Bouchon François, Chupin Laurent
We consider the exterior free boundary Bernoulli problem in the case of a rough given domain. An asymptotic analysis shows that the solution of the initial problem can be approximated by the solution of a non-rough Bernoulli problem at order 2. Numerical tests confirm these theoretical results.
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Nontrivial domains for the Bernoulli's overdetermined problem with small constant vorticity
本文应用隐函数定理求解小常数涡度的伯努利超定问题，还通过分岔分析构造出满足该问题解的周期性径向无界区域。
We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn-Minkowski inequality for the Ber…
We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions. We also prove that in very general situations the free boundary is necessarily the graph of a function.
The main contribution of this paper is the extension of the well-known boundary-control strategy based on structural invariants to the control of infinite-dimensional systems with in-domain actuation. The systems under consideration, governed by partial differential equations, are described in a port-Hamiltonian setting making heavy use of the underlying jet-bundle structure, where we restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian. To show the applicability of the proposed approach, we develop a dynamic controller for an Euler-Bernoulli beam actuated …
In this paper we consider a scalar transport equation with constant coefficients on domains with discrete space and continuous, discrete or general time. We show that on all these underlying domains solutions of the transport equation can conserve sign and integrals both in time and space. Detailed analysis reveals that, under some initial conditions, the solutions correspond to counting stochastic processes and related probability distributions. Consequently, the transport equation could generate various modifications of these processes and distributions and provide some insights into corresp…
Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss o…
In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler- Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on t…
A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic propertie…
This is a survey of results mostly relating elliptic equations and systems of arbitrary even order with rough coefficients in Lipschitz graph domains. Asymptotic properties of solutions at a point of a Lipschitz boundary are also discussed.
We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the domain, including a very weak compatibility condition between the Dirichlet boundary part and its complement, we prove Hölder continuity of the solution in space and time.
In this work we use reiterated homogenization and unfolding operator approach to study the asymptotic behavior of the solutions of the $p$-Laplacian equation with Neumann boundary conditions set in a rough thin domain with concentrated terms on the boundary. We study weak, resonant and high roughness, respectively. In the three cases, we deduce the effective equation capturing the dependence on the geometry of the thin channel and the neighborhood where the concentrations take place.
Roughness-induced effects on the thermomicropolar fluid flow through a thin domain
本文研究了粗糙边界薄通道中热微极流体流动的渐近行为，通过推导不同渐近模型揭示粗糙度对平均速度和微旋转的影响。
In this work we analyze the solutions of a $p$-Laplacian equation with homogeneous Neumann boundary conditions set in a family of rough domains with a nonlinear term concentrated on the boundary. At the limit, we get a nonlinear boundary condition capturing the oscillatory geometry of the strip where the reactions take place.
In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in \cite{HS1} to $\mathcal{A}$-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modeled on the $p$-Laplace equation for a fixed $1<p<\infty$. In particular, we show that if $K$ is a bounded convex set satisfying the interior ball condition and $c>0$ is a given constant, then there exists a unique convex domain $Ω$ with $K\subset Ω$ and a function $u$ which is $\mathcal{A}$-harmonic in $Ω\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on …
Revisited for existence proof of optimal solution in Bernoulli free boundary problem using an energy-gap cost functional
本文纠正了文献中关于 Bernoulli 自由边界问题最优解存在性证明的勘误，指出解序列有界性不能直接用 Cauchy-Schwarz 不等式估计，并改用 Poincaré-Friedrichs 不等式完成了证明。
In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous Dirichlet boundary condition is imposed; we assume boundedness of the embedding $i_{1}:H^{1}(D)\to L^{2}(D)$ when the Neumann boundary condition is imposed; we assume boundedness of the embeddings $i_{1}$ and of $i_{2}:H^{1}(D)\to L^{2}(\partial D)$ when the Robin boundary condition is imposed, and, if, in addition, $i_{1}$ and $i_{2}$ are compact, then the boun…
Motivated by lubrication problems, we consider a micropolar uid ow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < " << 1. We prove the existence and uniqueness of a solution of this problem for any value of " and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the limit problem when " tends to zero. Moreover we show that the limit velocity and micro-rotation elds are uniquely determined via auxiliary well-posed problems and the limit pressure is given as the unique solution of a…
A mathematical method for through-wall imaging via wave phenomena in the time domain is introduced. The method makes use of a single reflected wave over a finite time interval and gives us a criterion whether a penetrable obstacle exists or not in a general rough background medium. Moreover, if the obstacle exists, the lower and upper estimates of the distance between the obstacle and the center point of the support of the initial data are given. As an evidence of the potential of the method two applications are also given.
