In this talk we consider differential operators of the form $\text{div}(A \, \text{grad})$ where the coefficients of $A$ are measurable and bounded functions and the eigenvalues of $A$ are positive and uniformly bounded from $0$. A solution on a domain $D \subset \mathbb{R}^d$ is a function that satisfies\[ \operatorname{div}(A \, \operatorname{grad}) u = 0 \text{ on } D. \]

We examine boundary behavior of the gradient of solutions to such differential operators. The classical case, where the differential operator in question is the Laplacian, is well understood for $C^2$ domains due to Havin and Mozolyako [1] and later also for Lipschitz domains due to Müller and Riegler [2]. This talk mentions techniques to obtain estimates on the radial variation of positve harmonic functions and also gives an outlook of how to adapt the existing techniques to general uniformly elliptic divergence-form operators or solutions on more complicated domains.

1. Pavel A. Mozolyako and Viktor P. Khavin. Boundedness of variation of a positive harmonic function along the normals to the boundary. Algebra i Analiz 28.3 (2016), pp. 67–110. issn: 0234-0852. doi: 10.1090/spmj/1454. url: https://doi.org/10.1090/spmj/1454.
2. Paul F. X. Müller and Katharina Riegler. Radial variation of Bloch functions on the unit ball of Rd. Ark. Mat. 58.1 (2020), pp. 161–178. issn: 0004-2080,1871-2487. doi: 10.4310/ARKIV.2020.v58.n1.a10. url: https://doi.org/10.4310/ARKIV.2020.v58.n1.a10.