Existence results for diffuse interface models describing phase separation and damage

In this paper, we analytically investigate multi-component Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form $\int_\Omega\frac{1}{2}\mathbf\Gamma\nabla c:\nabla c+\frac{1}{2}|\nabla z|^2+W^$\mathrm{c}\mathrm{h}$©+W^$\mathrm{e}\mathrm{l}$(e,c,z),\mathrm dx$ with a polynomial or logarithmic chemical energy density $W^{\mathrm{c}\mathrm{h}}$, an inhomogeneous elastic energy density $W^{\mathrm{e}\mathrm{l}}$ and a quadratic structure of the gradient of the damage variable $z$. For the corresponding elastic Cahn-Hilliard and Allen-Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for the strain $e$.