Multiplicity of Solutions to Nonlinear Boundary Value Problem with Nonlocal Boundary Condition

In this paper the author considers the following nonlinear boundary value problem with nonlocal boundary conditions $[\left\{ \begin{array}{l} Lu \equiv - \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ij}}(x)\frac{{\partial u}}{{\partial {x_j}}}) = f(x,u,t)} \u{|_\Gamma } = const, - \int_\Gamma {\sum\limits_{i,j = 1}^n {{a_{ij}}\frac{{\partial u}}{{\partial {x_j}}}\cos (n,{x_i})ds = 0} } \end{array} \right.\]$ Under suitable assumptions on f it is proved that there exists $t_0\in R,-\infinityt_0, at least one solution at t=t_0 at least two solutions as t相似文献   

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).     

The Existence Theorem for Solutions of General Boundary Value Problem of Quasi-Linear Second Order Elliptic Differential Equations

In this paper, we provide the existence theorem for solutions of general boundary value problem of quasi-linear second order elliptic differential equations in the following form: $\[\sum\limits_{i,j = 1}^n {({a_{ij}}(x,u)\frac{{\partial u}}{{\partial {x_j}}}) + a(x,u,{u_{{x_k}}}),{\rm{ }}in} {\rm{ }}\Omega \]$, $\[\alpha (x,u)\frac{{\partial u}}{{\partial \gamma }} + \beta (x,u) = 0,{\rm{ on }}\partial \Omega \]$, where \alpha(x, u) \geq 0,\alpha_u(x, u) \leq 0 and \gamma is some direction, defining on $\[\partial \Omega \]$.     

The Initial Value Problems and Nonlinear Boundary Value Problems for a Class of the Second order Quasilinear Parabolio Systems

In this paper initial value problems and nonlinear mixed boundary value problems for the quasilinear parabolic systems below $\[\frac{{\partial {u_k}}}{{\partial t}} - \sum\limits_{i,j = 1}^n {a_{ij}^{(k)}} (x,t)\frac{{{\partial ^2}{u_k}}}{{\partial {x_i}\partial {x_j}}} = {f_k}(x,t,u,{u_x}),k = 1, \cdots ,N\]$ are discussed.The boundary value conditions are $\[{u_k}{|_{\partial \Omega }} = {g_k}(x,t),k = 1, \cdots ,s,\]$ $\[\sum\limits_{i = 1}^n {b_i^{(k)}} (x,t)\frac{{\partial {u_k}}}{{\partial {x_i}}}{|_{\partial \Omega }} = {h_k}(x,t,u),k = s + 1, \cdots N.\]$ Under some "basically natural" assumptions it is shown by means of the Schauder type estimates of the linear parabolic equations and the embedding inequalities in Nikol'skii spaces,these problems have solutions in the spaces $\[{H^{2 + \alpha ,1 + \frac{\alpha }{2}}}(0 < \alpha < 1)\]$.For the boundary value problem with $\[b_i^{(k)}(x,t) = \sum\limits_{j = 1}^n {a_{ij}^{(k)}} (x,t)\cos (n,{x_j})\]$ uniqueness theorem is proved.     

TIME OPTIMAL BOUNDARY CONTROL FOR SYSTEMS GOVERNED BY PARABOLIC EQUATIONS

In this paper we consider the systems governed, by parabolioc equations \[\frac{{\partial y}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}} ({a_{ij}}(x,t)\frac{{\partial y}}{{\partial {x_j}}}) - ay + f(x,t)\] subject to the boundary control \[\frac{{\partial y}}{{\partial {\nu _A}}}{|_\sum } = u(x,t)\] with the initial condition \[y(x,0) = {y_0}(x)\] We suppose that U is a compact set but may not be convex in \[{H^{ - \frac{1}{2}}}(\Gamma )\], Given \[{y_1}( \cdot ) \in {L^2}(\Omega )\] and d>0, the time optimal control problem requiers to find the control \[u( \cdot ,t) \in U\] for steering the initial state {y_0}( \cdot )\] the final state \[\left\| {{y_1}( \cdot ) - y( \cdot ,t)} \right\| \le d\] in a minimum, time. The following maximum principle is proved: Theorem. If \[{u^*}(x,t)\] is the optimal control and \[{t^*}\] the optimal time, then there is a solution to the equation \[\left\{ {\begin{array}{*{20}{c}} { - \frac{{\partial p}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ji}}(x,t)\frac{{\partial p}}{{\partial {x_j}}}) - \alpha p,} }\{\frac{{\partial p}}{{\partial {\nu _{{A^'}}}}}{|_\sum } = 0} \end{array}} \right.\] with the final condition \[p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)\], such that \[\int_\Gamma {p(x,t){u^*}} (x,t)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \]     

