Cristian Bereanu, Petru Jebelean, Calin Serban
Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.
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The Dirichlet problem for the prescribed curvature equations in Minkowski space
针对Minkowski空间中预定曲率类空超曲面，本文研究其Dirichlet问题，并为对应完全非线性椭圆PDE的容许解建立新的内部二阶估计。
We establish pointwise estimates for the Green function to the Dirichlet problem for parabolic equation with coefficients measurable in time variable. Using these estimate we obtain coercive estimates for this problem in anisotropic weighted Lebesgue spaces and prove the solvability theorems.
In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the degree theory after turning the problem into a Hammerstein equation. The right hand side $g$ is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions..
We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted $L_{p,q}$-spaces. The concept of "critical exponent" introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi-linear non-divergence parabolic equations with discontinuous in time coefficients in cylinders $Ω\times(0,T)$, where $Ω$ is a bounded domain with an edge or with a conical point.
The Dirichlet problem for a class of curvature equations in Minkowski space
针对 Minkowski 空间中的一类给定曲率方程，本文通过建立先验 C² 估计，证明了一般边界数据下光滑类空超曲面解的存在性。
The Dirichlet problem for Hessian quotient type curvature equations in Minkowski space
针对非凸区域，在不假设下解和Serrin型条件下建立先验估计，证明闵可夫斯基空间中一类Hessian商型曲率方程Dirichlet问题的存在性。
We prove the homogenization of fully nonlinear parabolic equations with periodic oscillating Dirichlet boundary conditions on certain general prescribed space-time domains. It was proved in [9,10] that for elliptic equations, the homogenized boundary data exists at boundary points with irrational normal directions, and it is generically discontinuous elsewhere. However for parabolic problems, on a flat moving part of the boundary, we prove the existence of continuous homogenized boundary data $\bar{g}$. We also show that, unlike the elliptic case, $\bar{g}$ can be discontinuous even if the ope…
We prove an existence result for a class of Dirichlet boundary value problems with discontinuous nonlinearity and involving a Leray-Lions operator. The proof combines monotonicity methods for elliptic problems, variational inequality techniques and basic tools related to monotone operators.
In this article we prove that the singular set of Dirichlet-minimizing $Q$-valued functions is countably $(m-2)$-rectifiable and we give upper bounds for the $(m-2)$-dimensional Minkowski content of the set of singular points with multiplicity $Q$.
On singular $p$-Laplacian problems with discontinuous convection terms
研究了具有在零处奇异且高度不连续的非线性反应项及不连续对流项的p-拉普拉斯问题，证明了正强解的存在性，采用局部论证和合适的测度理论结果。
Strong solutions to singular discontinuous $p$-Laplacian problems
本文研究了Dirichlet型$p$-拉普拉斯问题正强解的存在性，其反应项在零点奇异且高度不连续，仅要求不连续点集勒贝格测度为零。
The article considers the Dirichlet problem for a high-order mixed-type equation that splits into factors, each of which is a Lavrentiev-Bitsadze equation with its own excellent coefficient. Sufficient conditions are found for the coefficients under which the problem has a classical solution.
We deal with the problem of determining a time varying inclusion within a thermal conductor. In particular we study the continuous dependance of the inclusion from the Dirichlet-to-Neumann map. Under a priori regularity assumptions on the unknown defect we establish logarithmic stability estimates.
