15.01.2026

Speaker: Leon Happ     Title:   Different notions of capacity

Abstract:   Even though, for the student embarking with the topic for the first time this fact is almost buried under the theory that has developed over the course of the last century, it is no coincidence that the word capacity, which is now established as a central concept in the study of many problems in analysis and functional analysis, is also a prominent notion in physics. In fact, it can be shown how the mathematical definition emerges from the problem of finding the equilibrium charge distribution and the capacity of an electric body, studied by C. F. Gauss as early as 1839. Later on, the mathematical theory of capacity and what we call potential theory nowadays was pioneered by N. Wiener in 1924. Subsequently the theory was further developed by de la Vallée-Poussin, O. Frostman, G. Choquet, J. Serrin, V. G. Maz'ya and many others.

Starting from the motivational example in Coulomb (Newtonian) potential theory above, I will then introduce different definitions of capacity and discuss their connection. Afterwards, I will turn to the comparison between capacity and the Hausdorff measure, which allows to gain some geometrical intuition for capacity. These connections are then used to tackle questions about fine properties of Sobolev functions and other related subjects, for example, certain approximation properties. Under the given format of the presentation, I am looking for the balance between richness of the presented results and some in-depth calculations illuminating the power of the capacity concept. The main goal is to give an idea why capacity is such a natural notion in the study of Sobolev spaces.

It turns out that this fruitful connection can be extended to Bessel potential spaces, Besov and Triebel-Lizorkin spaces. I will briefly hint at these generalizations and also elaborate on how this could possibly give rise to a potential and capacity theory for function spaces based on general nonlocal gradients. 

In addition, I will recall a different notion of capacity that is closely intertwined with information theory. I will briefly comment on how this can be fruitfully applied in the theory of inverse problems and why the two different notions of capacity are connected through the name of A. P. Calderon.