Visa Latvala

Critical variable exponent functionals in image restoration

Abstract We study a variable exponent model for image restoration in the case that the exponent attains the critical value one. We prove existence and Γ -convergence. The results answer an open question by Li, Li and Pi [F. Li, Z. Li, L. Pi, Ling, Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (3) (2010) 870–882]

The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.

A Theorem on Fine Connectedness

We prove that Ω ∖ E is a p-fine domain whenever Ω ⊂ Rn is a p-fine domain, E ⊂ Rn is p-polar, and 1 < p ≤ n. By a p-fine domain we understand an open connected set in the p-fine topology, i.e. in the coarsest topology making all p-superharmonic functions continuous. As an application of our main result, we establish a general version of minimum principle.

Boundedness of solutions of the non-uniformly convex, non-standard growth Laplacian

We study the variable exponent p(·)-Laplace equation when p attains the value 1. We prove a removability result and local boundedness for its solutions and obtain a weak continuity result. Our results also apply to (𝒜, ℬ)-solutions of p(·)-growth.