Existence of C1-solutions to Certain Non-uniformly Degenerate Elliptic Equations

We are concerned with the Dirichlet problem of {div A(x, Du) + B(z) = 0 \qquad in Ω u= u_0 \qquad \qquad on ∂ Ω Here Ω ⊂ R^N is a bounded domain, A(x, p) = (A¹ (x, p), ... >A^N (x, p}) satisfies min{|p|^{1+α}, |p|^{1+β}} ≤ A(x, p) ⋅ p ≤ α_0(|p|^{1+α}+|p|^{1+β}) with 0 < α ≤ β. We show that if A is Lipschitz, B and u_0 are bounded and β < max {\frac{N+2}{N}α + \frac{2}{N},α + 2}, then there exists a C¹-weak solution of (0.1).     

A Singular Trudinger-Moser Inequality in Hyperbolic Space

In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space $$H^n: sup_{u∈W^{1,n}(H^n),∫_{H^n}|∇_H^nu|^ndμ ≤ 1}∫_{H^n}\frac{e^{α|u|\frac{n}{n-1}}-Σ^{n-2}_{k=0}\frac{α^k|u|^\frac{nk}{n-1}}{k!}}{ρ^β}dμ‹∞ ⇔ \frac{α}{α_n}+\frac{β}{n} ≤ 1,$$ where α>0,α ∈ [0,n), ρ and dμ are the distance function and volume element of $H^n$ respectively.     

Global W2,p (2<=p

This paper studies the initial-boundary value problem of GBBM equations u_t - Δu_t = div f(u) \qquad\qquad\qquad(a) u(x, 0) = u_0(x)\qquad\qquad\qquad(b) u |∂Ω = 0 \qquad\qquad\qquad(c) in arbitrary dimensions, Ω ⊂ R^n. Suppose that. f(s) ∈ C¹ and |f'(s)| ≤ C (1+|s|^ϒ), 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3, 0 ≤ ϒ < ∞ if n = 2, u_0 (x) ∈ W^{2⋅p}(Ω) ∩ W^{1⋅p}_0(Ω) (2 ≤ p < ∞), then ∀T > 0 there exists a unique global W^{2⋅p} solution u ∈ W^{1,∞}(0, T; W{2⋅p}(Ω)∩ W^{1⋅p}_0(Ω)), so the known results are generalized and improved essentially.     

Positive Solution of a Semilinear Elliptic Equation on RN

In this paper, we obtain the existence of positive solution of {-Δu = b(x)(u - λ)^p_+,\qquad x ∈ R^N λ > 0, |∇ u| ∈ L² (R^N),\qquad u ∈ L\frac{2N}{N-2} (R^N) under the assumptions that 1 < p < \frac{N+2}{N-2}, N ≥ 3, b(x) satisfies b(x) ∈ C(R^N), b(x) > 0 in R^N b(x) →_{|x|→∞}b^∞ and b(x) > \frac{4}{p+3}b^∞ for x ∈ R^N     

THE FIRST BOUNDARY VALUE PROBLEM FOR SOLUTIONS OF DEGENERATE QUASUJNEAR PARABOLIC EQUATIONS

In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation $$\[\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}} {}&{(x,t) \in [0,T]} \end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T} \end{array}} \right.\]$$ $$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}} {and}&{v(u) \to 0\begin{array}{*{20}{c}} {as}&{u \to 0} \end{array}} \end{array}} \right)\]$$ under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$     

Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source

In this paper, we study the asymptotic behavior of solutions to a quasilinear fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n$ $(n ≥1)$, where $b ≥0$, $γ ≥1$, $a_i ≥1$, $µ$, $b_i >0$ $(i =1,2)$, $D$, $S∈ C^2([0,∞))$ fulfilling $D(s) ≥ a_0(s+1)^{−α}$, $0 ≤ S(s) ≤ b_0(s+1)^β$ for all $s ≥ 0,$ where $a_0,b_0 > 0$ and $α,β ∈ \mathbb{R}$ are constants. The purpose of this paper is to prove that if $b ≥ 0$ and $µ > 0$ sufficiently large, the globally bounded solution $(u,v,w)$ with nonnegative initial data $(u_0,v_0,w_0)$ satisfies $$\Big\| u(·,t)− \Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\|_{L^∞(Ω)}+\Big\| v(·,t)−\frac{b_1b_2}{a_1a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)} +\Big\| w(·,t)−\frac{b_2}{a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)}→0$$ as $t→∞$.     

