Lipschitz Domains Research Articles

Korn and Poincaré-Korn inequalities: A different perspective

We present a concise point of view on the first and the second Korn’s inequality for general exponent p p and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for p = 2 p = 2 , uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent 1 > p > ∞ 1>p>\infty remains the same, the only change being invoking now the q q -Riesz representation theorem (with q q the harmonic conjugate of p p ). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains

For bounded domains Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} with Lipschitz boundary Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type A=B∗B0-NbBγ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathcal {A}}=\\begin{pmatrix} B^*B & 0 \\\\ -{\\mathscr {N}}^{\\mathfrak {b}}B & \\gamma \\end{pmatrix}, \\end{aligned}$$\\end{document}where B is associated to a quadratic form b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathfrak {b}}$$\\end{document} and Nb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {N}}^{{\\mathfrak {b}}}$$\\end{document} an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space H:=L2(Ω)×L2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {H}}:=L^2(\\Omega ) \ imes L^2(\\Gamma )$$\\end{document}. Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form B=-divQ∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B=-\\mathop {{div} }Q \ abla $$\\end{document} with Q∈C1,1(Ω¯,Rd×d),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q \\in C^{1,1}({\\overline{\\Omega }},{\\mathbb {R}}^{d\ imes d}),$$\\end{document} also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.

First mixed Laplace eigenfunctions with no hot spots

The hot spots conjecture of J. Rauch states that the second Neumann eigenfunction of the Laplace operator on a bounded Lipschitz domain in R n \mathbb {R}^n attains its extrema only on the boundary of the domain. We present an analogous problem for domains with mixed Dirichlet-Neumann boundary conditions. We then solve this problem for Euclidean triangles and a class of planar domains bounded by the graphs of certain piecewise smooth functions.

On a Dirichlet problem related to anisotropic fluid flow in bidisperse porous media

This paper is concerned with the study of a Dirichlet boundary value problem for a system of two coupled anisotropic Darcy–Forchheimer–Brinkman equations on a bounded Lipschitz domain in . Using variational methods and fixed point techniques, we obtain a well‐posedness result in ‐based Sobolev spaces for sufficiently small data. As an application, we investigate numerically the lid‐driven flow problem in a square cavity saturated with a bidisperse porous medium and analyze the effect of various physical parameters on the fluid motion.

Numerical solution to the Neumann problem in a Lipschitz domain, based on random walks

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods.

Gaussian estimates vs. elliptic regularity on open sets

AbstractGiven an elliptic operator $$L= - {{\,\textrm{div}\,}}(A \nabla \cdot )$$ L = - div ( A ∇ · ) subject to mixed boundary conditions on an open subset of $$\mathbb {R}^d$$ R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Nitsche method for Navier–Stokes equations with slip boundary conditions: convergence analysis and VMS-LES stabilization

In this paper, we analyze Nitsche’s method for the stationary Navier–Stokes equations on Lipschitz domains under minimal regularity assumptions. Our analysis provides a robust formulation for implementing slip (i.e., Navier) boundary conditions in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using the Banach Nečas–Babuška and Banach fixed point theorems under standard small data assumptions. We also provide optimal convergence rates for the approximation error. Furthermore, we propose a quasi-static VMS-LES formulation with Nitsche for the Navier–Stokes equations with slip boundary conditions to address the simulation of incompressible fluids at high Reynolds numbers. We validate our theory through several numerical tests in well-established benchmark problems.

Parabolic Boundary Harnack Inequalities with Right-Hand Side

We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side f∈Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f \\in L^q$$\\end{document} for q>n+2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q > n+2$$\\end{document}. In the case of the heat equation, we also show the optimal C1-ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1-\\varepsilon }$$\\end{document} regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} in the parabolic obstacle problem and in the parabolic Signorini problem.

Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian

For a given constant λ>0 and a bounded Lipschitz domain D⊂Rn (n≥2), we establish that almost-minimizers of the functionalJ(v;D)=∫D∑i=1m|∇vi(x)|p+λχ{|v|>0}(x)dx,1<p<∞, where v=(v1,⋯,vm), and m∈N, exhibit optimal Lipschitz continuity in compact sets of D. Furthermore, assuming p≥2 and employing a distinctly different methodology, we tackle the issue of boundary Lipschitz regularity for v. This approach simultaneously yields an alternative proof for the optimal local Lipschitz regularity for the interior case.

Modified Neumann–Neumann methods for semi- and quasilinear elliptic equations

The Neumann–Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann–Neumann methods that have better convergence properties and require fewer computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov–Poincaré operators.

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

Sublinear Elliptic Equations with Unbounded Coefficients in Lipschitz Domains

This paper is concerned with the homogeneous Dirichlet problem for a sublinear elliptic equation with unbounded coefficients in a Lipschitz domain. Bilateral a priori estimates for positive solutions and a priori upper estimates for their gradients are presented as a byproduct of the boundary Harnack principle. These estimates allow us to show the uniqueness of a positive solution of the homogeneous Dirichlet problem under no information about normal derivatives unlike in smooth domains.