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.     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. I

This paper is concerned with the existence and uniqueness of variational solutions of the strongly nonlinear equation $$ - \sum\limits_1^m {_i \frac{\partial }{{\partial x_i }}\left( {\sum\limits_1^m {_j a_{i,j} (x, u(x))\frac{{\partial u(x)}}{{\partial x_j }}} } \right) + g(x, u(x)) = f(x)} $$ with the coefficients a _i,j (x, s) satisfying an eHipticity degenerate condition and hypotheses weaker than the continuity with respect to the variable s. Furthermore, we establish a condition on f under which the solution is bounded in a bounded open subset Ω of R^m.     

DIRICHLET FORMS AND SYMMETRIC DIFFUSIONS ON A BOUNDED DOMAIN IN R~d

Let D be a bounded C~3-domain in R~d and(a_(ij))be a bounded symmetric matrixdefined on D.Consider the symmetric form(u,v)=1/2∫_D a_(ij)(x)(u(x))/(x_i) (v(x))/(x_j)dx,u,v∈H~1(D).Under some assumptions it is shown that the diffusion process associated with the regularDirichlet space(,(H~1(D))on L~2(D)can be characterized as a unique solution of acertain stochastic differential equation.     

Behavior of Solutions of Quasilinear Elliptic Inequalities

The paper considers solutions of the coercive inequalities

Oscillation of solutions of nonlinear partial differential equations of neutral type

In this paper, we deal with the oscillatory behavior of solutions of the neutral partial differential equation of the form $$\begin{gathered} \frac{\partial }{{\partial t}}\left[ {p\left( t \right)\frac{\partial }{{\partial t}}(u\left( {x,t} \right) + \sum\limits_{i = 1}^t {p_i \left( t \right)u\left( {x,t - \tau _i } \right)} )} \right] + q\left( {x,t} \right)f_j (u(x,\sigma _j (t))) \hfill \\ = a\left( t \right)\Delta u\left( {x,t} \right) + \sum\limits_{k = 1}^n {a_k \left( t \right)} \Delta u\left( {x,\rho _k \left( t \right)} \right), \left( {x,t} \right) \in \Omega \times R_ + \equiv G \hfill \\ \end{gathered} $$ where Δ is the Laplacian in EuclideanN-spaceR ^N, R₊=(0, ∞) and Ω is a bounded domain inR ^N with a piecewise smooth boundary δΩ.     

Behavior at infinity of solutions of second-order nonlinear equations of a particular class

Let Ω be an arbitrary, possibly unbounded, open subset of ? ⁿ , and letL be an elliptic operator of the form $$L = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{\partial }{{\partial x_j }}} \right)} $$ . The behavior at infinity of the solutions of the equationLu=?(¦u¦)signu in Ω is studied, where? is a measurable function. In particular, given certain conditions at infinity, the uniqueness theorem for the solution of the first boundary value problem is proved.     

Some Coneral Results on the First Boundry Value Problem for Quasiliear Degenerate Parabolic Equation

In this paper, the authors investigate the first boundary value problem for equations of the form $\[Lu = \frac{{\partial u}}{{\partial t}} - \frac{\partial }{{\partial {x_i}}}({a^{ij}}(u,x,t)\frac{{\partial u}}{{\partial {x_j}}}) - \frac{{\partial {f^i}(u,x,t)}}{{\partial {x_i}}} = g(u,x,t)\]$ with $a^ij(u,x,t)\xi_i\xi_j\geq 0$ An existence theorem of solution in BV_1,1/2(Q_T) is proved. The principal condition is that there exists \delta>0 such that for any (x, t)\in Q_T,|u|\geq M $a^ij(u,x,t)\xi_i\xi_j-\delta\sum\limits_i,j=1^m(a_x^ij(u,x,t)\xi_i)^2\geq 0$     

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. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$     