A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS

ＡＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＯＦＥＸＩＳＴＥＮＣＥＯＦＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＦＯＲＳＯＭＥＮＯＮＬＩＮＥＡＲＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺＨＡＮＧＱＵＡＮＤＥ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

SINGULAR PERTURBATIONS FOR QUASILINEAR HYPERBOLIC EQUATIONS

This paper deals with the following mixed problem for Quasilinear hyperbolic equationsThe M order uniformly valid asymptotic solutions are obtained and there errors areestimated.     

Space-Time Regularity for the Three Dimensional Navier–Stokes and MHD Equations

In this paper, we investigate the space-time regularity of solutions to (1) the three dimensional incompressible Navier–Stokes equations for initial data \(u_{0}=(u_{0}^{h},u_{0}^{3}) \in \dot{B}_{p,r}^{ \frac{3}{p}-1} (\mathbb{R}^{3})\) with large initial vertical velocity component; and (2) the three dimensional incompressible magneto-hydrodynamic equations for initial datum \(u_{0}=(u_{0}^{h},u _{0}^{3})\in \dot{B}_{p,r}^{\frac{3}{p}-1} (\mathbb{R}^{3})\) with large initial vertical velocity component and \(b_{0}=(b_{0}^{h},b_{0}^{3}) \in \dot{B}_{p,r}^{\frac{3}{p}-1} (\mathbb{R}^{3})\) with large initial vertical magnetic field component.

     

Global W2,p Solutions of GBBM Equations on Unbounded Domain

In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential.     

THROREMS OF DISTORTION FOR SCHLICHT FUNCTIONS

Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l} {\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m}, \end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m. Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l} \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3, \end{array}\], where \[\begin{array}{l} {F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}). \end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun. Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l} P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2}, \end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing.     

Supercritical Elliptic Equation in Hyperbolic Space

In this paper, we study the following semi-linear elliptic equation $$-Δ_H^nu=|u|^{p-2}u,\qquad\qquad (0.1)$$ in the whole Hyperbolic space $\mathbb{H}^n$,where n ≥ 3, p › 2n/(n-2). We obtain some regularity results for the radial singular solutions of problem (0.1). We show that the singular solution $u^∗$ with $lim_{t → 0}(sinht)^{\frac{2}{p-2}}⋅u(t)=±(\frac{2}{p-2}(n-2-\frac{2}{p-2})^{\frac{1}{p-2}}$ belongs to the closure (in the natural topology given by $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N))$ of the set of smooth classical solutions to the Eq. (0.1). In contrast, we also prove that any oscillating radial solutions of (0.1) on $\mathbb{H}^N$\{0} fails to be in the space $H¹_{loc}(\mathbb{H}^N)∩L^p_{loc}(H^N)$.     

On the Existence of Positive Solutions of Quasilinear Elliptic Equations with Mixed Boundary Conditions

