Regularity results for a class of obstacle problems

We prove some optimal regularity results for minimizers of the integral functional ∫ f(x, u, Du) dx belonging to the class K ≔ {u ∈ W ^1,p (Ω): u ⩾ ψ, where ψ is a fixed function, under standard growth conditions of p-type, i.e.

LOCAL REGULARITY RESULT IN OBSTACLE PROBLEMS

We obtain a local regularity result for solutions to Kψ,θ-obstacle problem of A-harmonic equation divA(x,u(x),▽u(x))=0,where A:Ω× R×Rn → Rn is a Carath'eodory function satisfying some coercivity and growth conditions with the natural exponent 1pn,the obstacle function ψ≥0,and the boundary data θ∈W1,p(?).     

LOCAL REGULARITY RESULT FOR SOLUTIONS OF OBSTACLE PROBLEMS

This paper gives the local regularity result for solutions to obstacle problems of A-harmonic equation divA(x, ξu(x)) = 0, |A.(x,ξ)|≈|?|p-1, when 1 < p < n and the obstacle function (?)≥0.     

Regularity for obstacle problems without structure conditions

This paper deals with the Lipschitz regularity of minimizers for a class of variational obstacle problems with possible occurrence of the Lavrentiev phenomenon. In order to overcome this problem, the availment of the notions of relaxed functional and Lavrentiev gap is needed. The main tool used here is a crucial Lemma which reveals to be needed because it allows us to move from the variational obstacle problem to the relaxed-functional-related one. This is fundamental in order to find the solutions’ regularity that we intended to study. We assume the same Sobolev regularity both for the gradient of the obstacle and for the coefficients.     

DIRICHLET PROBLEM FOR THE IMPLICIT OBSTACLE PROBLEMS

In this paper the implicit obstac|e problem of fully nonlinear second-order elliptic equations associated with impulsive control problem are investigated. The comparlon principle for viscosity solutions is proved,the existence and uniqueness results are disscussed.     

A-调和方程很弱解的正则性

本文证明了二阶拟线性偏微分方程很弱解的正则性．若ｕ是（１）的一个很弱解并属于一个合适的包含Ｗ１,ｐ　ｌｏｃ（ ）的空间,则ｕ属于 （ ）,即ｕ是（１）通常意义下的弱解．变分积分弱极值的同样结果被得到．     

The Regularity for Solutions of Variational Inequalities

The regularity for solutions of elliptic equations is rather perfectly solved. But it does not so perfect for that of elliptic variational inequalities. In literature only different special situations are considered. Now the boundedness, C^{0,λ} continuity and C^{1,α} regularity are proved for solutions of one-sided obstacle problems under more general structural conditions, in which the growth orders of u are permitted to reach the critical exponents and the growth order ϒ of the gradient in D is permitted to be super critical as 1 < p < n.     

Global higher integrability of solutions to subelliptic double obstacle problems

In this paper we consider the double obstacle problems associated with nonlinear subelliptic equation \[X^*A(x,u,Xu)+ B(x,u,Xu)=0, \ \ x\in\Omega,\] where $X=(X_1,\ldots,X_m)$ is a system of smooth vector fields defined in $\mathbb{R}^n$ satisfying H\"{o}rmander"s condition. The global higher integrability for the gradients of the solutions is obtained under a capacitary assumption on the complement of the domain $\Omega$.     

Regularity for anisotropic solutions to some nonlinear elliptic system

This paper deals with anisotropic solutions u∈W1,(pi)(Ω,?N) to the nonlinear elliptic system −Σi=1nDi(aiα(χ,Du(χ)))=−Σi=1nDiFiα(χ), α=1,2,...,N, We present a monotonicity inequality for the matrix a=(aiα)∈?N×n,whichguarantees global pointwise bounds for anisotropic solutionsu.     

Regularity Results for Minimizers of Certain Functional Having Nonquadratic Growth with Obstacles

We prove partial regularity for minimizers of degenerate variational integrals ∫_Ω F(x, u, Du)dx with obstacles of either the form (i) μ_f = {u ∈ H^{1,m} (Ω,\mathbb{R}^N)|u^N ≥ f_1(u¹, ... ,u^{N-1}) + f_2(x) a.e.} or (ii) μ_N = {u ∈ H^{1,m}(Ω,\mathbb{R}^N)|u^i(x) ≥ h^i(x), a.e.; i=1, ... ,N} The typical mode of variational integrals is given by ∫_Ω [a^{αβ}(x, u)b_{ij}(x, u)D_αu^i D_βu^i]^{\frac{m}{2}}dx, m ≥ 2     

泛函极小与椭圆型方程组解的正则性

研究定义在向量u=(u~1,…,u~N):Ω■R~n→R~N上的各项异性积分泛函F(u)=∫_Ωf(x,Du(x))dx和非线性椭圆型方程组-Σi=1nDi(aiα(x,Du(x)))=-Σi=1nDiFiα(x),α=1,2,…,N.在密度函数f:Ω×R~(N×n)→R和矩阵a=(a_i~α):Ω×R~(N×n)→R~(N×n)满足某单调不等式条件下,得到u整体有界.     

The factorization method for inverse acoustic scattering by a penetrable anisotropic obstacle

This paper is devoted to studying the factorization method applied to the inverse problem of reconstructing a penetrable anisotropic obstacle from far field patterns. We proved the validity of the factorization method. Copyright © 2013 John Wiley & Sons, Ltd.     

Principal frequency and existence of solutions for quasilinear elliptic obstacle problems

In this paper, we are interested in the first eigenvalue of p-Laplacian and the relation between the first eigenvalue and the existence (or nonexistence) of nontrivial (positive) solution for quasilinear elliptic obstacle problems. Utilizing the fact that obstacle problem have consanguineous relations with corresponding equation, we get a simple approach to study the properties of solutions of obstacle problems, such as existence and nonexistence, regularity and stability, etc. In this paper we are mainly concerned with the existence and nonexistence.     

A-调和方程很弱解的正则性

In this paper, the following result is given by using Hodge decomposition and weak reverse H(o)lder inequality: For every r1 with p-(2n 1100n2p(23 n/(p-1) 1)b/a)-1＜r1＜p,there exists the exponent r2 = r2(n,r1,p) ＞ p, such that for every very weak solution u ∈W1r1,loc(Ω) to A-harmonic equation, u also belongs to W1r2,loc(Ω) . In particular, u is the weak solution to A-harmonic equation in the usual sense.     

RBSDE''''s with jumps and the related obstacle problems for integral-partial differential equations

The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.     

三维各向异性Navier-Stokes方程一类大解的整体正则性——献给陆善镇教授75华诞

利用解析性估计和方程非线性项的特殊结构,本文证明了三维各向异Navier-Stokes方程对一类在垂直方向慢变的大初值的整体适定性.