Asymptotic behavior of a Bingham Flow in thin domains with rough boundary
该研究针对带粗糙边界的薄区域中不可压缩Bingham流问题，用适配线性展开算子分析小参数趋于零时的渐近行为，得到保持非线性的均匀化极限问题并给出非线性Darcy定律解释。
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued coefficients. Our main result gives a sharp condition on the local mean oscillations of the coefficients of the differential operator and the unit normal to the boundary (which is automatically satisfied if these functions belong to space VMO) guaranteeing that the solution operator associated with this problem is an isomorphism.
Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth
本文研究了具有非标准增长的单相 Bernoulli 型泛函非负几乎极小子的最优 Lipschitz 正则性，并在 Orlicz 模背景下通过 Campanato 型估计提出了新的 Höder 正则性刻画。
In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is actually sufficient to provide an estimate on suitable projection operators. Whereas this lemma could be applied under different regularity assumptions on the domain, here we use it to estimate the spectrum in Lipschitz and in so-called Reifenberg-flat domains. Our argument also relies on suitable extension techniques and on an estimate on the decay of the e…
In this paper, we consider viscoelastic flows in a rough domain (with typical roughness patterns of size $ε$ < 1). We present and rigorously justify an asymptotic expansion with respect to $ε$, at any order, based upon the definition of elementary problems: Oldroyd-type problems at the global scale defined on a smoothened domain and boundary-layer corrector problems. The resulting analysis guarantees optimality with respect to the truncation error.
Asymptotic limit of the compressible Navier-Stokes system on domains with rough boundaries
本文研究可压缩Navier-Stokes系统在具有快速振荡粗糙边界区域上的渐近极限，证明在全滑移边界条件下，非退化振荡会使流体在极限中完全粘附于边界。
A Schauder theory for the Stokes equations in rough domains
针对带散度型强迫和无滑移边界条件的稳态 Stokes 方程，本文利用 Campanato 空间逐点乘子，在粗糙区域的最低正则性假设下建立涵盖 BMO 与 C^{0,α} 的极大正则性理论。
Two-dimensional Carreau law for a quasi-newtonian fluid flow through a thin domain with a slightly rough boundary
针对薄域内具有Carreau定律粘度的稳态准牛顿Stokes流，考虑粗糙边界影响，通过渐近分析推导出有效非线性二维Reynolds模型，并利用先验估计和紧致性证明其合理性。
We settle the issue of well-posedness for the Dirichlet problem for a higher order elliptic system ${\mathcal L}(x,D_x)$ with complex-valued, bounded, measurable coefficients in a Lipschitz domain $Ω$, with boundary data in Besov spaces. The main hypothesis under which our principal result is established is in the nature of best possible and requires that, at small scales, the mean oscillations of the unit normal to $\partialΩ$ and of the coefficients of the differential operator ${\mathcal L}(x,D_x)$ are not too large.
We consider a bounded connected open set $Ω\subset {\rm R}^d$ whose boundary $Γ$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(Γ)$ by form methods. The operator $-D_0$ is self-adjoint and generates a contractive $C_0$-semigroup $S = (S_t)_{t > 0}$ on $L_2(Γ)$. We show that the asymptotic behaviour of $S_t$ as $t \to \infty$ is related to properties of the trace of functions in $H^1(Ω)$ which $Ω$ may or may not have.
