Fine regularity for elliptic systems with discontinuous ingredients
Dian Palagachev, Lubomira Softova
本文研究了具有不连续系数的二阶线性椭圆系统强解在Morrey、BMO和Hölder空间中的正则性问题。
我们提出了关于二阶线性椭圆系统强解在内部的Morrey、BMO和Hölder正则性的结果，这些系统具有不连续系数和属于Morrey空间的右端项。
We propose results on interior Morrey, BMO and Hölder regularity for the strong solutions to linear elliptic systems of order 2b with discontinuous coefficients and right-hand sides belonging to the Morrey spaces.
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In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
We study nonlocal conservation laws with a discontinuous flux function of regularity $\mathsf{L}^{\infty}(\mathbb{R})$ in the spatial variable and show existence and uniqueness of weak solutions in $\mathsf{C}\big([0,T];\mathsf{L}^{1}_{\text{loc}}(\mathbb{R})\big)$, as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fixed-point problem itself necessitates the study of existence, uniqueness and stability of a class of discontinuous ordinary differential equations. On the ODE level, we compare the solution type …
Higher regularity for entropy solutions of conservation laws with geometrically constrained discontinuous flux
本文针对具有间断通量的守恒律熵解正则性低于光滑通量情形的问题，提出在间断通量上施加最优几何条件，从而恢复与同类非线性光滑通量相同的正则性。
In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.
This article deals with the regularity of the entropy solutions of scalar conservation laws with discontinuous flux. It is well-known [Adimurthi et al., Comm. Pure Appl. Math. 2011] that the entropy solution for such equation does not admit BV regularity in general, even when the initial data belongs to BV. Due to this phenomenon fractional BVs spaces wider than BV are required, where the exponent 0<s\leq 1 and BV = BV1. It is a long standing open question to find the optimal regularizing effect for the discontinuous flux with L^\infty initial data. The optimal regularizing effect in BVs is pr…
We prove the partial Hölder continuity for minimizers of quasiconvex functionals \[ \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \] where $f$ satisfies a uniform VMO condition with respect to the $x$-variable and is continuous with respect to ${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
A'priori estimates and precise regularity for parabolic systems with discontinuous data
本文研究了具有不连续主系数的抛物系统，通过势分析和奇异积分算子的有界性，证明了Sobolev和Sobolev-Morrey空间中的先验估计，并给出了解及其导数的精确正则性刻画。
Precise Morrey regularity of the weak solutions to a kind of quasilinear systems with discontinuous data
本文研究了一类具有不连续数据的拟线性系统弱解的精确Morrey正则性，通过迭代方法获得梯度的最优正则性。
We prove the partial Hölder continuity on boundary points for minimizers of quasiconvex non-degenerate functionals \begin{equation*} \mathcal{F}({\bf u}) \colon =\int_Ω f(x,{\bf u},D{\bf u})\,\mathrm{d}x, \end{equation*} where $f$ satisfies a uniform VMO condition with respect to the $x$-variable, is continuous with respect to ${\bf u}$ and has a general growth with respect to the gradient variable.
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally …
The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field ar…
This article studies the partial Hölder continuity of weak solutions to certain degenerate parabolic systems whose model is the differentiable parabolic $p(x,t)$-Laplacian system, \begin{equation*}\partial_t u-\operatorname{div}[μ(z)(1+|Du|^2)^{\frac{p(z)-2}{2}}Du]=0,\qquad p(z)\geq2.\end{equation*} Here, the exponential function $p(z)$ satisfies a logarithmic continuity condition. We show that if $μ(z)$ satisfies a certain VMO-type condition, then $u$ is locally Hölder continuous except for a measure zero set.
In this paper we study nonnegative minimizers of general degenerate elliptic functionals, $\int F(X,u,Du) dX \to \min$, for variational kernels $F$ that are discontinuous in $u$ with discontinuity of order $\sim χ_{\{u > 0 \}}$. The Euler-Lagrange equation is therefore governed by a non-homogeneous, degenerate elliptic equation with free boundary between the positive and the zero phases of the minimizer. We show optimal gradient estimate and nondegeneracy of minima. We also address weak and strong regularity properties of free boundary. We show the set $\{u > 0 \}$ has locally finite perimeter…
In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition obtained from Bernoulli's principle as a nonlocal differential equation. This enables us to establish the existence of weak solutions of the problem by using elliptic estimates and bifurcation theory. Secondly, we investigate the a priori regularity of these weak solutions and prove that they are in fact strong solutions of the problem, describing waves with a …
Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a certain regularity condition on the constraint is satisfied. If this regularity condition fails, then solutions obtained from game theory and $L^p$-approximation need not coincide.