Positive solutions to semilinear Dirichlet problems with general boundary data
本文利用概率表示方法研究具有一般不连续边界数据的半线性椭圆Dirichlet问题，在Cornea受控收敛意义下刻画解的边界行为，并证明了解的唯一性。
Lp-integrability of the gradient of solutions to quasilinear systems with discontinuous coefficients
本文研究了 quasilinear 系统中不连续系数下解梯度的 Lp-可积性，得出了 p>2 的 Lp-可积性结果，并扩展到双曲系统情形。
The Dirichlet problem in a class of generalized weighted spaces
本文研究了在一类广义加权空间中，由经典积分算子和交换子生成的亚线性积分算子的连续性，并利用所得估计研究线性均匀椭圆算子的Dirichlet问题的全局正则性。
Existence of multiple radial solutions for nonlinear equation involving the mean curvature operator in Lorentz-Minkowski space
该研究利用Szulkin非光滑泛函临界点理论，证明了Lorentz-Minkowski空间中含平均曲率算子的凹凸型非线性方程Dirichlet问题多径向解的存在性，并拓展至非线性项依赖解梯度的情形。
Recovering discontinuous viscosity coefficients for inverse Stokes problems by boundary measurements
本文通过边界测量恢复逆斯托克斯问题中不连续粘性系数的唯一性。
Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities
本文研究了椭圆偏微分方程解的几何性质，证明了相关Brunn-Minkowski型不等式，并探讨了凸领域中特征函数的对数凹性。
We consider a parabolic PDE with Dirichlet boundary condition and monotone operator $A$ with non-standard growth controlled by an $N$-function depending on time and spatial variable. We do not assume continuity in time for the $N$-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic $N$-function, we also prove their uniqueness. This general result applies to equations studied in the literature like $p(t,x)$-Laplacian and double-phase problems.
We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in $\mathbb R^n$ of the union of a finite number of bounded Lipschitz domains. The mean curvature $H=H(x,t)$ is assumed to have absolute value controlled from above by a locally bounded, $L^p$-function, $p\in [1,2n/(n+2)]$, $n\geq 3$.
A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials
本文证明了Schrödinger型算子$\mathcal{H}_V$的第一Dirichlet特征值的Brunn-Minkowski不等式，利用生成半群的迹类性质，并通过半群的超收缩性得到基态的对数凹性。
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.
This paper concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of RN with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the in…
We show the obstacle version of the Strauss conjecture holds when the spatial dimension is equal to 4. We also show that an almost global existence theorem of Hörmander for (4+1)-dimensional Minkowski space holds in the obstacle setting. We use weighed space-time variants of the energy inequality and a variant of the classical Hardy inequality.
Quasilinear Cauchy-Dirichlet problem for parabolic equations with $VMO_x$ coefficients
研究具有不连续系数的抛物型拟线性方程Cauchy-Dirichlet问题，在适当结构条件下证明了强解的存在性、唯一性及Hölder连续性。
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 investigate the problem of identifying discontinuous doping profiles in semiconductor devices from data obtained by the stationary voltage-current (VC) map. The related inverse problem correspond to the inverse problem for the Dirichlet-to-Neumann (DN) map with partial data.
Nonlinear Dirichlet problem of non-local branching processes
本文针对不连续边界数据的非线性Dirichlet问题，提出利用非局部分支过程给出解的随机表示，并以受控收敛替代逐点收敛来处理边界不连续性。
We consider the Dirichlet-to-Neumann map $Λ$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, a magnetic field $A$ and a potential $q$. We show that we can recover the jet of $g,A,q$ on the boundary from $Λ$ up to a gauge transformation in a stable way. We also show that $Λ$ recovers the following three invariants in a stable way: the lens relation of $g$, and the light ray transforms of $A$ and $q$. Moreover, $Λ$ is an FIO away from the diagonal with a canonical relation given by the lens relation. We present applications for recovery of $A$ and $…
We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This problem is related by a generalized coarea formula to a Dirichlet energy functional in which the energy density is the local oscillation of a function. These two nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing wh…
In 1976, Leon Simon showed that if a compact subset of the boundary of a domain is smooth and has negative mean curvature, then the non-parametric least area problem with Lipschitz continuous Dirichlet boundary data has a generalized solution which is continuous on the union of the domain and this compact subset of the boundary, even if the generalized solution does not take on the prescribed boundary data. Simon's result has been extended to boundary value problems for prescribed mean curvature equations by other authors. In this note, we construct Dirichlet problems in domains with corners a…
