Lipschitz Domain In Rn Research Articles

Regularity of variational solutions to linear boundary value problems in Lipschitz domains

In a bounded Lipschitz domain in ℝn, we consider a second-order strongly elliptic system with symmetric principal part written in divergent form. We study the Neumann boundary value problem in a generalized variational (or weak) setting using the Lebesgue spaces Hpσ(Ω) for solutions, where p can differ from 2 and σ can differ from 1. Using the tools of interpolation theory, we generalize the known theorem on the regularity of solutions, in which p = 2 and {σ − 1} < 1/2, and the corresponding theorem on the unique solvability of the problem (Savare, 1998) to p close to 2. We compare this approach with the nonvariational approach accepted in numerous papers of the modern theory of boundary value problems in Lipschitz domains. We discuss the regularity of eigenfunctions of the Dirichlet, Neumann, and Poincare-Steklov spectral problems.

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝn. We deduce a sharp variant of the Brezis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brezis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].

Decompositions of the Hardy space ℋ1 z (Ω)

We give a decomposition of the Hardy space ℋ1z(Ω) into "div-curl" quantities for Lipschitz domains in ℝn. We also prove a decomposition of ℋ1z(Ω) into Jacobians det Du, u∈ W1,20 (Ω,ℝ2) for Ω in ℝ2. This partially answers a well-known open problem.

The 3G-inequality for general Schr�dinger operators on Lipschitz domains

In this paper we prove a new 3G-inequality for the Laplacian Green function on a bounded Lipschitz domain in ℝn, n≥3. We exploit this inequality to prove the existence and comparison of perturbed continuous Green functions associated with Δ−μ where μ is in a general class of signed Radon measures covering the well known Kato class.

Square roots of elliptic second order divergence operators on strongly lipschitz domains:L 2 theory

We prove the Kato conjecture for square roots of elliptic second order non-self-adjoint operators in divergence formL = -div(A∇) on strongly Lipschitz domains in ℝn, n≥2, subject to Dirichlet or to Neumann boundary conditions. The method relies on a transference procedure from the recent positive result on ℝn in [2].

On convexity and weak closeness for the set of Φ-superharmonic functions

AbstractConvexity and weak closeness of the set of Φ-superharmonic functions in a bounded Lipschitz domain in Rn is considered. By using the fact of that Φ-superharmonic functions are just the solutions to an obstacle problem and establishing some special properties of the obstacle problem, it is shown that if Φ satisfies Δ2-condition, then the set is not convex unless Φ(r) = Cr2 or n = 1. Nevertheless, it is found that the set is still weakly closed in the corresponding Orlicz-Sobolev space.

Maximum estimates for oblique derivative problems with right hand side in L p , p &lt; n

If u is a solution of the second order elliptic differential equation Lu=f in some Lipschitz domain in ℝn, we estimate the maximum of u in terms of some oblique derivative prescribed on the boundary of the domain and in terms of the Lp norm of f with p<n.

Boundary value problems in Morrey spaces for elliptic systems on Lipschitz domains

Let Ω be a bounded Lipschitz domain in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], n ≥ 3. Let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be a second order elliptic system with constant coefficients satisfying the Legendre-Hadamard condition. We consider the Dirichlet problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] = 0 in Ω, u = f on ∂Ω with boundary data f in the Morrey space L 2,λ (∂Ω). Assume that 0 ≤ λ &lt; 2 + ε for n ≥ 4 where ε &gt; 0 depends on Ω, and 0 ≤ λ ≤ 2 for n = 3. We obtain existence and uniqueness results with nontangential maximal function estimate ║( u )*║ L 2,λ(∂Ω) ≤ C ║ f ║ L 2,λ(∂Ω) . If [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] satisfies the strong elliptic condition and 0 ≤ λ &lt; min ( n -1, 2+ε), we show that the Neumann type problem [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] = 0 in Ω, ∂ u / ∂ν = g ∈ H 2,λ (∂Ω) on ∂Ω, ║(∇ u )*║ H 2,λ (∂Ω) &lt; ∞ has a unique solution. Here H 2,λ (∂Ω) is an atomic space with the property ( H 2,λ (∂Ω))* = L 2,λ (∂Ω). The invertibility of layer potentials on L 2,λ (∂Ω) and H 2,λ (∂Ω) is also obtained. Finally we study the Dirichlet problem for the biharmonic equation. We establish a similar estimate in L 2,λ for the biharmonic equation, in which case the range 0 ≤ λ &lt; 2 + ε is sharp for n = 4 or 5.

