Lipschitz Cylinders Research Articles

A refined convergence result in homogenization of second order parabolic systems

We derive the sharp O(ε) convergence rate in L2(0,T;Lq0(Ω)),q0=2d/(d−1) in periodic homogenization of second order parabolic systems with bounded measurable coefficients in Lipschitz cylinders. This extends the corresponding result for elliptic systems established in [20] to parabolic systems and improves the corresponding result in L2 settings derived in [7,28] for second order parabolic systems with time-dependent coefficients.

Boundary estimates for non-negative solutions to non-linear parabolic equations

This paper is mainly devoted to the boundary behavior of non-negative solutions to the equation $$\begin{aligned} \mathrm{H }u =\partial _tu-\nabla \cdot \mathrm{A }(x,t,\nabla u) = 0 \end{aligned}$$in domains of the form \(\Omega _T=\Omega \times (0,T)\) where \(\Omega \subset \mathbb R^n\) is a bounded non-tangentially accessible (NTA) domain and \(T>0\). The assumptions we impose on \(A\) imply that \(H\) is a non-linear parabolic operator with linear growth. Our main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerning the boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type \(\partial _tu-\nabla \cdot (\mathrm{A }(x,t)\nabla u)=0\), where \(\mathrm{A }\) is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green’s function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

Asymptotic Behavior of Positive Harmonic Functions in Certain Unbounded Domains

We derive asymptotic estimates at infinity for positive harmonic functions in a large class of non-smooth unbounded domains. These include domains whose sections, after rescaling, resemble a Lipschitz cylinder or a Lipschitz cone, e.g., various paraboloids and horns.

Boundary estimates for certain degenerate and singular parabolic equations

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p -Laplacian equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S_T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

On Initial-Boundary Value Problem of the Stochastic Heat Equation in Lipschitz Cylinders

We consider the initial boundary value problem of the non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random noises. The space boundary is Lipschitz, and we impose nonzero condition on the parabolic boundary. We prove a regularity result by finding appropriate spaces for solutions and pre-assigned data in the problem. We use a collection of tools from potential theory, harmonic analysis, and probability. Some lemmas are as important as the main theorem.

$H^\infty$-Calculus for cylindrical boundary value problems

In this note an ${\mathcal{R}}$-bounded ${\mathcal{H}}^\infty$-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis; cf. [28]. Cylindrical in this context means that both domain and differential operator possess a certain cylindrical structure. In comparison to standard methods (e.g. localization procedures), our approach appears less technical and provides short proofs. Besides, we are even able to deal with some classes of equations on rough domains. For instance, we can extend the well-known (and in general sharp) range for $p$ such that the (weak) Dirichlet Laplacian admits an ${\mathcal{H}}^\infty$-calculus on $L^p(\Omega)$, from $(3+\varepsilon)'<p<3+\varepsilon$ to $(4+\varepsilon)'<p<4+\varepsilon$ for three-dimensional bounded or unbounded Lipschitz cylinders $\Omega$. Our approach even admits mixed Dirichlet Neumann boundary conditions in this situation.

Boundary value problems for heat equation with certain potentials in Lipschitz cylinders

Abstract The Dirichlet and Neumann problems of heat equation with certain potentials of the form ∂ t - Δ + V = 0 $\partial _t-\Delta +V=0$ in Lipschitz cylinders are studied, where V is a nonnegative potential satisfying certain conditions.

Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H p σ and B p σ

In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel-Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations.

Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains

We consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz cylinders. We derive an optimal elliptic-type Harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times. We show several nontrivial applications of both the exponential estimate and the derived Harnack inequality.

Optimal limiting absorption principle for a Schrödinger type operator on a Lipschitz cylinder

Spectral properties of a Schrodinger type operator on a cylinder is established by a version of Mourre's conjugate operator method. In particular, the limiting absorption principle is proved in an optimal scale of Besov spaces.

