Most equations in physics assume ‘nice’ spaces where movement is smooth in all directions, but in life, it is rarely that simple. Some directions and environments facilitate normal flow, while others can slow things down drastically. For example, initially while trekking along a valley floor that narrows toward the centre, there is freedom to move left or right. Gradually, the path becomes so tight that the only possible movement is to walk straight ahead. Researchers have explored these ‘degenerate’ situations using a special mathematical tool called the
Concentration-Compactness principle.
The type of equation that was studied, exhibits contradicting behaviours. While it can easily be tamed, it can also be fragile, causing it to easily break down or fail to exist. In the case of the latter, called ‘critical exponent’, it is challenging to apply classical mathematical techniques, since in this case, solutions tend to ‘escape’ or fail to exist. Such responses tend to happen in degenerate environments that can be modelled by the Grushin operator.
Motivated by the Brézis–Nirenberg problem that was studied in the 1980s, the team has tried to prove that stable, meaningful solutions still exist even in degeneracy. Through some careful analysis, along with other powerful methods from modern mathematics, every possible favourable investigation was done to find the most appropriate solution.
Owing to how real-world systems can behave unevenly, having reliable mathematical models that capture their system effectively can open doors to better simulations, predictions and technologies.
The team includes Prof Claudianor O. Alves, Dr Somnath Gandal, Dr Annunziata Loiudice, and Prof Jagmohan Tyagi.
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