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?