Continuity of the double layer potential of a second order elliptic differential operator in Schauder spaces on the boundary

We prove the validity of a regularizing property on the boundary of the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M.~G\"{u}nter, S.~Mikhlin, V.D.~Kupradze, T.G.~Gegelia, M.O.~Basheleishvili and T.V.~Bur\-chuladze, U.~Heinemann and extends the work of A.~Kirsch who has considered the case of the Helmholtz operator.Comment: arXiv admin note: text overlap with arXiv:2103.0697

Poisson problems for semilinear Brinkman systems on Lipschitz domains in Rn

The purpose of this paper is to combine a layer potential analysis with the Schauder fixed point theorem to show the existence of solutions of the Poisson problem for a semilinear Brinkman system on bounded Lipschitz domains in Rn (n 65 2) with Dirichlet or Robin boundary conditions and data in L2-based Sobolev spaces. We also obtain an existence and uniqueness result for the Dirichlet problem for a special semilinear elliptic system, called the Darcy\u2013Forchheimer\u2013 Brinkman system

Two-parameter anisotropic homogenization for a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain:A functional analytic approach

We consider a Dirichlet problem for the Poisson equation in an unbounded period- ically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter , and the level of anisotropy of the cell is determined by a diagonal matrix with positive diagonal entries. The relative size of each periodic perforation is instead determined by a positive parameter . For a given value ̃ of , we analyze the behavior of the unique solution of the problem as (, , ) tends to (0, 0, ̃ ) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory

Two-parameter homogenization for a nonlinear periodic Robin problem for the Poisson equation:a functional analytic approach

We consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter delta. The relative size of each periodic perforation is instead determined by a positive parameter epsilon. We prove the existence of a family of solutions which depends on epsilon and delta and we analyze the behavior of such a family as (epsilon,delta) tends to (0,0 ) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory

A functional decomposition theorem for the conformal representation

n/

Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems

The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces

The large deformation ofnonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations of an elastic ring under the action of an external two-dimensional uniformly subsonic potential barotropic steady-state gas flow is considered. The configurations of the elastic ring are identified by a pair of functions (zeta, psi). The simple curve zeta represents the shape of the ring and the real-valued function psi identifies the orientation of the material sections of the ring. The pressure field on the ring depends nonlocally on zeta, and on two parameters U and P which represent the pressure and the velocity at infinity. The system is shown to be equivalent to a fixed-point problem, which is then treated with continuation methods. It is shown that the solution branch ensuing from certain equilibrium states ((zeta0, psi0), 0, P0) in the solution-parameter space of ((zeta0, psi0), 0, P0) either approaches the boundary of the admissible ((zeta, psi), U, p)'s in a well-defined sense, or is unbounded, or is homotopically nontrivial in the sense that there exists a continuous map sigma from the branch to a two-dimensional sphere which is not homotopic in the sphere to a constant, while sigma restricted to the branch minus ((zeta0, psi0), 0, P0) is homotopic to a constant in the sphere. Furthermore, by fixing the pressure parameter at P0 and by considering the one-parameter problem in ((zeta, psi), U), the following holds. Every hyperplane in the solution-parameter space of the ((zeta, psi), U)'s which contains the equilibrium state ((zeta0, psi0), 0) and does not include a well-determined one-dimensional subspace intersects the solution branch above at a point different from ((zeta0, psi0), 0)