THE GLOBAL SMOOTH SOLUTIONS OF SECOND ORDER QUASILINEAR HYPERBOLIC EQUATIONS WITH DISSIPATIVE BOUNDARY CONDITIONS

The paper deals with the following boundary problem of the second order quasilinear hyperbolic equation with a dissipative boundary condition on a part of the boundary:u_(tt)-sum from i,j=1 to n a_(ij)(Du)u_(x_ix_j)=0, in (0, ∞)×Ω,u|Γ_0=0,sum from i,j=1 to n, a_(ij)(Du)n_ju_x_i+b(Du)u_t|Γ_1=0,u|t=0=φ(x), u_t|t=0=ψ(x), in Ω, where Ω=Γ_0∪Γ_1, b(Du)≥b_0>0. Under some assumptions on the equation and domain, the author proves that there exists a global smooth solution for above problem with small data.     

The Gradient Theory of the Phase Transitions with Dirichlet Boundary Condition

§ 1.Introduction　 In this paper,we study the asymptotic behavior of minimizers{ uε} ε>0 (ε→ 0 ) of thevariational probleminf∫Ω[εp- 1 | Du| p + 1εW(x,u) ] dx:u∈ W1 ,p(Ω ) ,u =g,x∈ Ω (Pε)where p>1 ,N≥ 2 ,andΩ RN is a bounded open domain with Ω∈C2 .　 This kind of problems comes from the theory of phase transitions and are widelystudied in recent years.In [1 ]— [1 4] ,the authors investigated these problems underthe integral constraint∫Ωudx =constant,In [1 5]— [1 6] ,…     

Regularity for solutions to anisotropic obstacle problems

For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.     

退缩拟线性椭圆型方程某类边值问题的无穷多解

文[1]证明了如下D氏问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, u=0,x∈Ω存在非平凡解,本文讨论上述方程的另一类边界问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, g(|Du|~2)D_iu(0)(n,x_i)+h(x,u)=0,x∈Ω, (1)其中Ω∈R~n是具有光滑边界的有界区域,n(x)是Ω在x点的外法向,D_iu=u/x_i,Du=gradu=u,重复指标表示求和,与问题(1)相应的泛函为:     

一类椭圆型变分不等式的非平凡广义解的存在性

本文研究了下列障碍问题的非平凡解的存在性u∈K∶∫Ωu.(u-u)dx ∫Ωa(x)u.(v-u)dx∫Ωp(x,u)(v-u)dx,v∈K.其中K={v∈H01(Ω)∶vψa.e.onΩ}.利用关于不等式推广的山路引理,在a(x)和障碍p(x,ξ)满足适当的假设下,我们证明了上述不等式存在非平凡解.     

凸区域上拟线性椭圆型方程的尖峰解

本文讨论奇异扰动的拟线性椭圆型方程-ε△pu(x)=f(u(x)),u(x)≥0,x∈Ω;u=0,x∈Ω在Dirichlet边值条件下极小能量解的存在性和结构.其中ε＞0是小参数,p＞2,△pu=div(|Du|p-2Du),f(s)=sq-sp-1,p-1＜q＜Np/N-p-1.Ω RN(N≥2)是有界光滑区域.当ε→0时,方程存在一个极小能量解,应用移动平面方法可以证明此解在凸区域上会变成一个尖峰解.     

一类拟线性退化抛物方程Dirichlet问题粘性解的正则性

本文考虑下面的Dirichlet问题ut一Tr[a(x,t)D2u]+H(x,t,u,Du)=0,(x,t)∈QT=Ω×(0,T),u(x,t)=ψ(x,t), (x,t)∈ГT. (DP)利用粘性解理论证明了当H,Г满足一定条件时,(DP)的粘性解u(x,t)满足如果ψ∈Ca2,则u(x,t)∈Cα,羞;若ψ=0,则u(x,t)是Lpschitz连续的.     

凸区域上拟线性椭圆型方程的尖峰解

本文讨论奇异扰动的拟线性椭圆型方程-εΔ_pu(x)=f(u(x)),u(x)≥0,x∈Ω;u=0,x∈Ω在 Dirichlet边值条件下极小能量解的存在性和结构。其中ε>0是小参数,p>2,Δ_pu=div(|Du|~(p-2)Du),f(s)=s~q-s~(p-1),p-1相似文献   

奇系数非线性方程组弱解的正则性

我们研究椭圆型方程组-D_a(|x|~(-r)A_i~a(x,u,Du))+B_i(x,u,Du)=0 (1)i=1,2,…,N 的弱解 u 的 L_p 估计与它在原点附近的 C~(0,α)性质。其中,x∈ΩR~n,0∈Ω,Ω有界.n-p_0≤r2.关于(1),当 A_i~α(x,μ,Du)=A(x,u)Du 时,文[1]得到了它的弱解的 L_p 估计与 C~(0,α)正则性,文[4]则进一步研究了它在Ω\{0}内的部分正则性。对(1)的类似的结果,尚未     

拟凸条件下变分问题解的正则性

本文研究变分问题(1)■(u:Ω)=∫_Ωf(x,u,Du)dx的极小函数的正则性,其中Ω■R~n是有界开域,u:Ω→R-~N,Du:Ω→R(nN),f: ×R~N×R(nN)→R。定义称函数f满足严格拟凸条件,是指存在常数v>0,使得对任意的(x_0,u_0,p_0)∈Ω×R~N×R(nN)和φ∈C~∞_0(Ω,R~N),都有■(2)其中|Ω|是Ω的Lebesgue测度。定理设u∈H~(1,2)(Ω,R~N)是泛函f的极小函数,即对任意的φ∈H_0~(1,2)(Ω,R~N),都有■而f(x,u,p)满足下列假设 (H1) f满足严格拟凸性,即(2)成立, (H2) f关于p的二阶导数存在,且存在常数L>0,使得■对任意的(x,u,p)∈Ω×R~N×R~(nN),都有■ (3)|f_(pp)(x,u,p)|≤L_0 (H3) 存在[0,∞]上的连续、有界、凹的函数∞(t),使得(4)■(5)■(6)■且ω(t)≤At~α,其中A,α是正常数。那么存在常数δ∈(0,1)和开集Ω_0Ω,使得|Ω-Ω_0|=0,Du∈C~6(Ω_0,R(nN))。