The Resolvent Method for PDEs

The standard theory of evolution equations puts the semigroup — the exponential of the generator — at its centre and treats the resolvent, the inverse of the shifted generator, as scaffolding. This book reverses the order. The resolvent is the Laplace transform of the semigroup: prior to it, and, unlike it, available even where no semigroup exists.

That is not a matter of taste. There is a ladder of problems on which the semigroup degrades or vanishes while the resolvent still carries the solution: • the spectrum betrays you — non-self-adjoint generators, where pseudospectra tell the truth the eigenvalues cannot; • time refuses to factor — non-autonomous and fractional-in-time problems, where the propagator is built from frozen resolvents; • nonlinearity — where the nonlinear resolvent solves the Hamilton–Jacobi–Bellman equation.

The book carries one thesis across all three, to the doubly-hard summit: a time-inhomogeneous control problem — correcting circadian misalignment by timed light — solved as a convergent product of frozen-time nonlinear resolvents. Rigorous throughout and exact about where the method stops, it includes the Python code that reproduces every table and figure.