We consider linear inhomogeneous non-autonomous parabolic problems associated to sesquilinear forms, with discontinuous dependence of time. We show that for these problems, the property of maximal parabolic regularity can be extrapolated to time integrability exponents $r\neq 2$. This allows us to prove maximal parabolic $L^r$-regularity for discontinuous non-autonomous second-order divergence form operators in very general geometric settings and to prove existence results for related quasilinear equations.
We develop an optimal regularity theory for $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly unbounded coefficients. Our approach treats both superlinear and sublinear gradient regimes in a unified way. We show $C^{0,α}$, $C^{0,\textrm{Log-Lip}}$, $C^{1,α}$, $ C^{1,\textrm{Log-Lip}}$ and $C^{2,α}$ regularity estimates, by displaying the growth allowed to the Hamiltonian in order to deal with an unbounded nonlinear gradient coefficient, whose integrabilit…
Regularity for one-phase Bernoulli problems with discontinuous weights and applications
本文研究权重沿光滑界面不连续的单相 Bernoulli 自由边界问题，证明自由边界在低维奇异集外具有 $C^{1, α}$ 正则性，并利用该结果构造了二维情形下权重任意接近常数时的奇异最小自由边界。
Global-in-time Discontinuous Solutions for the Two-Phase Model of Compressible Fluids with Density-Dependent Viscosity
本文研究具密度依赖粘性的可压缩两相流体模型，证明在初始密度跨界面存在不连续的情况下，全局时间弱解的存在唯一性，并确保密度分片Hölder连续性及尖锐界面正则性的时间持久性。
An elliptic problem involving critical Choquard and singular discontinuous nonlinearity
本文研究结合临界Hartree型与奇异间断非线性的椭圆问题，应用变分方法和广义梯度证明了弱解的存在性与多重性，并通过$L^\infty$估计确立了其正则性。
Traveling waves for bistable reaction-diffusion-convection equations with discontinuous density-dependent coefficients
本文研究了具有不连续密度依赖系数的双稳反应-扩散-对流方程的行波解存在性，扩展了之前在更高正则性假设下的结果到弱正则性设置中。
For $Ω\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(Ω;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation} \nabla\cdot\left(a(x)|Du|^{p(x)-2}Du\right)=0\text{, a.e. }x\inΩ,\notag \end{equation} where concerning the coefficient function $x\mapsto a(x)$ we assume only that \begin{equation} a\in W^{1,q}(Ω)\cap L^{\infty}(Ω),\notag \end{equation} where $q>1$ is essentially arbitrary. This implies that the coefficient in the PDE can be highly irregular, and yet in spite of this we still recover that \begin{eq…
Partial regularity for minimizers of a class of discontinuous Lagrangians
本文研究含离散 Monge-Ampère 引力模型变分形式的一维不连续 Lagrangian 问题，借助 Voronoi 分解的几何结构证明局部极小元在有限冲击时刻外具有 C^{1,1} 正则性。
Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems
本文针对Helmholtz方程的内部罚间断Galerkin离散，利用最小正则性下的Aubin–Nitsche对偶论证，建立并推广了最优先验误差估计与可显式控制误差的后验估计。
In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [FVW].
We study linear and quasilinear Venttsel boundary value problems involving elliptic operators with discontinuous coefficients. On the base of the a priori estimates obtained, maximal regularity and strong solvability in Sobolev spaces are proved.
We study the obstacle problem with an elliptic operator in divergence form. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in VMO.
We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We briefly survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from -- and are inspired by -- fundamental facts in the theory of $L^p$-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi an…
In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result in [BL] and the arguments in [DcFLVW], we present an elementary method to derive the Carleman estimate under the optimal regularity assumption on the coefficients.
Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients
在Besov正则性假设下，通过有限差分商技术证明具有线性增长的椭圆方程的解具有几乎Lipschitz连续性，并给出Lebesgue空间尺度下的最优性例子。
We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect to the $x$ and continuity with respect to $u$ and prove Hölder continuity of the solutions outside of singular sets.
We obtain new partial Hölder continuity results for solutions to divergence form elliptic systems with discontinuous coefficients, obeying $p(x)$-type nonstandard growth conditions. By an application of the method of $\mathcal{A}$-harmonic approximation, we are able to allow for both VMO-discontinuities in the coefficients, and the minimal $\log$-Hölder regularity assumption on the exponent function. In doing so, we recover a local version of the quantisation phenomenon characteristic continuous coefficient case.