Potential theoretic methods for the hyperbolic Maxwell system in rough cylinders

AbstractThis paper is concerned with the solvability of the hyperbolic Maxwell system in Lipschitz cylinders via integral methods. By means of the Laplace transform, the original hyperbolic Maxwell system is reduced to a family of stationary Maxwell equations indexed by ω, the Laplace variable. The crux of the matter is establishing how all relevant constants in the main estimates for this family of Maxwell equations depend on ω. Among other things, appropriate norms and weighted Sobolev spaces are considered so that the integral equation method works. The solution is expressed in terms of retarded potentials. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Initial boundary-value problems for the parabolic Maxwell system in Lipschitz cylinders

Initial boundary-value problems for the parabolic Maxwell system in Lipschitz cylinders

A Note on Boundary Value Problems for the Heat Equation in Lipschitz Cylinders

We study the initial Dirichlet problem and the initial Neumann problem for the heat equation in Lipschitz cylinders, with boundary data in mixed norm spaces ${L^q}(0,T,{L^p}(\partial \Omega ))$.

A note on boundary value problems for the heat equation in Lipschitz cylinders

We study the initial Dirichlet problem and the initial Neumann problem for the heat equation in Lipschitz cylinders, with boundary data in mixed norm spaces L q ( 0 , T , L p ( ∂ Ω ) ) {L^q}(0,T,{L^p}(\partial \Omega )) .

Boundary value problems in Lipschitz cylinders for three-dimensional parabolic systems

We consider initial-boundary value problems for a parabolic system in a Lipschitz cylinder. When the space dimension is three, we obtain estimates for the solutions when the lateral data taken from the best possible range of L^p -spaces.

Boundary Value Problems for Parabolic Lame Systems and a Nonstationary Linearized System of Navier-Stokes Equations in Lipschitz Cylinders

Boundary Value Problems for Parabolic Lame Systems and a Nonstationary Linearized System of Navier-Stokes Equations in Lipschitz Cylinders

The Initial-Neumann Problem for the Heat Equation in Lipschitz Cylinders

We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in ${L^p}$, $1 < p < 2+\varepsilon$, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space ${H^1}$. Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and half of a time derivative in ${L^p}$, $1 < p < 2+\varepsilon$, with a corresponding Hardy space result when $p = 1$. Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in ${L^q}$, $2 - \varepsilon ’ < q < \infty$ and BMO may be represented as double-layer heat potentials.

Boundary integral operators for the heat equation

We study the integral operators on the lateral boundary of a space-time cylinder that are given by the boundary values and the normal derivatives of the single and double layer potentials defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show that the 2×2 matrix of these operators defines a bounded and positive definite bilinear form on certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive error estimates for various Galerkin methods. An example is the numerical approximation of an eddy current problem which is an interface problem with the heat equation in one domain and the Laplace equation in a second domain. Results of numerical computations for this problem are presented.

The initial-Neumann problem for the heat equation in Lipschitz cylinders

We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in L p {L^p} , 1 &gt; p &gt; 2 + ε 1 &gt; p &gt; 2+\varepsilon , with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space H 1 {H^1} . Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and half of a time derivative in L p {L^p} , 1 &gt; p &gt; 2 + ε 1 &gt; p &gt; 2+\varepsilon , with a corresponding Hardy space result when p = 1 p = 1 . Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in L q {L^q} , 2 − ε ′ &gt; q &gt; ∞ 2 - \varepsilon ’ &gt; q &gt; \infty and BMO may be represented as double-layer heat potentials.

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 $(0.T) \times \Omega$ with $\Omega$ a bounded Lipschitz domain in ${{\mathbf {R}}^n}$. On the lateral boundary, $(0,T) \times \partial \Omega = {\Sigma _T}$, we specify $\left \langle {\alpha ,\nabla u} \right \rangle$ where $\nabla u$ denotes the spatial gradient of the solution and $\alpha :{\Sigma _T} \to \{ x:|x| = 1\}$ is a continuous vector field satisfying $\left \langle {\alpha ,\nu } \right \rangle \geq \mu > 0$ with $\nu$ the unit normal to $\partial \Omega$. On the initial surface, $\{ 0\} \times \Omega$, we require that the solution vanish. The lateral data is taken from ${L^p}({\Sigma _T})$. For $p \in (2 - ,\infty )$, we show existence and uniqueness of solutions to this problem with estimates for the parabolic maximal function of the spatial gradient of the solution.