Zhongwei Shen
We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-ε<p<2$, we obtain the solvability of the L^p Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established.
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We prove Lipschitz continuity of viscosity solutions to a class of two-phase free boundary problems governed by fully nonlinear operators.
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 provide a new and simple proof based on Harnack's inequality to the Lipschitz continuity of the solutions of a class of free boundary problems.
In this paper we prove local interior and boundary Lipschitz continuity of solutions of a free boundary problem involving the $A$-Laplacian. We also show that the free boundary is represented locally by graphs of a family of lower semi-continuous functions.
We derive Lipschitz stability estimates for the Hausdorff distance of polygonal conductivity inclusions in terms of the Dirichlet-to-Neumann map.
In this paper, we study the nonhomogeneous Dirichlet problem concerning general semilinear elliptic equations in divergence form. We establish that the boundary Lipschitz regularity of solutions under some more weaker conditions on the coefficients, the boundary, the boundary function and the nonhomogeneous term. In particular, we assume that the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, which will be the optimal conditions to obtain the boundary Lipschitz regularity of solutions.
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with $L^2$ boundary data in a periodically perforated Lipschitz domain.
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 $p\geq 1$.
We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable.
We consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. We prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map extending in a highly nontrivial way the results obtained in [18] and [19] in the two-dimensional case to the three-dimensional setting.
In this paper we consider the problem of determining an unknown pair $λ$, $μ$ of piecewise constant Lamé parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $λ$ and $μ$ from the Dirichlet to Neumann map.
We consider relativistic plasma particles subjected to an external gravitation force in a $3$D half space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the collision effect is negligible. As an effective PDE, we study the relativistic Vlasov-Maxwell system and its local-in-time unique solvability in the space-time locally Lipschitz space, for several basic mesoscopic (kinetic) boundary conditions: the inflow, diffuse, and specular reflection boundary conditions. We construct weak solutions to these initial-boundary v…
We prove forward and backward parabolic boundary Harnack principles for nonnegative solutions of the heat equation in the complements of thin parabolic Lipschitz sets given as subgraphs $E=\{(x,t): x_{n-1}\leq f(x'',t),x_n=0\}\subset \mathbb{R}^{n-1}\times\mathbb{R} $ for parabolically Lipschitz functions $f$ on $\mathbb{R}^{n-2}\times\mathbb{R}$. We are motivated by applications to parabolic free boundary problems with thin (i.e co-dimension two) free boundaries. In particular, at the end of the paper we show how to prove the spatial $C^{1,α}$ regularity of the free boundary in the parabolic …
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。
This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this result is that it does not assume more than Lipschitz regularity on the boundary. Our Theorem, which is a significant improvement of our previous work on Lipschitz estimates in bumpy domains, should be read as an improved regularity result for an elliptic system over a Lipschitz boundary. Our progress in this direction is made possible by an estimate for a bounda…
We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero cylinder condition. We prove a regularity result after finding suitable spaces for the solution and the pre-assigned datum in the problem. The tools from potential theory, harmonic analysis and probability are used. Some Lemmas are as important as the main Theorem.
For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain $Ω$ we show that solutions of the corresponding elliptic problem with Robin and thus in particular with Neumann boundary conditions are Hoelder continuous for sufficiently $L^p$-regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell-Robin boundary conditions are well-posed on $\mathrm{C}(\barΩ)$.
In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.
We are concerned with the problem of determining the damping boundary coefficient appearing in a dissipative wave equation from a single boundary measurement. We prove that the uniqueness holds at the origin provided that the initial condition is appropriately chosen. We show that the choice of the initial condition leading to uniqueness is related to a fine version of unique continuation property for elliptic operators. We also establish a Lipschitz directional stability estimate at the origin, which is obtained by a linearization process.
We prove a global version of the so-called div-curl-lemma, a crucial result for compensated compactness and in homogenization theory, for mixed tangential and normal boundary conditions in bounded weak Lipschitz domains in 3D and weak Lipschitz interfaces. The crucial tools and the core of our arguments are the de Rham complex and Weck's selection theorem, the essential compact embedding result for Maxwell's equations.
We show that the elliptic equation with a non-Lipschitz right-hand side, $-Δu = λ|u|^{β-1}u - |u|^{α-1}u$ with $λ>0$ and $0<α<β<1$, considered on a smooth star-shaped domain $Ω$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset $Γ$ of the boundary $\partialΩ$ such that $Γ\neq \partialΩ$.
