Phd Thesis

Inserted: 24 jun 2025
Last Updated: 24 jun 2025

Year: 2025
Doi: https://dx.doi.org/10.13140/RG.2.2.22693.61926
Links: HAL science ouverte

Abstract:

The aim of this thesis is to describe the regularity results that I obtained during my PhD journey for the gradient of solutions to some classes of strongly degenerate elliptic and parabolic problems. Regarding the elliptic problem in question, this arises as the Euler-Lagrange equation of an integral functional of the Calculus of Variations. The energy density of this functional satisfies p-growth and p-ellipticity conditions with respect to the gradient variable, but only outside a ball of radius λ > 0 centered at the origin. As for the parabolic problems in question, a motivation for studying them can be found in gas filtration problems taking into account the initial pressure gradient. I conclude the thesis with the discussion of a physics-informed deep learning approach for solving a certain type of strongly degenerate parabolic problems.

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