Sylvie Monniaux
Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.
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The magnetohydrodynamical system in bounded non smooth domains of dimension $n\ge3$
本文研究了在有界非光滑域中三维及以上维度磁流体动力学系统在临界空间中的温和解存在性，利用斯托克斯算子的正则性结果和微分形式的利布尼茨型公式进行证明。
Several formulations of a local geometric measure-type condition are imposed on super-level sets of mild solutions to the homogeneous incompressible 3D magnetohydrodynamical system with bounded initial data to prevent finite-time singularity formation. Supporting this, results regarding the existence, uniqueness, and real analyticity of mild solutions are established as is a sharp lower bound on the radius of analyticity.
We study the boundary integral operator induced from fractional Laplace equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace equation.
We construct a Lipschitz truncation which approximates functions of bounded variation in the area-strict metric. The Lipschitz truncation changes the original function only on a small set similar to Lusin's theorem. Previous results could only give estimates on the Lebesgue measure of the set where the Lipschitz approximations differ from the original function.
Lipschitz regularity for a priori bounded minimizers of integral functionals with nonstandard growth
该研究在增长指数与椭圆性指数之差满足严格界的条件下，建立了满足非标准增长条件的非自治积分泛函先验有界局部极小元的Lipschitz正则性。
We prove local Lipschitz regularity for bounded minimizers of functionals with nonstandard $p,q$-growth with the source term in the Lorentz space $L(N,1)$ under the restriction $q<p+1+p\,\min\left\{\frac 1N,\frac{2(p-1)}{Np-2p+2}\right\}$. This extends the recent work by Beck-Mingione to bounded minimizers under weaker hypothesis and is sharp for some special ranges of $p$, $q$ and $N$.
In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, without any assumption of homogeneity, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-Laplacian and the mean-curvature operator, where the nonlinearity satisfies mild assumptions.
In this paper, we study a Dirichlet problem of a fractional Laplace equation in a bounded Lipschitz domain in $ \R, n \geq 2$. Our main result is that for the given data $F \in \dot H^s(\Om^c), 0 < s<1$, we find the function which satisfies that $\De^s u =0$ in $\Om$, $u|_{\Om^c} =F$ and $|u|_{\dot{H}^s(\R)} \leq c |F|_{\dot H^s(\Om^c)}$. Furthermore, we represent the solution with an integral operator.
Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains
本文证明了一般有界凸区域上Monge-Ampère特征函数及一类退化Monge-Ampère方程凸解的全局Lipschitz性质，并由此导出了全局$W^{2,1}$估计。
Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition
本文在$g$有界且$ϕ$满足下有界斜率条件的情况下，证明了非一致凸泛函极小值点的局部Lipschitz正则性，并作为副产品证明了允许$f$非凸时该类泛函局部Lipschitz极小值点的存在性。
We use the scale $B^s_τ(L_τ(Ω))$, $1/τ=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $Ω\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.
On fractional Orlicz boundary Hardy inequalities
本文研究有界Lipschitz域上的分数阶Orlicz边界Hardy型不等式，并在多类域中对特定临界情形建立了带对数修正的不等式。
In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in critical Besov spaces, which satisfies a non-linear smallness condition. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity field. Furthermore, with additional regularity assumption on the initial velocity or on the initial density, we can also prove the uniqueness of such solution. We should mention that the classical maximal regularity theorem f…
The paper extends the results obtained by C. Kenig, F. Lin and Z. Shen in \cite{SZW2} to more general elliptic homogenization problems in two perspectives: lower order terms in the operator and no smoothness on the coefficients. We do not repeat their arguments. Instead we find the new weighted-type estimates for the smoothing operator at scale $\varepsilon$, and combining some techniques developed by Z. Shen in \cite{SZW12} leads to our main results. In addition, we also obtain sharp $O(\varepsilon)$ convergence rates in $L^{p}$ with $p=2d/(d-1)$, which were originally established by Z. Shen …
We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard $(p,q)$-growth satisfying the dimension-independent restriction $q<p+2$ with $p \geq 2$. This relation improves existing restrictions when $p \leq N-1$, moreover our result is sharp in the range $N > \frac{p(2+p)}{2} + 1$. The standard Lipschitz regularity takes the form $W^{1,\infty}_{\text{loc}} - W^{1,p}_{\text{loc}}$, whereas we obtain $W^{1,\infty}_{\text{loc}} - L^{\infty}_{\text{loc}}$ regularity estimate and then make use of existing sharp $L^{\infty}_{\text{loc}}$ bounds to obtain the required conclus…
Lipschitz stability for an elliptic inverse problem with two measurements
本文研究通过包围区域边界上的测量值反演椭圆方程外部未知边界值的问题，在可容许函数类上证明了Lipschitz稳定性，并将结果推广到内部问题和抛物情形。
We study Robin-to-Robin maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$, with generalized Robin boundary conditions.
