Modulus of continuity of weak solutions to a class of singular elliptic equations

TL;DR

This paper investigates the regularity of weak solutions to a class of singular elliptic equations in the plane, showing that their modulus of continuity can be characterized by a reciprocal logarithmic function with an arbitrarily large power.

Contribution

It establishes a novel description of the modulus of continuity for solutions to singular elliptic equations under weak integrability conditions, contrasting with previous results on degenerate equations.

Findings

Modulus of continuity is described by reciprocal logarithmic functions.

The power in the logarithmic expression can be arbitrarily large.

Results differ significantly from degenerate elliptic equations.

Abstract

In this paper we study the modulus of continuity of weak solutions to a singular elliptic equation in the plane under very weak assumption on the integrability of the elliptic coefficients. Our investigation reveals that the modulus of continuity can be described by the reciprocal of the logarithmic function raised to a power. However, the power can be arbitrarily large. This is in sharp contrast with a result by J. Onninen and X. Zhong for a degenerate elliptic equation in the plane, in which the power must be suitably small.