We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain $Ω\subset\mathbb{R}^n$ and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true.
In this paper we study the inverse boundary value problem of determining the potential in the Schrödinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values.
A mixed boundary value problem for the partial differential equation of diffusion in an inhomogeneous medium in a Lipschitz domain is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed in the papers by Mikhailov (2002, 2006) and Chkadua, Mikhailov, Natroshvili (2009). We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.
We consider the inverse problem of determining the Lamé parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition.
We consider the inverse problem of determining the Lamé moduli for a piecewise constant elasticity tensor ${\mathbb C}= \sum_{j} {\mathbb C}_j χ_{D_j}$, where $\{D_j\}$ is a known finite partition of the body $Ω$, from the Dirichlet-to-Neumann map. We prove that Lipschitz stability estimates can be derived under $C^{1,α}$ regularity assumptions on the interfaces.
In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior which is uniformly Lipschitz and nonlinear terms are concentrated in a region which neighbors the boundary domain. We prove that this family of solutions converges to the solutions of a limit problem in H^1 , an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boun…
This paper develops a new Hilbert space method to characterize a family of reproducing kernel Hilbert spaces of real harmonic functions in a bounded Lipschitz domain $Ω\subset \mathbb R^d, d\geq 2$ involving some families of positive self-adjoint operators and making use of characterizations of their trace data and of a special inner product on $H^1(Ω).$ We also establish boundary representation results for this family in terms of the $L^2-$ Bergman kernel. In particular, a boundary integral representation for the very weak solution of the Dirichlet problem for Laplace's equation with $L^2- $ …
We consider an inverse boundary value problem for Maxwell's equations, which aims to recover the electromagnetic material properties of a body from measurements on the boundary. We show that a Lipschitz continuous conductivity, electric permittivity, and magnetic permeability are uniquely determined by knowledge of all tangential electric and magnetic fields on the boundary of the body at a fixed frequency.
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.
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.
In this article, for the radiative transport equation, we study inverse problems of determining a time independent scattering coefficient or total attenuation by boundary data on the complementary sub-boundary after making one time input of a pair of a positive initial value and boundary data on a suitable sub-boundary. The main results are Lipschitz stability estimates. We can also prove the reverse inequality, which means that our estimates for the inverse problems are the best possible. The proof is based on a Carleman estimate with a linear weight function.
We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.
Existence of mild solutions for the 3D MHD system in bounded Lipschitz domains is established in critical spaces with the absolute boundary conditions.
Continuity of attractors of parabolic equations with nonlinear boundary conditions and rapidly varying boundaries. The case of a Lipschitz deformation
本文研究了带非线性边界条件的抛物型方程在边界随参数$ε$趋于零时快速振荡变化下吸引子的连续性，通过引入变化域函数与Banach空间算子族的收敛性，证明了解收敛至含振荡因子的极限方程。
We give a very short proof of the fact that H1 functions on Lipschitz domains have L2 traces on the boundary of the domain.
We study the generalized boundary value problem for nonnegative solutions of of $-Δu+g(u)=0$ in a bounded Lipschitz domain $Ω$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $Ω$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $Ω$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $Ω$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and uniqueness …
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 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.
We study the generalized boundary value problem for nonnegative solutions of $-Δu+g(u)=0$ in a bounded Lipschitz domain $\Gw$, when $g$ is continuous and nondecreasing. Using the harmonic measure of $\Gw$, we define a trace in the class of outer regular Borel measures. We amphasize the case where $g(u)=|u|^{q-1}u$, $q>1$. When $\Gw$ is (locally) a cone with vertex $y$, we prove sharp results of removability and characterization of singular behavior. In the general case, assuming that $\Gw$ possesses a tangent cone at every boundary point and $q$ is subcritical, we prove an existence and unique…
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 establish two commutator estimates for the Dirichlet-to-Neumann map associated with a second-order elliptic system in divergence form in Lipschitz domains. Our approach is based on Dahlberg's bilinear estimates.
We develop various quantitative estimates for the anisotropic Maxwell system in Lipschitz domains, with a focus on how the estimates precisely depend on the Lipschitz character of the domain. We pay special attention to trace operators and extension operators over certain Sobolev spaces. Finally, we provide a weak formulation of the interior scattering problem in terms of the exterior Calderón operator, and provide explicit bounds for the solution of the interior problem in terms of the incident fields and the Lipschitz character of the domain.