On the Formation of Microstructure for Singularly Perturbed Problems with Two, Three, or Four Preferred Gradients

In this manuscript, singularly perturbed energies with 2, 3, or 4 preferred gradients subject to incompatible Dirichlet boundary conditions are studied. This extends results on models for martensitic microstructures in shape memory alloys (N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2$$\\end{document}), a continuum approximation for the J1-J3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_1-J_3$$\\end{document}-model for discrete spin systems (N=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=4$$\\end{document}), and models for crystalline surfaces with N different facets (general N). On a unit square, scaling laws are proved with respect to two parameters, one measuring the transition cost between different preferred gradients and the other measuring the incompatibility of the set of preferred gradients and the boundary conditions. By a change of coordinates, the latter can also be understood as an incompatibility of a variable domain with a fixed set of preferred gradients. Moreover, it is shown how simple building blocks and covering arguments lead to upper bounds on the energy and solutions to the differential inclusion problem on general Lipschitz domains.

An essay on the foundations of variational methods: Exploring Sobolev Spaces for boundary integral equations

This work addresses the uniqueness and regularity of solutions to integral equations associated with elliptic boundary value problems in irregular domains. Traditional results often assume smooth (Lipschitz) boundaries, but this study extends these results to more general domains with irregular boundaries. By leveraging Sobolev spaces, particularly fractional Sobolev spaces , and the properties of the Slobodetskii norm, we develop a robust theoretical framework. Our main theorem demonstrates that, under suitable conditions, has a unique solution in , and this solution inherits the regularity properties from the function . The results provide significant advancements in the mathematical understanding of boundary value problems in non-smooth domains, with potential applications in various fields of physics and engineering.

Weighted maximal regularity estimates of higher order elliptic systems on Lipschitz domains

For the homogeneous biharmonic equation Δ2u=0 in Ω with boundary data in WAk(∂Ω,ωσdS), the Banach space of the kth order Whitney Arrays on ∂Ω, we investigate values for σ in the power weightωσ=|z|σforσ>1−d such that the weighted L2 Dirichlet or regularity problem is solvable. Specifically, we show that there exists α>0 such that the Dirichlet and regularity problems are uniquely solvable for−min⁡(2+α,d−1)<σ<α, and−α<σ<2+α respectively. This gives a partially affirmative answer to a question posed by Shen (2005) [15].

The mixed problem for the Darcy-Forchheimer-Brinkman system

This paper studies the mixed problems for the Brinkman system in Hs(Ω;Rm)×Hs−1(Ω) with 1≤s≤2. Here Ω⊂Rm is a bounded domain with Lipschitz boundary (not necessarily connected). Then, using a fixed point theorem we prove the existence of solutions of the mixed problems for the Darcy-Forchheimer-Brinkman system in these spaces.

Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains

We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace’s equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document}, d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}, in the space L2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\Gamma )$$\\end{document}, where Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace’s equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space L2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\Gamma )$$\\end{document}. Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in L2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L^2(\\Gamma )}$$\\end{document} do not converge when applied to the standard second-kind formulations, but do converge for the new formulations.

On some Robin–transmission problems for the Brinkman system and a Navier–Stokes type system

The aim of this paper is twofold. The first goal is to give well‐posedness results for a Robin–transmission problem and a Robin–Dirichlet problem for the Brinkman system in bounded Lipschitz domains in . The Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by employing the layer potential theoretical methods. The second goal is to provide a well‐posedness result for a Robin–transmission problem and an existence result for the Robin–Dirichlet problem for a Navier–Stokes type system in bounded Lipschitz domains in . Again, the Robin–Dirichlet problem appears as a special limit case of the Robin–transmission problem. This aim is achieved by using the well‐posedness results in the linear case combined with a fixed point argument. In order to illustrate our theoretical results, we also consider a special case of boundary value problems of Robin–Dirichlet type, which models a fluid flow in a porous square cavity. We solve numerically this problem.

S-asymptotically ω periodic solutions of heat equation

We mean by an S-asymptotically ω-periodic function any continuous and bounded function from the real axis to a Banach space that converges to a periodic function as t tends to infinity. We study in this paper the existence, uniqueness and differentiable dependence on the S-asymptotically ω-periodic mild solution of a heat equation on the space of continuous functions from a non-empty n-dimensional bounded domain with a Lipschitz boundary to the real axis. The dependence of the solution concerns the initial conditions, more precisely when the initial conditions is a S-asymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of heat equation. To show our main result in this work we introduce the properties of the superposition operator, or also called Nemytskii operator, in the space of S-asymptotically ω-periodic functions. The notion of derivation for the last operator will also be highlighted. We also use in this paper semi-groups which have become important tools for differential equations. In this study, our focus shifted from seeking an s-asymptotically w-periodic solution for our heat equation, which is a problem in dynamic systems, to one in functional analysis. More precisely, our strategy consists of applying the implicit function theorem on a certain operator that we constructed on our workspaces in order to achieve the objective described in our main theorem. In fact our theorem gives conditions to ensure that around a mild S-asymptotically ω-periodic solution of our heat equation with an initial value, there exists a regular (in the usual sens) mild S-asymptotically ω-periodic solution which depends on a neighboring initial condition.