For a fixed bounded Lipschitz domain and a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we investigate a necessary and sufficient condition that an A1 weight ...ω must satisfy in order for the weighted W1,2(ω) estimates for weak solutions of Neumann problems to be true.
Moreover, in any Lipschitz domain, under the assumption that the coefficient A is Hölder continuous, we prove that the uniform W1,p estimates for solutions to the Neumann problem hold for 2dd+1−δ<p<2dd−1+δ. As a by-product, in non-periodic setting with A∈VMO, we are able to show that the W1,p estimates hold for 2dd+1−δ<p<2dd−1+δ. The ranges are sharp for d=2,3.
Finally, we prove an extrapolation result for Lp Dirichlet problems for systems of linear elasticity. Specifically, we extrapolate from solvability for 1<p0<2(d−1)d−2 to the range p0<p<2(d−1)d−2+δ. The novelty is that the method avoids using the L2 regularity estimate.
We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain Ω subject to homogeneous Dirichlet boundary conditions. We prove Lp-resolvent estimates for p satisfying the ...condition |1/p−1/2|<1/4+ε for some ε>0. We further show that the Stokes operator admits the property of maximal regularity and that its H∞-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutions to the Navier–Stokes equations in bounded planar Lipschitz domains.
Let n≥2, w be a Muckenhoupt A2(Rn) weight, Ω a bounded Lipschitz domain of Rn, and L:=−w−1div(A∇⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this ...article, the authors establish the following weighted Lp estimate for the Kato square root of L:‖L1/2(f)‖Lp(Ω,vw)∼‖∇f‖Lp(Ω,vw) for any f∈W01,p(Ω,vw) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,p(Ω,vw) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w, W01,p(Ω,vw) and W1,p(Ω,vw) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w, via letting v:=w−1, the unweighted L2 estimate for the Kato square root of L that ‖L1/2(f)‖L2(Ω)∼‖∇f‖L2(Ω) for any f∈W01,2(Ω) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,2(Ω) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L2 estimates, the unweighted L2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established.
The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this ...paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains.
Inequalities for the eigenvalues of the (negative) Laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established. The eigenvalues subject to a ...Dirichlet boundary condition on a part of the boundary and a Neumann boundary condition on the remainder of the boundary are estimated in terms of either Dirichlet or Neumann eigenvalues. The results complement several classical inequalities between Dirichlet and Neumann eigenvalues due to Pólya, Payne, Levine and Weinberger, Friedlander, and others.
This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi-boundary triples and associated Weyl functions is developed further in such a way that it ...can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kreĭn type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results by Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.
Let n≥2 and Ω be a bounded Lipschitz domain in Rn. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second ...order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p∈(2,∞), two necessary and sufficient conditions for W1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W1,q estimates of solutions with q∈2,p and some Muckenhoupt weights, are obtained. As applications, for any given p∈(1,∞) and ω∈Ap(Rn) (the class of Muckenhoupt weights), the authors establish weighted Wω1,p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.
The square root of the heat operator ∂t−Δ, can be realized as the Dirichlet to Neumann map of the heat extension of data on Rn+1 to R+n+2. In this note we obtain similar characterizations for general ...fractional powers of the heat operator, (∂t−Δ)s, s∈(0,1). Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.
•A Dirichlet boundary value problem for the nonlinear D-F-B system on a bounded Lipschitz domain in Rnn=2,3, is considered.•The existence and uniqueness of a weak solution of the linear Brinkman ...system is proved using the potential theory technique.•The well-posed result is extended to the nonlinear D-F-B system using Banach Contraction Principle.•A numerical simulation of the flow in a two dimensional lid-driven porous cavity with internal square block is performed taking into account different parameters.
In this paper we are concerned with both theoretical and numerical study of a Dirichlet boundary value problem for the nonlinear Darcy-Forchheimer-Brinkman system on a bounded Lipschitz domain in Rn(n=2,3). Using the potential theory technique we obtain a well-posed theorem which implies the existence and uniqueness of a weak solution for the aforementioned Dirichlet problem when the boundary data belongs to a L2-based Sobolev space. A numerical investigation of the flow of a viscous fluid through a two dimensional lid-driven porous cavity with a solid square block is performed. The effect of the dimension and position of the internal obstacle on the flow behaviour is analysed.
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