We define a homogeneous De Giorgi class of order $p \geqslant 2$ that contains the solutions of two evolution equations of elliptic-parabolic and forward-backward parabolic type like $\rho (x,t) u_t + A u = 0$ and $(\rho (x,t) u)_t + A u = 0$, where $\rho$, for simplicity, takes values in the set $\{ -1, 0, 1 \}$, and $A$ a suitable monotone operator. For functions belonging to this class, we prove an unusual local boundedness result.

We establish the existence of ground state and least positive energy solutions to some problems involving mixed fractional $p$-Laplacians using some abstract critical point theorems based on the $\mathbb Z_2$-cohomological index.

We consider the homogenization of an optimal control problem in which the control $v$ is placed on a part $\Gamma _{0}$ of the boundary and the spatial domain contains a thin layer of “small particles”, very close to the controlling boundary, and a Robin boundary condition is assumed on the boundary of those “small particles”. This problem can be associated with the climatization modeling of Bioclimatic Double Skin Façades which was developed in modern architecture as a tool for energy optimization. We assume that the size of the particles and the parameters involved in the Robin boundary condition are critical (and so they justify the occurrence of some “strange terms” in the homogenized problem). The cost functional is given by a weighted balance of the distance (in a $H^{1}$-type metric) to a prescribed target internal temperature $u_{T}$ and the proper cost of the control $v$ (given by its $L^{2}(\Gamma _{0})$ norm). We prove the (weak) convergence of states ${u_{\varepsilon }}$ and of the controls $ v_{\varepsilon }$ to some functions, ${u_{0}}$ and $v_{0}$, respectively, which are completely identified: ${u_{0}}$ satisfies an artificial boundary condition on $\Gamma _{0}$ and $v_{0}$ is the optimal control associated to a limit cost functional $J_{0}$ in which the “boundary strange term” on $\Gamma _{0}$ arises. This information on the limit problem makes much more manageable the study of the optimal climatization of such double skin structures.

Local and global well-posedness of the coagulation-fragmentation equation with size diffusion are investigated. Owing to the semilinear structure of the equation, a semigroup approach is used, building upon generation results previously derived for the linear fragmentation-diffusion operator in suitable weighted $L_1$-spaces.