Serrin's overdetermined problem in rough domains
本文证明了 Serrin 超定定理在 Lipschitz 域及更一般的有限周长集（允许裂缝）上成立，通过新方法肯定解决了长期未决的开放问题。
The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniquen…
Higher-order regularity for a structurally damped plate equation on rough domains
本文研究了粗糙域上结构阻尼板方程的高阶正则性，通过加权Sobolev空间和最大正则性性质，处理非齐次边界条件下的适定性问题。
Optimal domains for elliptic eigenvalue problems with rough coefficients
该研究证明了给定测度下、带粗糙系数的椭圆算子第一Dirichlet特征值最小开集的存在性，采用自由边界方法结合特征函数Hölder估计推导开性，还得到特征函数在自由边界处的Lipschitz连续性。
A Rough Divergence Theorem
本文建立了允许内边界的广义散度定理，通过引入有界涨落函数空间及精确范数，给出了定理成立的充要条件，即积分域特征函数范数有限。
We study a coupled fluid-structure system involving boundary conditions on the pressure. The fluid is described by the incompressible Navier--Stokes equations in a 2D rectangular type domain where the upper part of the domain is described by a damped Euler--Bernoulli beam equation. Existence and uniqueness of local strong solutions without assumptions of smallness on the initial data is proved.
We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calderón-Zygmund theory for such systems. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integr…
In this paper, we study irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as …
This paper is concerned with the mathematical analysis of time-dependent fluid-solid interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above a local rough surface. We reformulate the unbounded scattering problem into an equivalent initial-boundary value problem defined in a bounded domain by proposing a transparent boundary condition (TBC) on a hemisphere. Analyzing the reduced problem with Lax-Milgram lemma and abstract inversion theorem of Laplace transform, we prove the well-posedness and stability for the reduced problem. Moreover, an a prior…
Weakly harmonic maps from a domain $Ω$ (the upper half-space $\Rd$ or a bounded $C^{1,α}$ domain, $α\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes $L^{\infty}(\partialΩ)$ or $BMO(\partialΩ)$, we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, $C^{\infty}_{loc}$. Moreover, we show that boundary data can be chosen large in the underlying topologies if $Ω$ is smooth and bounded by perturbing strictly stable smooth harmonic…
Uniqueness of the 2D Euler equation on rough domains
该研究针对二维不可压缩Euler方程在粗糙有界单连通区域上的弱解唯一性问题，通过构造新变量替换与能量泛函突破正则性障碍，得到更具一般性的唯一性充分条件及相关推论。
New classes of non-smooth bounded domains D, for which the embedding operator from H^1(D) into L^2(D) is compact are introduced. Examples are given and applications to scattering by rough obstacles are mentioned.
Let $Ω$ be a domain in $\mathbf R^d$ and $h(φ)=\sum^d_{k,l=1}(\partial_kφ, c_{kl}\partial_lφ)$ a quadratic form on $L_2(Ω)$ with domain $C_c^\infty(Ω)$ where the $c_{kl}$ are real symmetric $L_\infty(Ω)$-functions with $C(x)=(c_{kl}(x))>0$ for almost all $x\in Ω$. Further assume there are $a, δ>0$ such that $a^{-1}d_Γ^δ\,I\le C\le a\,d_Γ^δ\,I$ for $d_Γ\le 1$ where $d_Γ$ is the Euclidean distance to the boundary $Γ$ of $Ω$. We assume that $Γ$ is Ahlfors $s$-regular and if $s$, the Hausdorff dimension of $Γ$, is larger or equal to $d-1$ we also assume a mild uniformity property for $Ω$ in the ne…
On scattering behavior of corner domains with anisotropic inhomogeneities
该研究将各向异性非均匀介质的非散射问题关联至Bernoulli型自由边界问题，证明了边界为Lipschitz而非$C^{1,α}$的各向异性介质会散射所有满足非退化条件的入射波。
In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi's theorem to give an improved bound on the Hausdorff dimension of the singular set, assuming that the map has bounded energy at all scales. This last assumption can be removed when the target manifold is simply-connected.
We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.
