H\"{o}lder continuity of weak solutions to an elliptic-parabolic system modeling biological transportation network

TL;DR

This paper proves that weak solutions to a singular elliptic-parabolic system modeling biological networks are H"{o}lder continuous in two dimensions, using advanced inequalities and refined mathematical lemmas.

Contribution

It establishes the regularity of solutions for a complex nonlinear system, extending understanding of biological network models in mathematical analysis.

Findings

Weak solutions are H"{o}lder continuous in 2D.

Uses inequality related to Stummel-Kato class.

Refines classical lemmas for regularity proof.

Abstract

In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are H\"{o}lder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).