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

Loewner PDE in infinite dimensions

In this paper, we prove the existence and uniqueness of the solution $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach space $X$. We also give improvements of the results obtained recently by Hamada and Kohr, but we omit their proofs for the sake of brevity. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings $v(z,s,t)$ with normalization $Dv(0,s,t)=e^{-(t-s)A}$ for $t\geq s\geq 0$, where $m(A)>0$, which satisfy the semigroup property on the unit ball of a complex Banach space $X$. We further obtain the biholomorphicity of $A$-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space $X$. We prove the existence of the biholomorphic solutions $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$ on the unit ball of a separable reflexive complex Banach space $X$. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013

The L2L^2-unique continuation property on manifolds with bounded geometry and the deformation operator

A differential operator $T$ satisfies the $L^2$-unique continuation property if every $L^2$-solution of $T$ that vanishes on an open subset vanishes identically. We study the $L^2$-unique continuation property of an operator $T$ acting on a manifold with bounded geometry. In particular, we establish some connections between this property and the regularity properties of $T$. As an application, we prove that the deformation operator on a manifold with bounded geometry satisfies regularity and $L^2$-unique continuation properties. As another application, we prove that suitable elliptic operators are invertible (Hadamard well-posedness). Our results apply to compact manifolds, which have bounded geometry

Essentially translation invariant pseudodifferential operators on manifolds with cylindrical ends

We study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of pseudodifferential operators that are ``translation invariant at infinity'' and the class of ``essentially translation invariant operators'' that have appeared in the study of layer potential operators on manifolds with straight cylindrical ends. Both classes are close to the $b$-calculus considered by Melrose and Schulze and to the $c$-calculus considered by Melrose and Mazzeo-Melrose. Our calculi, however, are different and, while some of their properties follow from those of the $b$- or $c$-calculi, many of their properties do not. In particular, the ``essentially translation invariant calculus'' is spectrally invariant, a property not enjoyed by the ``translation invariant at infinity'' calculus or the $b$-calculus. For our calculi, we provide easy, intuitive proofs of the usual properties: stability for products and adjoints, mapping and boundedness properties for operators acting between Sobolev spaces, regularity properties, existence of a quantization map and topological properties of our algebras, the Fredholm property. Since our applications will be to the Stokes operator, we systematically work in the setting of Agmon-Douglis-Nirenberg-elliptic operators.Comment: 39 page

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

Transmission Problems for the Navier–Stokes and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains on Compact Riemannian Manifolds

M. Kohr acknowledges the support of the Grant PN-II-ID-PCE-2011-3-0994 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI. The research has been also partially supported by the Grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK

Certain partial differential subordinations on some Reinhardt domains in Cnℂ^n

We obtain an extension of Jack-Miller-Mocanu's Lemma for holomorphic mappings defined in some Reinhardt domains in $ℂ^n$. Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain $B_{2p}$ with p ≥ 1

An indirect boundary integral method for an oscillatory Stokes flow problem

The purpose of this paper is to present an indirect boundary integral method for the oscillatory Stokes flow provided by the translational oscillations of two rigid spheres in an incompressible Newtonian fluid of infinite expanse