Homogenization and Asymptotics for Small Transaction Costs: The Multidimensional Case

In the context of the multi-dimensional infinite horizon optimal consumption investment problem with small proportional transaction costs, we prove an asymptotic expansion. Similar to the one-dimensional derivation in our accompanying paper, the first order term is expressed in terms of a singular ergodic control problem. Our arguments are based on the theory of viscosity solutions and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available and we also prove the existence of a corrector and its properties. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.     

A Free Boundary Problem with Curvature

Abstract

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.     

Homogenization of a Model Problem on Contact Angle Dynamics

In this paper we consider homogenization of oscillating free boundary velocities in periodic media, with general initial data. We prove that there is a unique and stable effective free boundary velocity in the homogenization limit.     

Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L~2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator.Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.     

Reflected BSDE and Locally Periodic Homogenization of Semilinear PDEs with Nonlinear Neumann Boundary Condition

We study the homogenization of semilinear partial differential equations (PDEs) with nonlinear Neumann boundary condition, locally periodic coefficients, and highly oscillating drift and nonlinear term. Our method is entirely probabilistic, as in a periodic case by Ouknine and Pardoux [14 Ouknine , Y. , and Pardoux , É. 2002 . Homogenization of PDEs with non linear boundary condition, Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999). Progresses of Probability, 52, Birkhäuser, Basel , pp. 229 – 242 . [Google Scholar]] and builds on our earlier work [5 Diakhaby , A. , and Ouknine , Y. 2006 . Locally periodic homogenization of reflected diffusion . Journal of Applied Mathematics and Stochastic Analysis . [Google Scholar]], which gives us the locally periodic counterpart of Theorem 2.2 in Tanaka [21 Tanaka , H. 1984 . Homogenization of diffusion processes with boundary conditions . Stochastic Analysis and Applications 7 : 411 – 437 . Advanced Probability and Related Topics 7, Dekker, New York . [Google Scholar]].     

Nonuniqueness in a Free Boundary Problem from Combustion

We study a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. Vázquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion. A. Petrosyan was supported in part by NSF grant DMS-0701015. N.K. Yip was supported in part by NSF grant DMS-0406033.     

带自由边界的调和映射存在性

本文研究带自由边界的调和映射的存在性.用Sack-Uhlenbeck的挠动Morse理论及Micallef和Moore的Morse指标的估计方法,我们得到了从圆盘到Riemann流形的带自由边界的调和映射的存在性.     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

振荡Robin混合边值齐次化问题

该文研究了振荡Robin混合边值齐次化问题解的收敛率.该工作的困难之处在于Robin边值上出现的振荡因子以及边界交叉项的处理.该文利用对偶方法巧妙得对振荡积分进行了估计.文中建立了解的H1和L2收敛率,所得结果明显地依赖于维数.该文可以视为将对偶方法和光滑算子,延拓到处理振荡Robin混合边值问题的情形.     

On Regularity and Singularity of Free Boundaries in Obstacle Problems

The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems. This approach is based on the monotonicity of several variational integrals, the Federer-Almgren dimension reduction and stratification theorems, and some simple PDE arguments.     

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L _p -spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.     

连续铸钢中的一类周期自由边界问题

本文讨论了连续铸钢中的一类周期自由边界问题,证明了周期古典解的存在性,唯一性和稳定性。     

Homogenization and correctors for monotone problems in cylinders of small diameter

In this paper we study the homogenization of monotone diffusion equations posed in an N  -dimensional cylinder which converges to a (one-dimensional) segment line. In other terms, we pass to the limit in diffusion monotone equations posed in a cylinder whose diameter tends to zero, when simultaneously the coefficients of the equations (which are not necessarily periodic) are also varying. We obtain a limit system in both the macroscopic (one-dimensional) variable and the microscopic variable. This system is nonlocal. From this system we obtain by elimination an equation in the macroscopic variable which is local, but in contrast with usual results, the operator depends on the right-hand side of the equations. We also obtain a corrector result, i.e. an approximation of the gradients of the solutions in the strong topology of the space L^p$L^p$ in which the monotone operators are defined.     

The fresh-salt water interface in a semi-pervious aquifer

The paper determines the location of the steady state interface between fresh and saltwater in a plane coastal aquifer. The lower boundary is totally impervious while the upper one is impervious below land and semi-pervious below the sea allowing an outflow through this part of the upper boundary. The model equations reduce to two boundary value problems, one valid in x < 0 and the other in x > 0 here x is measured along the aquifer with the origin at the coast. In each region unknown boundaries have to be determined as part of the solution using boundary and continuity conditions. Two cases are presented using the Dupuit approximation. One where the solution can be written down in terms of elementary functions and the other in which we have to use a phase space analysis.     

非线性椭圆方程自由边值问题球对称解的存在性

给出了如下的非线性椭圆方程自由边值问题-Δu=λu+(1+ε)u+p,x∈B Rn,u|Ω=μ,∫Ωnu=-M(1)在C[0,1]中的球对称解的存在性.并得到比上述问题更一般的非线性椭圆方程自由边值问题-Δu=h(u),x∈B Rn,u|Ω=μ,∫Ωun=-M,在C[0,1]中的球对称解的存在性,其中B为Rn中的单位球,p>1,λ>0,μ<0,M>0,ε>0;λ,μ,M,ε均为常数,n为正整数.     

Heat Conduction with Flux Condition on a Free Patch

A new free boundary or free patch problem for the heat equation is presented. In the problem a nonlinear heat flux condition is prescribed on a free portion of the boundary, the patch, the position of which depends on the solution. The existence of a weak solution is established using the theory of set-valued pseudomonotone operators.     

FREE BOUNDARY PROBLEM ARISING FROM EVAPORATION FROM POROUS MEDIUM

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Error estimates on homogenization of free boundary velocities in periodic media

In this paper we consider a free boundary problem which describes contact angle dynamics on inhomogeneous surface. We obtain an estimate on convergence rate of the free boundaries to the homogenization limit in periodic media. The method presented here also applies to more general class of free boundary problems with oscillating boundary velocities.