In this paper, the existence of positive solutions for the mixed boundary problem of quasilinear elliptic equation {-div (|∇u|^{p-2}∇u) = |u|^{p^∗-2}u + f(x, u), \quad u > 0, \quad x ∈ Ω u|_Γ_0 = 0, \frac{∂u}{∂\overrightarrow{n}}|_Γ_1 = 0 is obtained, where Ω is a bounded smooth domain in R^N, ∂Ω = \overrightarrow{Γ}_0 ∪ \overrightarrow{Γ}_1, 2 ≤ p < N, p^∗ = \frac{Np}{N-p}, Γ_0 and Γ_1 are disjoint open subsets of ∂Ω.     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      16. NONLINEAR BOUNDARY PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS      Zheng Songmu 《数学年刊B辑(英文版)》1983,4(2):177-186 By means of the supersolution and subsolution method and monotone iteration technique, the following nonlinear elliptic boundary problem with the nonlocal boundary conditions is considerd. The sufficient conditions which ensure at least one solution are given. Furthermore, the estimate of the first nonzero eigenvalue for the following linear eigenproblem is obtained, that is λ_1≥2α/(nd~2).      17. Regularity Results for Minimizers of Certain Functional Having Nonquadratic Growth with Obstacles          下载免费PDF全文 Minchun Hong 《偏微分方程(英文版)》1997,10(1):65-84 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      18. PROPAGATION OF SINGULARITES FOR SOLUTION OF SEMILINER WAVE EQUATIONS      Liu Linqi 《数学年刊B辑(英文版)》1988,9(4):379-389 In order better to research the singularities of the solutions $\[u \in H_{loc}^s(\Omega ),\Omega \subset {R^n},s > \frac{n}{2} + 1\]$ , for semilinear hyperbolic equations $\[u = f(u,Du)\]$, in this paper, a kind of weighted Sobolev space $\[({H^s})_{{P_\mu }}^\alpha \],\[\mu = 1,2,{p_1} = {D_i} - \left| {{D_x}} \right|,{P_2} = {D_i} + \left| {{D_x}} \right|\]$, closely related with the solutions of the equations, is presented. It is discussed that their products tacitly keep roughly $\[{H^{3x - n}}\]$ microlocal regularity on the characteristic directions for $\[{P_\mu }\]$ and invariance under nonlinear maps. Then it is obtained that roughly $\[{H^{3x - n}}\]$ propagation of singularities theorem is valid for $\[u = f(u)\]$.      19. Ginzburg-Landau Vortices in Inhomogeneous Superconductors          下载免费PDF全文 Huaiyu Jian &  Youde Wang 《偏微分方程(英文版)》2002,15(3):45-60 We study the vortex convergence for an inhomogeneous Ginzburg-Landau equation, -Δu = ∈^{-2}u(a(x) - |u|²), and prove that the vortices are attracted to the minimum point b of a(x) as ∈ → 0. Moreover, we show that there exists a subsequence ∈ → 0 such that u_∈ converges to u strongly in H¹_{loc}(\overline{Ω} \ {b}).      20. ON THE GENERALIZED POLYNOMIALS OF L. V. KANTOROVITCH AND THEIR ASYMPTOTIC BEHAVIORS      Cao Jiading 《数学年刊B辑(英文版)》1981,2(2):243-255 In this article we generahze the polynomials of Kantorovitch \({P_n}(f)\) . Let \({B_n}\) be a sequence of linear operators from C[a,b] into \({H_n}\), if \[f(t) \in L[a,b],F(u) = \int_a^u {f(t)dt} ,{A_n}(f(t),x) = \frac{d}{{dx}}{B_{n + 1}}(F(u),x)\], here \({B_n}\)satisfy\[\begin{array}{l} (a):{B_n}(1,x) \equiv 1,{B_n}(u,x) \equiv x;\(b):for{\kern 1pt} {\kern 1pt} g(u) \in C[a,b]{\kern 1pt} {\kern 1pt} we{\kern 1pt} {\kern 1pt} have{\kern 1pt} {\kern 1pt} {B_n}(g(u),b) = g(b). \end{array}\]. we call such \({A_n}(f)\) generalized polynomials of Kantorovitch (denoted by \({A_n}(f) \in K\) ). Let \[\begin{array}{l} {\varepsilon _n}({W^2};x)\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}} \left| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right|,\{\varepsilon _n}{({W^2}{L^p})_{{L^p}}}\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}{L^p}} {\left\| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right\|_p}. \end{array}\] We have proved the following results: Let An he a sequence of linear continuous operators of type \[C[a,b] \Rightarrow C[a,b],{D_n}(x,z)\mathop = \limits^{def} {A_n}(\left| {t - z} \right|,x) - \left| {x - z} \right| - ({A_n}(t,x) - x)Sgn(x - z),{A_n}(1,x) = 1\] then (1):\({\varepsilon _n}({W^2};x) = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dz\), (2): Moreover, if \({A_n}\) be a sequence of linear positive operators, then for \(\left[ {\begin{array}{*{20}{c}} {a \le x \le b}\{a \le z \le b} \end{array}} \right]\) ,we have \({D_n}(x,z) \ge 0\), and \({\varepsilon _n}({W^2};x) = \frac{1}{2}{A_n}({(t - x)^2},x)\). Let \({A_n}(f) \in K\) be a sequence of linear positive operators,\[{R_n}{(z)_L} = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dx\],then \[{R_n}{(z)_L} = \frac{1}{2}\left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right]\] and \[{\varepsilon _n}{({W^2}L)_L}{\rm{ = }}\frac{1}{2}\left\| {{B_{n + 1}}({u^2},z) - {z^2}} \right\|\]. Let \[{g_n} = \frac{1}{2}\mathop {\max }\limits_{a \le x \le b} {A_n}({(t - x)^2},x),{h_n} = \frac{1}{2}\mathop {\max }\limits_{a \le z \le b} \left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right],\] then \[{\varepsilon _n}{({W^2}{L^p})_{{L^p}}} \le {g_n}^{1 - \frac{1}{p}}{h_n}^{\frac{1}{p}}(1 < p < \infty ).\]

NONLINEAR BOUNDARY PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS

By means of the supersolution and subsolution method and monotone iteration technique, the following nonlinear elliptic boundary problem with the nonlocal boundary conditions is considerd. The sufficient conditions which ensure at least one solution are given. Furthermore, the estimate of the first nonzero eigenvalue for the following linear eigenproblem is obtained, that is λ_1≥2α/(nd~2).     

