Lipschitz Domains Research Articles

Decay estimates of gradient of the Stokes semigroup in exterior Lipschitz domains

This paper develops Lp-Lq decay estimates of the gradient of the Stokes semigroup (T(t))t≥0 generated by the negative of the Stokes operator in exterior Lipschitz domains Ω⊂Rn, n≥3. More precisely, the Lp-Lq estimates of ∇T(t) with optimal rates are proved if p and q satisfy |1/p−1/2|<1/(2n)+ε, |1/q−1/2|<1/(2n)+ε, and p≤q≤n with some ε>0, which should be useful for the study of stability of the several physically relevant flows. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier–Stokes equations in the critical space L∞(0,∞;Lσ3(Ω)) provided that the initial velocity is small in the L3-norm.

Global well‐posedness and finite time blow‐up for a class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearity

AbstractIn this paper, we consider the following class of wave equation involving fractionalp‐Laplacian with logarithmic nonlinearitywhere is a bounded domain with Lipschitz boundary, , , and is the critical exponent in the Sobolev inequality. First, via the Galerkin approximations, the existence of local solutions are obtained when . Next, by combining the potential well theory with the Nehari manifold, we establish the existence of global solutions when . Then, via the Pohozaev manifold, the existence of global solutions are obtained when . By virtue of a differential inequality technique, we prove that the local solutions blow‐up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we discuss the asymptotic behavior of solutions as time tends to infinity. Here, we point out that the main difficulty is the lack of logarithmic Sobolev inequality concerning fractionalp‐Laplacian.

Error analysis of a fully discrete projection method for magnetohydrodynamic system

AbstractIn this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi‐implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.

Coercivity of the Dirichlet-to-Neumann Operator and Applications to the Muskat Problem

We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann operator controls some sharp homogeneous fractional Sobolev norms. As an application, we prove that the global Lipschitz solutions constructed in Dong et al. (2021) for the one-phase Muskat problem decays exponentially in time in any Hölder norm Cα, α ∈ (0,1).

Weighted $$L^2$$ Estimates for Elliptic Homogenization in Lipschitz Domains

We develop a new real-variable method for weighted \(L^p\) estimates. The method is applied to the study of weighted \(W^{1, 2}\) estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. It produces a necessary and sufficient condition, which depends on the weight function, for the weighted \(W^{1,2}\) estimate to hold in a fixed Lipschitz domain with a given weight. Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients, we reduce the problem of weighted estimates to the case of constant coefficients.

Error Estimates for Adaptive Spectral Decompositions

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L^2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

Adaptive observer design for uncertain nonlinear ODE‐PDE cascades

SummaryWe are revisiting the problem of adaptive observer design for systems that are constituted of an Ordinary Differential Equation (ODE), containing a globally Lipschitz function of the state, and a linear Partial Differential Equation (PDE) of a diffusion–reaction heat type. The ODE and PDE are connected in series and both are subject to parametric uncertainties. In addition to nonlinearity and uncertainty, the system complexity also lies in the fact that no sensor can be implemented at the junction point between the ODE and the PDE. In the absence of parameter uncertainty, nonadaptive state observers are available featuring exponential convergence. However, convergence is guaranteed only under the condition that either the Lipschitz coefficient is sufficiently small or the PDE domain length is sufficiently small. To get around this limitation, and also to account for parameter uncertainty, we develop a design that involves two concatenated adaptive observers, covering the two subintervals of the PDE domain. The proposed design employs one extra sensor, providing the measurement of the PDE state at an inner position close to the ODE‐PDE junction point. Both observers are shown to be exponentially convergent, under ad‐hoc persistent excitation (PE) conditions, with no limitation on the Lipschitz coefficient and domain length.

A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any mathsf {L}_{}^{2}-bounded sequence of vector fields with mathsf {L}_{}^{2}-bounded rotations and mathsf {L}_{}^{2}-bounded divergences as well as mathsf {L}_{}^{2}-bounded tangential traces on one part of the boundary and mathsf {L}_{}^{2}-bounded normal traces on the other part of the boundary, contains a strongly mathsf {L}_{}^{2}-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

On criticality theory for elliptic mixed boundary value problems in divergence form

The paper is devoted to the study of positive solutions of a second-order linear elliptic equation in divergence form in a domain [Formula: see text] that satisfy an oblique boundary condition on a portion of [Formula: see text]. First, we study weak solutions for the degenerate mixed boundary value problem [Formula: see text] where [Formula: see text] is a bounded Lipschitz domain, [Formula: see text] is a relatively open portion of [Formula: see text], and [Formula: see text] is an oblique (Robin) boundary operator defined on [Formula: see text] in a weak sense. In particular, we discuss the unique solvability of the above problem, the existence of a principal eigenvalue, and the existence of a minimal positive Green function. Then we establish a criticality theory for positive weak solutions of the operator [Formula: see text] in a general domain [Formula: see text] with no boundary condition on [Formula: see text] and no growth condition at infinity. The paper extends results obtained by Pinchover and Saadon for classical solutions of such a problem, where stronger regularity assumptions on the coefficients of [Formula: see text], and the boundary [Formula: see text] are assumed.