When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable?

We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in ℝN and classify the solutions whose spatial level surfaces are invariant with respect to the time variable. (Of course, the values of each solution on its spatial level surfaces vary with time.) The prototype of such classification is a result of Alessandrini, which proved a conjecture of Klamkin. He considered the initial-Dirichlet problem for the heat equation on bounded domains and showed that if all the spatial level surfaces of the solution are invariant with respect to the time variable under the homogeneous Dirichlet boundary condition, then either the initial data is an eigenfunction or the domain is a ball and the solution is radially symmetric with respect to the space variable. His proof is restricted to the initial-Dirichlet problem for the heat equation. In the present paper, in order to deal with the initial-Neumann problem, we overcome this obstruction by using the invariance condition of spatial level surfaces more intensively with the help of the classification theorem ofisoparametric hypersurfaces in Euclidean space of Levi-Civita and Segre. Furthermore, we can deal with nonlinear diffusion equations, such as the porous medium equation.

Inhomogeneous Calderón reproducing formula on spaces of homogeneous type

In this paaper we use the Calderon-Zygmund operator theory to prove an inhomogenous Calderon reproducing formula on spaces of homogeneous type with finite or infinite measures. Our formula is new even for classical spaces of homogeneous type such as the surface of the unit ball and then-torus inRn, compact Lie groups,C∞-compact Riemannian manifolds, and the boundary of any bounded Lipschitz domain inRn.

Resolvent Estimates in Lp for Elliptic Systems in Lipschitz Domains

We obtain the Lp resolvent estimates in Lipschitz domains in Rn for constant coefficient elliptic systems satisfying the Legendre-Hadamard condition, where 1 ≤ p ≤∞ for n = 3; and 2n/(n + 3) − δ < p < 2n/(n − 3) + δ for n ≥ 4 and some δ = δ(Ω) > 0.

A spline collocation method for multidimensional strongly elliptic pseudodifferential operators of order zero

In the present paper we prove the stability of a nodal spline collocation method for (locally) strongly elliptic zero order pseudodifferential equations inL2 (Ω), where Ω is a bounded Lipschitz domain inℝn. As trial functions we use multi-polynomial splines of odd multi-degree on a rectangular grid. The key of our analysis is the reduction to the case of an operator with frozen symbol by using a local principle due to one of the authors [30]. Moreover, for a right hand sightf ∈H3 (Ω),s>n/2, we obtain an asymptotic error estimate. Finally we extend these results to the case of (locally) strongly elliptic pseudodifferential equations on the torusTn.

The oblique derivative problem for the heat equation in Lipschitz cylinders

We consider a class of initial-boundary value problems for the heat equation on ( 0. T ) × Ω (0.T) \times \Omega with Ω \Omega a bounded Lipschitz domain in R n {{\mathbf {R}}^n} . On the lateral boundary, ( 0 , T ) × ∂ Ω = Σ T (0,T) \times \partial \Omega = {\Sigma _T} , we specify ⟨ α , ∇ u ⟩ \left \langle {\alpha ,\nabla u} \right \rangle where ∇ u \nabla u denotes the spatial gradient of the solution and α : Σ T → { x : | x | = 1 } \alpha :{\Sigma _T} \to \{ x:|x| = 1\} is a continuous vector field satisfying ⟨ α , ν ⟩ ≥ μ &gt; 0 \left \langle {\alpha ,\nu } \right \rangle \geq \mu &gt; 0 with ν \nu the unit normal to ∂ Ω \partial \Omega . On the initial surface, { 0 } × Ω \{ 0\} \times \Omega , we require that the solution vanish. The lateral data is taken from L p ( Σ T ) {L^p}({\Sigma _T}) . For p ∈ ( 2 − , ∞ ) p \in (2 - ,\infty ) , we show existence and uniqueness of solutions to this problem with estimates for the parabolic maximal function of the spatial gradient of the solution.

On functions subharmonic in a Lipschitz domain

Let D be a starlike Lipschitz domain in R n , n ⩾ 2 {R^n},n \geqslant 2 . If w is a subharmonic function in D with positive harmonic majorant, then at almost every point on the boundary of D (surface measure), w has radial limit. Results on limits along certain C 1 {C^1} curves in general Lipschitz domains are also obtained.