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     

PROPAGATION OF SINGULARITES FOR SOLUTION OF SEMILINER WAVE EQUATIONS

In order better to research the singularities of the solutions $\[u \in H_{loc}^s(\Omega ),\Omega \subset {R^n},s > \frac{n}{2} + 1\]$ , for semilinear hyperbolic equations $\[u = f(u,Du)\]$, in this paper, a kind of weighted Sobolev space $\[({H^s})_{{P_\mu }}^\alpha \],\[\mu = 1,2,{p_1} = {D_i} - \left| {{D_x}} \right|,{P_2} = {D_i} + \left| {{D_x}} \right|\]$, closely related with the solutions of the equations, is presented. It is discussed that their products tacitly keep roughly $\[{H^{3x - n}}\]$ microlocal regularity on the characteristic directions for $\[{P_\mu }\]$ and invariance under nonlinear maps. Then it is obtained that roughly $\[{H^{3x - n}}\]$ propagation of singularities theorem is valid for $\[u = f(u)\]$.     

Ginzburg-Landau Vortices in Inhomogeneous Superconductors

We study the vortex convergence for an inhomogeneous Ginzburg-Landau equation, -Δu = ∈^{-2}u(a(x) - |u|²), and prove that the vortices are attracted to the minimum point b of a(x) as ∈ → 0. Moreover, we show that there exists a subsequence ∈ → 0 such that u_∈ converges to u strongly in H¹_{loc}(\overline{Ω} \ {b}).     

ON THE GENERALIZED POLYNOMIALS OF L. V. KANTOROVITCH AND THEIR ASYMPTOTIC BEHAVIORS

In this article we generahze the polynomials of Kantorovitch \({P_n}(f)\) . Let \({B_n}\) be a sequence of linear operators from C[a,b] into \({H_n}\), if \[f(t) \in L[a,b],F(u) = \int_a^u {f(t)dt} ,{A_n}(f(t),x) = \frac{d}{{dx}}{B_{n + 1}}(F(u),x)\], here \({B_n}\)satisfy\[\begin{array}{l} (a):{B_n}(1,x) \equiv 1,{B_n}(u,x) \equiv x;\(b):for{\kern 1pt} {\kern 1pt} g(u) \in C[a,b]{\kern 1pt} {\kern 1pt} we{\kern 1pt} {\kern 1pt} have{\kern 1pt} {\kern 1pt} {B_n}(g(u),b) = g(b). \end{array}\]. we call such \({A_n}(f)\) generalized polynomials of Kantorovitch (denoted by \({A_n}(f) \in K\) ). Let \[\begin{array}{l} {\varepsilon _n}({W^2};x)\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}} \left| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right|,\{\varepsilon _n}{({W^2}{L^p})_{{L^p}}}\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}{L^p}} {\left\| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right\|_p}. \end{array}\] We have proved the following results: Let An he a sequence of linear continuous operators of type \[C[a,b] \Rightarrow C[a,b],{D_n}(x,z)\mathop = \limits^{def} {A_n}(\left| {t - z} \right|,x) - \left| {x - z} \right| - ({A_n}(t,x) - x)Sgn(x - z),{A_n}(1,x) = 1\] then (1):\({\varepsilon _n}({W^2};x) = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dz\), (2): Moreover, if \({A_n}\) be a sequence of linear positive operators, then for \(\left[ {\begin{array}{*{20}{c}} {a \le x \le b}\{a \le z \le b} \end{array}} \right]\) ,we have \({D_n}(x,z) \ge 0\), and \({\varepsilon _n}({W^2};x) = \frac{1}{2}{A_n}({(t - x)^2},x)\). Let \({A_n}(f) \in K\) be a sequence of linear positive operators,\[{R_n}{(z)_L} = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dx\],then \[{R_n}{(z)_L} = \frac{1}{2}\left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right]\] and \[{\varepsilon _n}{({W^2}L)_L}{\rm{ = }}\frac{1}{2}\left\| {{B_{n + 1}}({u^2},z) - {z^2}} \right\|\]. Let \[{g_n} = \frac{1}{2}\mathop {\max }\limits_{a \le x \le b} {A_n}({(t - x)^2},x),{h_n} = \frac{1}{2}\mathop {\max }\limits_{a \le z \le b} \left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right],\] then \[{\varepsilon _n}{({W^2}{L^p})_{{L^p}}} \le {g_n}^{1 - \frac{1}{p}}{h_n}^{\frac{1}{p}}(1 < p < \infty ).\]