Derivation of Darcy’s law in randomly perforated domains

We consider the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with Dirichlet boundary conditions. We assume that the holes are spherical and have random centres and radii. We impose that the average distance between the balls is of size $$\varepsilon $$ ε and their average radius is $$\varepsilon ^{\alpha }$$ ε α , $$\alpha \in (1; 3)$$ α ∈ ( 1 ; 3 ) . We prove that, as in the periodic case (Allaire, G., Arch. Rational Mech. Anal. 113(113):261–298, 1991), the solutions converge to the solution of Darcy’s law (or its scalar analogue in the case of Poisson). In the same spirit of (Giunti, A., Höfer, R., Ann. Inst. H. Poincare’- An. Nonl. 36(7):1829–1868, 2019; Giunti, A., Höfer, R., Velàzquez, J.J.L., Comm. PDEs 43(9):1377–1412, 2018), we work under minimal conditions on the integrability of the random radii. These ensure that the problem is well-defined but do not rule out the onset of clusters of holes.

Combined Effects in Singular Elliptic Problems in Punctured Domain

The paper deals with nonlinear elliptic differential equations subject to some boundary value conditions in a regular bounded punctured domain. By means of properties of slowly regularly varying functions at zero and the Schauder fixed-point theorem, we establish the existence of a positive continuous solution for the suggested problem. Global estimates on such solution, which could blow up at the origin, are also obtained.

Spectral properties of the iterated Laplacian with a potential in a punctured domain

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m 2 N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ? d ? 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ?nonharmonic? analysis.