A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
该研究通过有界解的Carleson测度估计，刻画了Lipschitz图区域上抛物型$L^p$Dirichlet问题的可解性，改进了需附加抛物一致可求长性等额外假设的现有结果。
In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\rm div}(A^t\nabla\cdot)-{\rm div}(b\,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${\rm div}b$ in order to obtain solvability. O…
We study generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains in $\bbR^n$, $n\ge 2$. We also discuss the case of bounded $C^{1,r}$-domains, $(1/2)<r<1$.
Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain
通过改进已有方法，建立三维Lipschitz区域Stokes算子的L^p预解估计，证明其生成一致有界解析半群，补充了插值相关定理，修正了大p下估计可能失效的传统认知。
Boundedness of fractional integrals and fractional derivatives on Laguerre Lipschitz spaces
该研究受高斯Lipschitz空间相关结果启发，通过发展拉盖尔框架下的泊松积分理论，证明了与拉盖尔多项式展开相关的一类分数次积分与导数在拉盖尔Lipschitz空间上的有界性。
Calderon-Zygmund estimates for stochastic elliptic systems on bounded Lipschitz domains
本文研究有界 Lipschitz 区域上具可积相关随机系数椭圆系统的 Calderón-Zygmund 估计，并借助新型最小半径与 Gloria-Neukamm-Otto 策略得到近乎最优的振荡与涨落均匀化误差。
In this paper we introduce Lipschitz spaces with respect to the Gaussian measure, and study the boundedness of the fractional integral and fractional derivative operators on them.The methods are general enough to provide alternative proofs of the ones given in the classical case and moreover can be extended to the case of Laguerre and Jacobi expansions too
In this paper we study the resolvent of wave operators on open and bounded Lipschitz domains of $\mathbb{R}^N$ with Dirichlet or Neumann boundary conditions. We give results on existence and estimates of the resolvent for the real and complex cases.
We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $Ω$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-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 solutio…
In this note we show that gradient of Harmonic functions on a smooth domain with Lipschitz boundary values is pointwise bounded by a universal function which is in $L^p$ for all finite $p\geq 1$.
Heat Kernel Estimates for Schrödinger Operators in the Domain Above a Bounded Lipschitz Function
本文研究有界 Lipschitz 图像上方区域内 Schrödinger 算子的 Dirichlet 热核，并在势函数远离边界呈超二次衰减时建立匹配的上下界。
Recently developed theoretical framework for analysis of structured population dynamics in the spaces of nonnegative Radon measures with a suitable metric provides a rigorous tool to study numerical schemes based on particle methods. The approach is based on the idea of tracing growth and transport of measures which approximate the solution of original partial differential equation. In this paper we present analytical and numerical study of two versions of Escalator Boxcar Train (EBT) algorithm which has been widely applied in theoretical biology, and compare it to the recently developed split…
We prove that for bounded Lipschitz domains in $\mathbb{R}^N$ Korn's first inequality holds for vector fields satisfying homogeneous mixed normal and tangential boundary conditions.
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
We consider the functional $\int_Ωg(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(Ω)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous.
