Widely degenerate partial differential equations

The considered partial differential equations model problems of optimal transport with congestion effects. The model is based on a game theoretic approach for traffic dynamics called the Wardrop equilibrium. The Wardrop equilibrium relies on two principles: User equilibrium, which assumes that each user chooses the best possible route, and system optimality, which assumes that users behave cooperatively so that average travel time is minimal. Rather than inspecting effects on real live traffic dynamics, we are interested in the associated partial differential equation and its solutions. In particular we will investigate regularity properties of solutions. Our objective is to develop a systematic approach to higher regularity properties, i.e. regularity beyond Lipschitz continuity. We will consider interior and boundary regularity, the scalar and the vectorial case and optimality aspects. The methods used to solve these problems are manifold. Deep knowledge in real analysis and regularity theory for nonlinear PDEs is necessary. The class of partial differential equations considered is called widely degenerate PDEs. There is also a time-dependent parabolic counterpart. This parabolic PDE appears in models of gas filtration with nonlinear effects, where the flow starts only above a certain critical pressure. There are many important examples of PDEs with a degenerate structure, such as the elliptic and parabolic p-Laplace equation, the porous medium equation, the Stefan problem, PDEs with vanishing coefficients, etc. Each of them has its own peculiarities. Deep analytical techniques are required to understand them. Over the past few decades, some understanding of regularity for these equations has been developed. On the other hand, regularity theory for widely degenerate PDEs is a largely open field. In this project, we systematically investigate the topic to provide a better understanding of PDEs with general degenerate structures.