We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequencies as the data. A conditional Lipschitz stability estimate for the inverse problem holds in the case of wavespeeds that are a linear combination of piecewise constant functions (following a domain partition) and gives a framework in which the scheme converges. The stability constant grows exponentially as the number of subdomains in the domain partition increases. We establish an order optimal upper bound for the stability constant. We eventually realize computational …
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.
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 construct solutions to p-Laplace type equations in unbounded Lipschitz domains in the plane with prescribed boundary data in appropriate fractional Sobolev spaces. Our approach builds on a Cauchy integral representation formula for solutions.
We prove wellposedness of the Cauchy problem for the nonlinear Schrodinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric.
Elliptic fourth-order operators with Wentzell boundary conditions on Lipschitz domains
本文利用形式方法和弱迹概念研究Lipschitz区域上带Wentzell边界条件的四阶椭圆算子，证明了解析半群生成并用Sobolev正则性刻画定义域，还讨论了渐近行为与正性。
In this paper, we consider the energy conservation of the Leray-Hopf weak solution $u$ to the Navier-Stokes equations on bounded domains $Ω$ with Lipschitz boundary $\partialΩ$. We prove that although the boundary effect appears, the Shinbrot's condition $u\in L^q_{loc}((0,T];L^p(Ω))$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{2},p\geq 4$ still guarantees the validity of energy conservation of $u$, no boundary layer assumptions are required when dealing with domains with Lipschitz boundary. Compared to the existed methods, our critical strategies are that we first separate the mollification of wea…
We study the inverse problem in Optical Tomography of determining the optical properties of a medium $Ω\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $Ω$ is probed with an input field that is modulated with a fixed harmonic frequency $ω=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. We prove a result of Lipschitz stability of the absorption coefficient $μ_a$ at the boundary $\partialΩ$ in terms of the measurements in the case when the scattering coefficient $μ_s$ is assumed to be known and $…
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 weighte…
Bounding Radial Variation of positive harmonic Functions on Lipschitz Domains
本文给出了高维 Lipschitz 域上正调和函数的径向变分估计，并记录了对 arXiv:2003.07176 的更新与精炼版本，修改了 Mozolyako 和 Havin 的证明。
In this paper we consider a minimization problem for the functional $$ J(u)=\int_{B_1^+}|\nabla u|\sp 2+λ_{+}^2χ_{\{u>0\}}+λ_{-}^2χ_{\{u\leq0\}}, $$ in the upper half ball $B_1^+\subset\R^n, n\geq 2$ subject to a Lipschitz continuous Dirichlet data on $\partial B_1^+$. More precisely we assume that $0\in \partial \{u>0\}$ and the derivative of the boundary data has a jump discontinuity. If $0\in \bar{\partial(\{u>0\} \cap B_1^+)}$ then (for $n=2$ or $n>3$ and one-phase case) we prove, among other things, that the free boundary $\partial \{u>0\}$ approaches the origin along one of the two possi…
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…
Boundary Lipschitz Regularity and the Hopf Lemma on Reifenberg Domains for Fully Nonlinear Elliptic Equations
该研究针对完全非线性椭圆方程，提出统一方法证明Reifenberg区域上的边界Lipschitz正则性与Hopf引理，明确内外Reifenberg $C^{1,\mathrm{Dini}}$条件对应的成立情形，拓展了常规条件下的已有结论。
Using Maz'ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in $R^n$. For $n\ge 8$, combined with a result in \cite{S2}, these estimates lead to the solvability of the $L^p$ Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of $p$. In the case of convex domains, the estimates allow us to show that the $L^p$ Dirichlet problem is uniquely solvable for any $2-\e<p<\infty$ and $n\ge 4$.
Boundary Lipschitz Regularity and the Hopf Lemma for Fully Nonlinear Elliptic Equations
针对完全非线性椭圆方程粘性解的边界正则性问题，提出将曲边界视为超平面扰动的统一简洁方法，证明内外C^{1,\text{Dini}}条件分别对应边界Lipschitz连续性与Hopf引理成立，并验证该条件的最优性。
We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +ε$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for $(3/2)-\varep < p< 3+ε$. This gives an affirmative answer to a conjecture of M. Taylor.
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