We introduce the class of quasiconvex Lipschitz domains, which covers both $C^1$ and convex domains, to the study of boundary unique continuation for elliptic operators. In particular, we prove the upper bound of the size of nodal sets for Dirichlet eigenfunctions of general elliptic equations in bounded quasiconvex Lipschitz domains. Our result is new even for Laplace operator in convex domains.
Given a semi-convex potential V on a convex and bounded domain $Ω$, we consider the Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation with potential V, which defines, for fixed time step $τ$ > 0, a sequence of densities $ρ$ k $\in$ P($Ω$). Supposing that V is $α$-convex, i.e. D 2 V $\ge$ $α$I, we prove that the Lipschitz constant of log $ρ$ + V satisfies the following inequality: Lip(log($ρ$ k+1) + V)(1 + $α$$τ$) $\le$ Lip(log($ρ$ k) + V). This provides exponential decay if $α$ > 0, Lipschitz bounds on bounded intervals of time, which is coherent with the results on the continuous…
Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains
本文在有界Lipschitz域中利用临界Besov空间和最优$\mathrm{L}^q$-正则性，证明了含一般耦合项的Keller-Segel-Navier-Stokes系统在小初值下存在局部与全局强解，并得到平衡态指数稳定性及解的正性保持。
In dimension two or three, the weak maximum principal for biharmonic equation is valid in any bounded Lipschitz domains. In higher dimensions (greater than three), it was only known that the weak maximum principle holds in convex domains or $C^1$ domains, and may fail in general Lipschitz domains. In this paper, we prove the weak maximum principle in higher dimensions in quasiconvex Lipschitz domains, which is a sharp condition in some sense and recovers both convex and $C^1$ domains.
We study spectral estimates of the divergence form uniform elliptic operators $-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in bounded non-Lipschitz simply connected domains $Ω\subset \mathbb C$. The suggested method is based on the quasiconformal composition operators on Sobolev spaces with applications to the weighted Poincaré-Sobolev inequalities.
The boundedness of rough generalized commutators with Lipschitz functions on homogeneous variable exponent Herz type spaces
针对经典函数空间难以处理非标准增长非线性问题的局限，本文利用Lipschitz函数与变指数性质，证明了一类粗糙广义交换子在齐次变指数Herz及Herz-Morrey空间上的有界性，丰富了变指数分析框架。
We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.
We develop a sharp boundary trace theory in arbitrary bounded Lipschitz domains which, in contrast to classical results, allows "forbidden" endpoints and permits the consideration of functions exhibiting very limited regularity. This is done at the (necessary) expense of stipulating an additional regularity condition involving the action of the Laplacian on the functions in question which, nonetheless, works perfectly with the Dirichlet and Neumann realizations of the Schrödinger differential expression $-Δ+V$. In turn, this boundary trace theory serves as a platform for developing a spectral …
A Compactness Result for the div-curl System with Inhomogeneous Mixed Boundary Conditions for Bounded Lipschitz Domains and Some Applications
本文证明有界 Lipschitz 区域上非齐次混合边界条件下 div-curl 向量场的 $L^2$ 紧性，并用于建立 Friedrichs/Poincaré 估计、div-curl 引理及 Maxwell 算子紧预解式。
We give a proof that the first eigenfunction of the $α$-symmetric stable process on a bounded Lipschitz domain in $\R^d$, $d\geq 1$, is superharmonic for $α=2/m$, where $m>2$ is an integer. This result was first proved for the ball by M. Kaßmann and L. Silvestre (personal communication) with different methods. For $α=1$, the result was proved in \cite[Theorem 4.7]{BanKul}.
Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
研究加权Morrey空间上Hardy-Littlewood极大函数与sharp极大函数交换子的有界性，并据此给出加权Lipschitz函数的新刻画。
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
In this paper we study the self-adjoint Krein-von Neumann realization $A_K$ of the perturbed Laplacian $-Δ+V$ in a bounded Lipschitz domain $Ω\subset\mathbb{R}^n$. We provide an explicit and self-contained description of the domain of $A_K$ in terms of Dirichlet and Neumann boundary traces, and we establish a Weyl asymptotic formula for the eigenvalues of $A_K$.