An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity

It is proved that the operator $$P \equiv - \frac{{\partial ^2 }}{{\partial x_1^2 }} - \sum\nolimits_{k = 2}^n {\frac{\partial }{{\partial x_k }}\varphi ^2 (x)} \frac{\partial }{{\partial x_k }},$$ where ? ε C^∞(Ω) (Ω is a domain in Rⁿ), {x: ?(x) = 0} is a compactun in Ω which is the closure of its internal points, has the property of global hypoellipticity in Ω, i.e., $$\begin{array}{*{20}c} {v \in D'(\Omega ),} & {Pv \in C^\infty } & {(\Omega ) \Rightarrow \upsilon \in C^\infty (\Omega ).} \\ \end{array} $$ . This operator is not hypoelliptic.     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

On the role of arbitrary order Bessel functions in higher dimensional Dirac type equations

This paper exhibits an interesting relationship between arbitrary order Bessel functions and Dirac type equations. Let be the Euclidean Dirac operator in the n-dimensional flat space the radial symmetric Euler operator and α and λ be arbitrary non-zero complex parameters. The goal of this paper is to describe explicitly the structure of the solutions to the PDE system in terms of arbitrary complex order Bessel functions and homogeneous monogenic polynomials. Received: 27 October 2005     

SINGULAR PERTURBATION PROBLEMS FOR A CLASS OF ELLIPTIC EQUATIONS OF SECOND ORDER AS DEGENERATED OPERATOR HAS SINGULAR POINTS

In this paper we study the first and tiie third boundary value problems for the elliptic equation \[\begin{array}{l} \varepsilon \left( {\sum\limits_{i,j = 1}^m {{d_{i,j}}(x)\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum\limits_{i = 1}^m {{d_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + d(x)u} } } \right) + \sum\limits_{i = 1}^m {{a_i}(x)\frac{{\partial u}}{{\partial {x_i}}} + b(x) + c} \ = f(x),x \in G(0 < \varepsilon \le 1), \end{array}\] as the degenerated operator bas singular points, where \[\sum\limits_{i,j = 1}^m {{d_{i,j}}(x){\xi _i}{\xi _j}} \ge {\delta _0}\sum\limits_{i = 1}^m {\xi _i^2} ,({\delta _0} > 0,x \in G).\] The uniformly valid asymptotic solutions of boundary value problems have been obtained under the condition of \[\sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} > 0,or} \sum\limits_{i = 1}^m {{a_i}(x){n_i}(x){|_{\partial G}} < 0} ,\] where \(n = ({n_1}(x),{n_2}(x), \cdots ,{n_m}(x))\) is the interior normal to \({\partial G}\).     

Existence of Infinitely Many Solutions for Δ<Subscript>γ</Subscript>-Laplace Problems

In this article, we study the existence of infinitelymany solutions for the boundary–value problem

$$ - {\Delta _\gamma }u + a\left( x \right)u = f\left( {x,u} \right)in\Omega ,u = 0on\partial \Omega $$

, where Ω is a bounded domain with smooth boundary in ? ^N (N ≥ 2) and Δ_γ is a subelliptic operator of the form

$${\Delta _\gamma }: = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}} \right)} ,{\partial _{{x_j}}}: = \frac{\partial }{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \cdots ,\gamma N} \right)$$

. Our main tools are the local linking and the symmetric mountain pass theorem in critical point theory.

     

Boundary-value problem in an infinite layer

We establish necessary and sufficient conditions for a nonlocal two-point boundary-value problem in an infinite layer for the equation $$\frac{{\partial ^2 u(x,t)}}{{\partial t^2 }} + P\left( {\frac{\partial }{{\partial x}}} \right)\frac{{\partial u(x + h_1 ,t)}}{{\partial t}} + Q\left( {\frac{\partial }{{\partial x}}} \right)u(x + h_2 ,t) = 0,$$ whereP(s) andQ(s) are polynomials ins∈? ^m with constant coefficients, to have infinite type and be degenerate.     

On the rate of convergence of some difference schemes for second order elliptic equations

The purpose of this paper is to estimate the rate of convergence for some natural difference analogues of Dirichlet's problem for uniformly elliptic differential equations, $$\begin{gathered} \sum\limits_{j,k = 1}^N {\frac{\partial }{{\partial x_j }}} \left( {a_{jk} \frac{{\partial u}}{{\partial x_k }}} \right) = F in R, \hfill \\ u = f on B, \hfill \\ \end{gathered}$$ in aN-dimensional domainR with boundaryB. These schemes will in general not be of positive type, and the analysis will therefore be carried out in discreteL ²-norms rather than in the maximum norm. Since our approximation of the boundary condition is rather crude, we will only arrive at a rate of convergence of first order for smoothF andf. Special emphasis will be put on appraising the dependence of the rate of convergence on the regularity ofF andf.