A quantitative symmetry result for $p$-Laplace equations with discontinuous nonlinearities
本文研究带有不连续非线性项的 $p$-Laplace 方程正解的 Gidas-Ni-Nirenberg 型对称性定量稳定性，利用 Pólya-Szegö 原理的定量版本证明了解与其 Schwarz 对称化的偏差可由区域的等周亏格进行估计。
Convexity inequalities for eigenvalues and log-concavity of eigenfunctions
本文通过简洁证明给出了施罗德inger算子的两个经典结果：特征值的布伦-闵科夫斯基不等式和第一特征函数的对数凹性。证明适用于包含C^{1,1}连通域和凸势能的领域。
We prove $L^{p}$-uniqueness of Dirichlet operators for Gibbs measures on the path space $C(\mathbb R, \mathbb R^{d})$ associated with exponential type interactions in infinite volume by extending an SPDE approach presented in previous work by the last two named authors. We also give an SPDE characterization of the corresponding dynamics. In particular, we prove existence and uniqueness of a strong solution for the SPDE, though the self-interaction potential is not assumed to be differentiable, hence the drift is possibly discontinuous. As examples, to which our results apply, we mention the st…
Lorentzian Calderón problem near the Minkowski geometry
本文研究 Lorentz 流形上由 Dirichlet-to-Neumann 映射反演波动方程零阶系数的问题，利用双零锥外部的最优唯一延拓原理，将唯一性推广到更弱曲率条件及其时空扰动，从而解决 Minkowski 几何附近的 Calderón 问题。
Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the Dirichlet problem for a class of Monge-Ampère type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Ampère type equation in the case $p\geq 1$ by studying variational properties for a family of Monge-Ampère functionals. Moreover, the existence and optimal global Hölder regularity in the case $p<1$ and $q\geq n$ is…
We consider an energy functional combining the square of the local oscillation of a one--dimensional function with a double well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of these heteroclinic connections. We show that these minimizers are monotone, and therefore they satisfy a suitable Euler-Lagrange equation. We also prove that, differently from the classical cases …
The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction
本文证明 Ornstein-Uhlenbeck 算子第一 Dirichlet 特征值关于 Minkowski 加法具有凸性，刻画了部分凸集类中的等号情形，并证明凸区域上相应正特征函数是对数凹的。
We consider a heat problem with discontinuous diffusion coefficientsand discontinuous transmission boundary conditions with a resistancecoefficient. For all compact $(ε,δ)$-domains $Ω\subset\mathbb{R}^n$ with a $d$-set boundary (for instance, aself-similar fractal), we find the first term of the small-timeasymptotic expansion of the heat content in the complement of$Ω$, and also the second-order term in the case of a regularboundary. The asymptotic expansion is different for the cases offinite and infinite resistance of the boundary. The derived formulasrelate the heat content to the volume of…
Let $Ω$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of $k$-convex solutions to a class of Hessian quotient equations defined over $Ω\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary condition.
In a convex domain $Ø\subset\R^3$, we consider the minimization of a 3D-Ginzburg-Landau type energy $E_\v(u)=1/2\int_Ø|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ with a discontinuous pinning term $a$ among $H^1(Ø,\C)$-maps subject to a Dirichlet boundary condition $g\in H^{1/2}(\pØ,§^1)$. The pinning term $a:\R^3\to\R^*_+$ takes a constant value $b\in(0,1)$ in $ø$, an inner strictly convex subdomain of $Ø$, and 1 outside $ø$. We prove energy estimates with various error terms depending on assumptions on $Ø,ø$ and $g$. In some special cases, we identify the vorticity defects via the concentration of…
Positive and nodal solutions for the Minkowski mean curvature equation: multiplicity and asymptotics
研究闵可夫斯基空间中平均曲率算子狄利克雷问题，通过Szulkin方法等，探讨参数λ、μ对解多重性的影响，给出解的极限轮廓、不存在性准则，在球域中用打靶法证明存在任意多个节点径向解。
Fredholm Alternative for Periodic-Dirichlet Problems for Linear Hyperbolic Systems
本文研究了线性双曲系统在周期-狄利克雷边界条件下的Fredholm替代定理，通过构建两个尺度的巴纳赫空间来建模问题。
Discontinuous eikonal equations in metric measure spaces
本文在度量测度空间中研究非齐次项不连续、无界且仅 p 次可积的程函方程，通过推广 Monge 解并引入基于最优控制的新度量，证明了 Dirichlet 边值问题解的存在唯一性，并分析了其 Hölder 连续性。
