We prove an explicit formula for the tangential derivatives of the double layer heat potential. By exploiting such a formula, we prove the validity of a regularizing property for the integral operator associated to the double layer heat potential in spaces of functions with high-order derivatives in parabolic Hölder spaces defined on the boundary of parabolic cylinders which are unbounded in the time variable.

We study the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. The periodicity cell is a rectangle of sides of length l and 1/l, where l is a positive parameter, and the shape of the cross section of the cylinders is determined by the image of a fixed domain through a diffeomorphism φ. We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair (l, φ), one defines the average of the longitudinal component of the flow velocity Σ[l, φ]. Here, we prove that the quantity Σ[l, φ] depends analytically on the pair (l, φ), which we consider as a point in a suitable Banach space.

We consider the effective conductivity λ eff of a periodic two-phase composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. Then we study the behavior of λ eff upon perturbation of the shape of the inclusions, of the periodicity structure, and of the conductivity of each material.

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.