The aim of our work is to give a well-posedness result for a boundary value problem of transmission-type for the nonlinear, generalized Darcy-Forchheimer-Brinkman and Stokes systems in complementary Lipschitz domains in R 3 . First, we introduce the Sobolev spaces in which we seek our solution, then we define the trace operators and conormal derivative operators that are involved in the boundary conditions of our treated problem. Next, we state a result that concerns the well-posedness of the transmission problem for the generalized Brinkman and Stokes system in complementary Lipschitz domains in R 3 . Afterwards, we state and prove an important lemma. Finally, we obtain our desired result by employing the well-posedness of the linearized version of our problem and Banach's fixed point theorem.
The aim of this paper is to establish a well‐posedness result for a boundary value problem of transmission‐type for the standard and generalized Brinkman systems in two Lipschitz domains in R3, the former being bounded, and the latter, its complement in R3. As a first step, we establish a well‐posedness result for a transmission problem for the standard Brinkman systems on complementary Lipschitz domains in R3 by making use of the Potential theory developed for such a system. As a second step, we prove our desired result (in L2‐based Sobolev spaces) by using a method based on Fredholm operator theory and the well‐posedness result from the previous step.
The purpose of this paper is to give a well-posedness result for a boundary value problem of transmission-type for the Stokes and generalized Brinkman systems in two complementary Lipschitz domains in R 3 . In the first part of the paper, we have introduced the classical and weighted L 2 -based Sobolev spaces on Lipschitz domains in R 3 . Afterwards, the trace and conormal derivative operators are defined in the case of both Stokes and generalized Brinkman systems. Also, a summary of the main properties of the layer potential operators for the Stokes system, is provided. In the second part of the work, we exploit the well-posedness of another transmission problem concerning the Stokes system on two complementary Lipschitz domains in R 3 which is based on the Potential Theory for the Stokes system. Then, certain properties of Fredholm operators will allow us to show our main well-posedness result in L 2 -based Sobolev spaces.
The aim of this paper is to state and prove certain inequalities that involve means (e.g., the arithmetic, geometric, logarithmic means) using a particular result. First of all we recall useful properties of a real-valued convex function that will be used in the proof of our inequalities. Further, we present three inequalities, the first involving the logarithmic mean, the second involving the classical arithmetical and geometrical means and in the last we introduce a new mean. Finally, we give alternate proofs to the Schweitzer’s inequality and Khanin’s inequality.
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