Modeling of a micropolar thin film flow with rapidly varying thickness and non-standard boundary conditions
本文研究了微极流体在薄层中流动的渐近行为，推导了考虑粗糙度影响的广义雷诺方程，并给出了平均速度和微旋转的表达式。
Singularity models for the Bernoulli free boundary problem from isoparametric hypersurfaces
本文从等参叶状结构构造 Bernoulli 自由边界问题的齐次解，产生无穷多新奇点模型，建立几何刚性定理，并猜想最低密度锥体的形式。
We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in $ \R^n$. Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a $ \Lip_α$-domain whenever $ (d-1)/α<d $, prove that the remainder in this for…
We consider the Bernoulli one-phase free boundary problem in a domain $Ω$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial Ω$. We achieve this by relating the behavior of $F$ near $\partial Ω$ to a Signorini-type obstacle problem.
Existence and uniqueness of the scattering solutions is proved for a class of bounded rough obstacles which is much larger than the class of Lipschitz obstacles. Integral equations method is not used. The approach is based on the representation theorem for selfadjoint operators via closed symmetric quadratic forms and on the limiting absorption principles. For the Dirichlet boundary condition no assumptions on the smoothness of the boundary are needed, for the Neumann boundary condition and the Robin boundary condition some assumptions on the smoothness of the boundary are needed.
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains $Ω\subset Ω'$. With a local spectral stability requirement on $Ω$, we show that the difference of the Dirichlet eigenvalues of $Ω'$ and $Ω$ is explicitly controlled from above in terms of the first eigenvalue of $Ω'\setminus\barΩ$ and of geometric constants depending on the inner domain $Ω$. In particular, $Ω'$ can be an arbitrary bounded domain.
Functional calculus on weighted Sobolev spaces for the Laplacian on rough domains
本文研究了在粗糙域上拉普拉斯算子的加权Sobolev空间上的函数 calculus，并建立了热方程的最大正则性，扩展了抛物线方程理论到最小光滑度的域。
Dimension and structure of the Robin Harmonic Measure on Rough Domains
研究粗糙区域Robin调和测度的维数与结构，证明其对Ahlfors正则集曲面测度具定量相互绝对连续性，无维数下降，预期相变通过非尺度不变权重估计体现，颠覆相关物理模拟预期。
We consider a non-Newtonian fluid flow in a thin domain with thickness $η_\varepsilon$ and an oscillating top boundary of period $\varepsilon$. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $p$, with $9/5\leq p<+\infty$. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness ($η_\varepsilon\approx \varepsilon$), Reynolds roughness ($η_\varepsilon\ll \var…
We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier-Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form $x_2 = \varepsilon^{1+α} η(x_1/\varepsilon)$, $α> 0$. Under suitable conditions on the oscillating parameter $\varepsilon$ and the viscosity $ν$, we show that solutions of the Navier-Stokes system converge to solutions of the Euler system in the vanishing limit of both $ν$ and $\varepsilon$. The main issue is that the curvature…
Gauss-Green formulas for divergence measure tensor fields on rough domains
研究粗糙区域上散度测度张量场的高斯 - 格林公式，通过引入新配对概念扩展经典理论，调整法向迹定义，建立低正则性集合上的公式，为不规则张量场和区域的分部积分提供统一强大框架。
In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded $H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall outside the classical class of $A_p$-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.
$H^\infty$-calculus for Stokes operators on rough and on unbounded domains
本文基于 Kunstmann–Weis 抽象比较原理，统一证明并概述粗糙及无界区域上 Stokes 算子的有界 $H^\infty$-演算，并新证有界凸区域中 Neumann 型边界条件情形。
Computing eigenvalues of the Laplacian on rough domains
本文证明了有界欧氏区域的Mosco收敛定理，发展了显式Poincaré型不等式，构造了计算粗糙区域上Dirichlet Laplacian特征值的通用算法，并通过反例证明此类算法无法推广至任意有界区域。
We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Holder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, unique…
Analysis and Simulation of a Fluid-Heat System in a Thin, Rough Layer in Contact With a Solid Bulk Domain
研究薄层与固体本体域间流-热耦合，考虑温度依赖粘度和对流热传输，分析非线性系统存在性、唯一性和能量估计，并通过多尺度收敛理论推导有效界面模型，通过数值模拟验证结果。
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