The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value pr…
Conditonal Lipschitz stability for the Inverse Problem of the 2D Navier-Stokes System in a Bounded Domain
本文研究了二维纳维-斯托克斯系统在有限域中的逆问题，通过条件Lipschitz稳定性与局部恢复方法，利用初始速度场和全局涡量不变量恢复局部速度场和边界涡量。
Boundary regularity for anisotropic minimal Lipschitz graphs
本文证明了在各向异性能量满足一致标量原子条件时，具有$C^{1,α}$边界且平均曲率在$L^p$中有界的任意余维数Lipschitz图像在特定密度边界点处的正则性。
For a bounded weak Lipschitz domain we show the so called `Maxwell compactness property', that is, the space of square integrable vector fields having square integrable weak rotation and divergence and satisfying mixed tangential and normal boundary conditions is compactly embedded into the space of square integrable vector fields. We will also prove some canonical applications, such as Maxwell estimates, Helmholtz decompositions and a static solution theory. Furthermore, a Fredholm alternative for the underlying time-harmonic Maxwell problem and all corresponding and related results for exter…
It is proved that the space of differential forms with weak exterior and co-derivative, is compactly embedded into the space of square integrable differential forms. Mixed boundary conditions on weak Lipschitz domains are considered. Furthermore, canonical applications such as Maxwell estimates, Helmholtz decompositions and a static solution theory are proved. As a side product and crucial tool for our proofs we show the existence of regular potentials and regular decompositions as well.
This work provides a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{dist(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset \mathbb R^{n+1}$. As a corollary, one o…
Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders
本文研究了抛物型 $L^p$ 霍尔德问题在有界利普希茨圆柱上的可解性外推问题，提出新的方法以解决该问题。
On a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$, $d \geq 3$, we continue the study of Shen and of Kunstmann and Weis of the Stokes operator on $\mathrm{L}^p_σ (Ω)$. We employ their results in order to determine the domain of the square root of the Stokes operator as the space $\mathrm{W}^{1 , p}_{0 , σ} (Ω)$ for $\lvert \frac{1}{p} - \frac{1}{2} \rvert < \frac{1}{d} + \varepsilon$ and some $\varepsilon > 0$. This characterization provides gradient estimates as well as $\mathrm{L}^p$-$\mathrm{L}^q$-mapping properties of the corresponding semigroup. In the three-dimensional case this provi…
Let Ω\subset\mathbb{R}^n be a bounded domain with C^\infty boundary. We show that a harmonic function in Ωthat is Lipschitz along a family of curves transversal to bΩis Lipschitz in Ω. The space of Lipschitz functions we consider is defined using the notion of a majorant which is a certain generalization of the power functions t^α, 0<α<1.
Weyl asymptotics for Poincaré-Steklov eigenvalues in a domain with Lipschitz boundary
本文证明了在Lipschitz边界区域中，Poincaré-Steklov谱问题特征值的Weyl渐近公式成立。
In this paper, we consider the characterizations of the Lipschitz spaces and homogeneous Lipschitz spaces associated to the biharmonic operator $Δ^2.$ With this characterizations, we prove the boundedness of the Bessel potentials, fractional integrals, fractional powers, Riesz transforms and multipliers of Laplace transforms type associated to $Δ^2$ on Lipschitz spaces and homogeneous Lipschitz spaces. The proofs of these results need the language of semigroups in an essential way.
We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.
The Hot Spots constant for bounded smooth domains was recently introduced by Steinerberger (2021) as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We generalize the Hot Spots constant to bounded Lipschitz domains and show that it leads to an if and only if condition for the weak Hot Spots conjecture HS2 from Bañuelos and Burdzy (1999). We also derive a new general formula for a dimension-dependent upper bound that can be tailored to any specific class of domains. This formula is then used to compute upper bounds fo…
Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps defined on R2 by Alberti, Bianchini and Crippa in [1].
Let $n\ge2$ and $Ω$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $Ω$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent $p$ or …
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