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth, the coefficients of $A$ are discontinuous and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
We study a Euler-Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $C_0$-semigroup $(S(t))_{t\geq0}$ is of Gevrey class $δ>24$ for $t>0$, hence immediately differentiable. Moreover, we show that $(S(t))_{t\geq0}$ is exponentially stable.
Elliptic Equations in Divergence Form with Discontinuous Coefficients in Domains with Corners
研究角域中具有分段$C^{\ga}$系数的散度型椭圆方程，获得跨不连续面的分段$C^{1,\ga}$估计，并通过例子揭示角处正则性挑战。
An Elliptic-Parabolic Free Boundary Problem with Discontinuous Data
针对模拟部分饱和多孔介质流体流动的1+1维椭圆-抛物型自由边界问题，研究边界和初始数据跳跃间断对解与自由边界正则性的影响，证明非饱和区弱解及自由边界的最优正则性，为高维研究奠基。
We study a class of nonlocal double phase problems with discontinuous coefficients. A local self-improving property and a higher Hölder continuity result for weak solutions to such problems are obtained under the assumptions that the associated coefficient functions are of type VMO (vanishing mean oscillation) and that the principal coefficient depends not only on the variables but also on the solution itself.
Interior $C^{2,α}$ Regularity for the Quadratic Hessian Equation
本文针对正分支上右端项为正$C^α$的二次Hessian方程容许解，建立内部$C^{2,α}$正则性，采用定量大迹传播论证与自适应Dirichlet比较法，明确了该方程解的正则性结论。
Generalized Morrey regularity for parabolic equations with discontinuity data
本文研究了抛物方程在不连续数据下的正则性问题，通过广义Morrey空间中的连续性估计，探讨了线性均匀抛物方程的全局解的正则性。
We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.
We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these …
In this paper we obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude Hölder estimates for bounded subsolutions. In many interesting situations, this gives way to a priori estimates for subsolutions. We apply this regularity results to obtain the ergodic asymptotic behavior of the associated evolution problem in the case of superlinear equations. One of the surprising features in our proof is that it avoids the key ingredient which…
Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains
本文证明了Reifenberg平坦域上非局部椭圆方程解的尖锐边界正则性，即当平坦度参数足够小时解具有$C^{s-\varepsilon}$正则性，证明基于障碍函数和比较原理的归纳论证。
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients $\mathbb{A}$ are discontinuous and singular in $(x,t)$-variables, and dependent on the solution $u$. Global and interior weighted $W^{1,p}(Ω, ω)$-regularity estimates are established for weak solutions of these equations, where $ω$ is a weight function in …
A Discontinuous Solution of the Critical n-Laplace System with Antisymmetric Potential
针对Rivière提出的正则性问题，构造了具反对称势的临界n-拉普拉斯方程组的不连续解，给出其否定回答，同时明确了相关空间正则性结果。
The shallow--water limit of linear water waves over a discontinuous bottom
该论文严格证明不连续底床上线性水波的浅水极限，提出抽象函数框架和显式共形映射方法，识别并数值验证了台阶诱导的色散边界层，并推导了描述该边界层的色散模型。
In this paper we study interior potential-theoretic properties of purely discontinuous Markov processes in proper open subsets $D\subset \mathbb{R}^d$. The jump kernels of the processes may be degenerate at the boundary in the sense that they may vanish or blow up at the boundary. Under certain natural conditions on the jump kernel, we show that the scale invariant Harnack inequality holds for any proper open subset $D\subset \mathbb{R}^d$ and prove some interior regularity of harmonic functions. We also prove a Dynkin-type formula and several other interior results.
A refined version of the strong maximum principle is proven for a class of second order ordinary differential equations with possibly discontinuous non-monotone nonlinearities. Then, exploiting this tool, some optimal regularity results recently established by Lopez-Gomez and Omari, for the bounded variation solutions of non-autonomous quasilinear equations driven by the one-dimensional curvature operator, are substantially improved by admitting general prescribed curvatures and by incorporating general boundary conditions. The new approach developed here yields a new, deeper, interpretation o…
In this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction. By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of regularity of the solution and in a migration of the total energy through the scales.
A Discontinuous Galerkin Method for H(curl)-Elliptic Hemivariational Inequalities
本文提出一种不连续Galerkin方法求解H(curl)-椭圆半变分不等式，通过选择合适数值通量构造内罚不连续Galerkin方案，并分析其一致性和稳定性，建立先验误差估计，展示数值解的最优收敛阶。
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven by Domokos & Manfredi (Cont. Math. 2005). In turn, the a priori estimates found are shown to imply the suitable loca…
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