Project: All physically meaningful solutions of the PDEs in Mathematical Physics are actually Hausdorff continuous functions.

Background: In the current state of the art any class of PDEs comes with own space, own concept of generalized solution and own method of singling out a unique physically meaningful solution. The formulated conjecture seems to be a major step towards building a unified approach or theory for PDEs.

H-continuous functions: Extended real interval valued functions with closed graph which are minimal with respect to inclusion. The space of nearly finite H-continuous functions provides completion of the set of continuous functions with respect to order and with respect to various topological processes. It is rationally complete as a ring and is the largest set of functions on which the algebraic operations on the space of continuous functions can be extended in an unambiguous way. Further, it is also the largest set of interval functions which is also a linear space. With all these properties this space is rather small particularly when compared with the than the typical PDE spaces like the ones of Lebesgue and Sobolev. Hence, any the proof of the project statement for any particular equation typically provides a significant improvement on the regularity of the generalized solution.

The project statement is motivated by an abstract existence results by the order completion method as well as results for viscosity solutions of Hamilton-Jacobi equations and for the entropy solution of scalar conservation laws.

We invite interested colleagues to prove or disprove the statement in their particular area of interest.

Is there Hille-Yosida Theorem for nonlinear semigroups on Vector Lattices?

Background: In the current state of the art any class of PDEs comes with own space, own concept of generalized solution and own method of singling out a unique physically meaningful solution. The formulated conjecture seems to be a major step towards building a unified approach or theory for PDEs.

H-continuous functions: Extended real interval valued functions with closed graph which are minimal with respect to inclusion. The space of nearly finite H-continuous functions provides completion of the set of continuous functions with respect to order and with respect to various topological processes. It is rationally complete as a ring and is the largest set of functions on which the algebraic operations on the space of continuous functions can be extended in an unambiguous way. Further, it is also the largest set of interval functions which is also a linear space. With all these properties this space is rather small particularly when compared with the than the typical PDE spaces like the ones of Lebesgue and Sobolev. Hence, any the proof of the project statement for any particular equation typically provides a significant improvement on the regularity of the generalized solution.

The project statement is motivated by an abstract existence results by the order completion method as well as results for viscosity solutions of Hamilton-Jacobi equations and for the entropy solution of scalar conservation laws.

We invite interested colleagues to prove or disprove the statement in their particular area of interest.

Is there Hille-Yosida Theorem for nonlinear semigroups on Vector Lattices?