We revisit the box Minkowski model [Phys. Rev. Lett. 109, 221101 (2012)] and provide a strong argument that, subject to the Dirichlet boundary condition, it is unstable toward black hole formation for arbitrarily small generic perturbations. Using weakly nonlinear perturbation theory, we derive the resonant system, which compared to systems with the anti-de Sitter asymptotics, has extra resonant terms, and study its properties, including conserved quantities. We find that the generic solution of the resonant system becomes singular in finite time. Surprisingly, the additional resonant interact…
Rigidity in the Lorentzian Calderón problem with formally determined data
本文研究洛伦兹Calderón问题，证明在紧集外与Minkowski度量一致且具有相同双曲Dirichlet-to-Neumann映射的全局双曲度量必为Minkowski度量，并用更少测量量解决了形式确定的反问题。
The paper provides two versions of nonlocal Poincaré-type inequalities for integral operators with a convolution-type structure and functions satisfying a zero-Dirichlet like condition. The inequalities extend existing results to a large class of nonhomogeneous kernels with supports that can vary discontinuously and need not contain a common set throughout the domain. The measure of the supports may even vanish allowing the zero-Dirichlet condition to be imposed on only a lower-dimensional manifold, with or without boundary. The conditions may be imposed on sets with co-dimension larger than o…
Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function
本文研究由时间不连续 N-函数刻画增长与强制性的单调随机抛物方程，证明弱解存在性，并建立 Orlicz 空间中的 Itô 公式，统一涵盖 p(t,x)-Laplacian 与双相位问题。
A high order discontinuous Galerkin method for the recovery of the conductivity in Electrical Impedance Tomography
本文提出一种基于高阶间断伽辽金方法和Dirichlet-to-Neumann映射的电阻抗层析成像电导率恢复优化方法，证明了极小化子存在性，并通过数值实验验证了其高精度与高效率。
A finite element analysis of a Dirichlet boundary control problem governed by the linear parabolic equation is presented in this article. The Dirichlet control is considered in a closed and convex subset of the energy space $H^1(Ω\times(0,T)).$ We prove well-posedness and discuss some regularity results for the control problem. We derive the optimality system for the optimal control problem. The first order necessary optimality condition results in a simplified Signorini type problem for control variable. The space discretization of the state variable is done using conforming finite elements, …
We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only uses the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as opposed to the more difficult $O(|t|^{-1})$ decay. A…
Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
本文在旋转对称性下，给出了凸旋转体Christoffel-Minkowski问题的显式解，并研究了混合面积测度的存在性问题，通过构造显式的凸解来解决混合Monge-Ampère方程，特别地处理了R^n上的k-Hessian方程的Dirichlet问题。
In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov.
Collapse and saturation for radial nodal solutions of a Lorentz--Minkowski mean curvature equation
研究Lorentz--Minkowski平均曲率方程径向结点解的渐近行为，证明解的坍塌与饱和二分法，并构造小解与大解分别收敛于零和饱和轮廓。
Homogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem
本文研究满足 Cordes 条件的不连续周期系数非散度型椭圆方程均匀化，证明不变测度、校正子与最优收敛率，并提出基于有限元逼近不变测度的有效系数及均匀化解的数值格式。
Time-optimal problem in the space of probabilities measures
本文研究了概率测度空间中连续方程的时间最优问题，推导了动态规划原理，并证明了价值函数的Kruzhkov变换为Hamilton-Jacobi方程的粘性解，同时建立了价值函数的Γ收敛性。
We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps from 3+1 dimensional Minkowski spacetime into the three-sphere. Using mixed analytical and numerical methods we show that, for a given topological degree of the map, all solutions starting from smooth finite energy initial data converge to the unique static solution (harmonic map). The asymptotics of this relaxation process is described in detail. We hope that our model will provide an attractive mathematical setting for gaining insight into dissipation-by-dispersion phenomena, in particular the soliton resolution conj…
This is the first of a series of papers devoted to the study of classical initial-boundary value problems of Dirichlet, Neumann and mixed type for the Nonlinear Schrödinger equation on the segment. Considering proper periodic discontinuous extensions of the profile, generated by suitable point-like sources, we show that the above boundary value problems can be rewritten as nonlinear dynamical systems for suitable sets of algebro-geometric spectral data, generalizing the classical Dubrovin equations. In this paper we consider, as a first illustration of the above method, the case of the Dirichl…
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