Geometric functional inequalities and their stability

• Applicant: Dr. Tobias König
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 555837013

Geometric functional inequalities such as Sobolev's or the isoperimetric inequality are central objects of contemporary Analysis. The application of such inequalities is decisive for many neighboring fields, ranging from determining the steady states and asymptotic convergence rates of time-dependent problems to the famous Yamabe problem in differential geometry. In recent years, research has become increasingly focused on the quantitative stability of functional inequalities. The goal of this research project is to make innovative and fundamental contributions to this highly active field. Unlike for many classical inequalities whose optimizers are classified, even the existence of optimizers is open for most stability inequalities. In a recent breakthrough, the existence of optimizers is proved for the Bianchi-Egnell (BE) stability inequality associated to Sobolev's inequality. One main objective of this project is a far-reaching development, in two different directions, of this fundamental result. On the one hand, we aim to substantially clarify the conditions for existence of stability optimizers by treating further examples, but also counterexamples. On the other hand, a central objective is to investigate the qualitative properties of the BE optimizers, about which to date almost nothing is known. We plan to develop symmetrization methods adapted to the surprisingly subtle behavior of the BE functional which permit to study the properties of its optimizers. Such results are highly relevant because they serve as a blueprint for the basic properties of a large class of stability inequalities of similar structure whose analysis is even more challenging. Motivated both by applications and as natural mathematical generalizations, variants of classical inequalities including higher-order derivatives as well as vector-valued functions are topics of high current interest. A notorious obstacle in their analysis is that fundamental tools like the maximum principle, rearrangement or ODE analysis techniques are usually lost. The second main objective of this project is to develop new methods to overcome these challenges by treating a number of exemplary unsolved instances, with a view towards stability. For instance, we aim to fully describe convergence rates for the Q-curvature flow, classify singular solutions to the higher-order Yamabe equation and investigate the stability and duality properties of reverse Sobolev inequalities. By proving a stability inequality for a certain recent inequality for matrix-valued functions, we plan to develop a unified approach to the stability of various inequalities contained therein as special cases.

• DFG Programme: Emmy Noether Independent Junior Research Groups