mappings

ABSTRACT

We study the Dirichlet problem for the semi-linear partial differential equations $\mathrm{d}\mathrm{i}\mathrm{v}\left(A\mathrm{\nabla }u\right)=f\left(u\right)$ in simply connected domains D of the complex plane $\mathbb{C}$ with continuous boundary data. We prove the existence of the weak solutions u in the class $C\cap W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}\left(D\right)$ if the Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring–Martio. An example of such a domain that fails to satisfy the standard (A)-condition by Ladyzhenskaya–Ural'tseva and the known outer cone condition is given. We also extend our results to simply connected non-Jordan domains formulated in terms of the prime ends by Carathéodory. Our approach is based on the theory of the logarithmic potential, singular integrals, the Leray–Schauder technique and a factorization theorem in Gutlyanskii et al. [On quasiconformal maps and semi-linear equations in the plane. Ukr Mat Visn. 2017;14(2):161–191]. This theorem allows us to represent u in the form $u=U\circ \omega ,$ where $\omega \left(z\right)$ stands for a quasiconformal mapping of D onto the unit disk $\mathbb{D}$, generated by the measurable matrix function $A\left(z\right),$ and U is a solution of the corresponding quasilinear Poisson equation in the unit disk $\mathbb{D}$. In the end, we give some applications of these results to various processes of diffusion and absorption in anisotropic and inhomogeneous media.

COMMUNICATED BY:

• A. Meziani

KEYWORDS:

• Semi-linear elliptic equations
• quasilinear Poisson equation
• anisotropic and inhomogeneous media
• conformal and quasiconformal mappings

AMS SUBJECT CLASSIFICATIONS:

• Primary: 30C62
• 31A05
• 31A20
• 31A25
• 31B25
• 35J61
• Secondary: 30E25
• 31C05
• 34M50
• 35F45
• 35Q15

Acknowledgments

We would like to thank our referees for a series of useful remarks that enabled us essentially to improve the text. Dedicated to the memory of Professor Bogdan Bojarski for his great contribution to the theory of quasiformal mappings.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by grants of Ministry of Education and Science of Ukraine, project number is 0119U100421.