**Title:** New advances for partial differential equations with spatially varying coefficients

**Abstract:**

Partial differential equations (PDEs) have a long tradition in modeling a wide variety of phenomena in physics. In order to facilitate the mathematical analysis of PDEs, one often makes simplifying assumptions, for instance, that some quantities do not vary in time and space and, hence, only enter as parameters. In recent years, with both, the rise of ever more sophisticated means of measuring and building and an increase in the use of modeling beyond physics, there is a need for developing analytical tools for more sophisticated models. A first step in this direction is the extension of classical tools for autonomous equations to the case of spatially varying coefficients. We give two PDE examples where such extensions were successful:

- one from photonics where the coefficients encode the structure of metamaterials
- and one from ecology where the coefficients account for varying terrain in vegetation models.