The elasticity complex: compact embeddings and regular decompositions

We investigate the Hilbert complex of elasticity involving spaces of symmetric tensor fields. For the involved tensor fields and operators we show closed ranges, Friedrichs/Poincaré type estimates, Helmholtz-type decompositions, regular decompositions, regular potentials, finite cohomology groups, and, most importantly, new compact embedding results. Our results hold for general bounded strong Lipschitz domains of arbitrary topology and rely on a general functional analysis framework (FA-ToolBox). Moreover, we present a simple technique to prove the compact embeddings based on regular decompositions/potentials and Rellich's selection theorem, which can be easily adapted to any Hilbert complex.

Penalization method for the Navier–Stokes–Fourier system

We apply the method of penalization to the Dirichlet problem for the Navier–Stokes–Fourier system governing the motion of a general viscous compressible fluid confined to a bounded Lipschitz domain. The physical domain is embedded into a large cube on which the periodic boundary conditions are imposed. The original boundary conditions are enforced through a singular friction term in the momentum equation and a heat source/sink term in the internal energy balance. The solutions of the penalized problem are shown to converge to the solution of the limit problem. In particular, we extend the available existence theory to domains with rough (Lipschitz) boundary. Numerical experiments are performed to illustrate the efficiency of the method.

The Stokes operator in two-dimensional bounded Lipschitz domains

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.

Two-scale cut-and-projection convergence for quasiperiodic monotone operators

Averaging a certain class of quasiperiodic monotone operators can be simplified to the periodic homogenization setting by mapping the original quasiperiodic structure onto a periodic structure in a higher dimensional space using the cut-and projection method. We characterize the cut-and-projection convergence limit of the nonlinear monotone partial differential operator −divσx,Rxη,∇uη for a bounded sequence uη in W01,p(Ω), where 1<p<∞, and Ω is a bounded open subset in Rn with Lipschitz boundary. We identify the homogenized problem with a local equation defined on a hyperplane, or a lower dimensional plane in the higher-dimensional space. A new corrector result is established.

On the Hölder continuity of ring Q-homeomorphisms

Abstract We prove the Hölder continuity of a homeomorphism f defined on a bounded domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} with Lipschitz boundary. The homeomorphism f is assumed to belong to a certain Orlicz–Sobolev class and to satisfy a distortion condition near the boundary.

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^infty -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in {mathbb {R}}^{ntimes n} with zero matrix traces. We analyze, in L^2-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in {mathbb {R}}^n, nge 2 (n=2,3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.

A phase-field approach for detecting cavities via a Kohn–Vogelius type functional

We deal with the geometrical inverse problem of the shape reconstruction of cavities in a bounded linear isotropic medium by means of boundary data. The problem is addressed from the point of view of optimal control: the goal is to minimize in the class of Lipschitz domains a Kohn–Vogelius type functional with a perimeter regularization term which penalizes the perimeter of the cavity to be reconstructed. To solve numerically the optimization problem, we use a phase-field approach, approximating the perimeter functional with a Modica–Mortola relaxation and modeling the cavity as an inclusion with a very small elastic tensor. We provide a detailed analysis showing the robustness of the algorithm through some numerical experiments.

On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting

We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain Omega subset {mathbb {R}}^2 and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function {text {dist}}(cdot ,partial Omega ).The results are extended to anisotropic problems without splitting structure under natural growth and ellipticity conditions. In both cases we argue with variants of Caccioppoli’s inequality involving small weights.

Hilbert complexes with mixed boundary conditions—Part 2: Elasticity complex

We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from Pauly and Schomburg (2021, 2022) and recent results on the elasticity Hilbert complex with empty or full boundary conditions from Pauly and Zulehner (2020, 2022).

Hardy spaces and the Neumann problem in $$L^p$$ for elliptic equation with periodic high-contrast coefficients in Lipschitz domains

Hardy spaces and the Neumann problem in $$L^p$$ for elliptic equation with periodic high-contrast coefficients in Lipschitz domains

Error analysis of a class of semi-discrete schemes for solving the Gross–Pitaevskii equation at low regularity

We analyze a class of time discretizations for solving the nonlinear Schrödinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain Ω⊂Rd, d≤3. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space Hr, r≥0, beyond the more typical L2 or Hσ(σ>d2) -error analysis. Numerical experiments illustrate our results.