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1.
In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem $$\left\{ \begin{gathered} - \Delta {\kern 1pt} {\kern 1pt} u = f\left( {x,u} \right){\text{in}}\Omega \hfill \\ u_{\left| {\wp \Omega } \right.} = 0 \hfill \\ \end{gathered} \right.$$ Positive answers to these problems would produce innovative multiplicity results on problem (Pf).  相似文献   

2.
Given aself similar fractal K ? ? n of Hausdorff dimension α>n?2, andc 1>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? h such that for every bounded open setΩ?? n and for everyf ∈ L2(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.  相似文献   

3.
LetΛ 1(Ω) be the first eigenvalue of the vector-valued problem $$\begin{gathered} \Delta u + \alpha grad div u + \Delta u = 0 in \Omega , \hfill \\ u = 0 in \partial \Omega , \hfill \\ \end{gathered} $$ , withα>0. Letλ 1(Ω) be the first eigenvalue of the scalar problem $$\begin{gathered} \Delta u + \lambda u = 0 in \Omega , \hfill \\ u = 0 on \partial \Omega . \hfill \\ \end{gathered} $$ . The paper contains a proof of the inequality $$\left( {1 + \frac{\alpha }{n}} \right)\lambda _1 \left( \Omega \right) > \Lambda _1 \left( \Omega \right) > \left( \Omega \right)$$ and improves recent estimates of Sprössig [15] and Levine and Protter [11]. Moreover we show, ifΩ is a ball, that an eigensolution u1, associated withΛ 1(Ω) is not unique and that the eigensolutions for this and higher eigenvalues are never rotationally invariant. Finally we calculate some eigensolutions explicitly.  相似文献   

4.
Rudykh  G. A.  Semenov  É. I. 《Mathematical Notes》2001,70(5-6):714-719
In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$\begin{gathered} {\text{ }}u_t = \Delta \ln u, \hfill \\ u \triangleq u\left( {x,t} \right):\Omega \times \mathbb{R}^ + \to \mathbb{R},{\text{ }} x \in \mathbb{R}^n , \hfill \\ \end{gathered} $$ where $\Omega \subset \mathbb{R}^n $ is the domain and $\mathbb{R}^ + = \left\{ {t:0 \leqslant t < + \infty } \right\},{\text{ }}u\left( {x,t} \right) \geqslant 0$ is the temperature of the medium.  相似文献   

5.
A difference scheme is constructed for the solution of the variational equation $$\begin{gathered} a\left( {u, v} \right)---u \geqslant \left( {f, v---u} \right)\forall v \varepsilon K,K \{ vv \varepsilon W_2^2 \left( \Omega \right) \cap \mathop {W_2^1 \left( \Omega \right)}\limits^0 ,\frac{{\partial v}}{{\partial u}} \geqslant 0 a.e. on \Gamma \} ; \hfill \\ \Omega = \{ x = (x_1 ,x_2 ):0 \leqslant x_\alpha< l_\alpha ,\alpha = 1, 2\} \Gamma = \bar \Omega - \Omega ,a(u, v) = \hfill \\ = \int\limits_\Omega {\Delta u\Delta } vdx \equiv (\Delta u,\Delta v, \hfill \\ \end{gathered} $$ The following bound is obtained for this scheme: $$\left\| {y - u} \right\|_{W_2 \left( \omega \right)}^2 = 0(h^{(2k - 5)/4} )u \in W_2^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0(h^{\min (k - 2;1,5)/2} ),u \in W_\infty ^k \left( \Omega \right) \cap W_2^3 \left( \Omega \right)$$ The following bounds are obtained for the mixed boundary-value problem: $$\begin{gathered} \left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{\min \left( {k - 2;1,5} \right)} } \right),u \in W_\infty ^k \left( \Omega \right),\left\| {y - u} \right\|_{W_2^2 \left( \omega \right)} = 0\left( {h^{k - 2,5} } \right), \hfill \\ u \in W_2^k \left( \Omega \right),k \in \left[ {3,4} \right] \hfill \\ \end{gathered} $$ .  相似文献   

6.
This paper is primarily concerned with the large time behaviour of solutions of the initial boundary value problem $$\begin{gathered} u_t = \Delta \phi (u) - \varphi (x,u)in\Omega \times (0,\infty ) \hfill \\ - \frac{{\partial \phi (u)}}{{\partial \eta }} \in \beta (u)on\partial \Omega \times (0,\infty ) \hfill \\ u(x,0) = u_0 (x)in\Omega . \hfill \\ \end{gathered} $$ Problems of this sort arise in a number of areas of science; for instance, in models for gas or fluid flows in porous media and for the spread of certain biological populations.  相似文献   

7.
8.
Consider the following Bolza problem: $$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$ We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.  相似文献   

9.
In this paper we extend the results of Brezis and Nirenberg in [4] to the problem $$\left\{ \begin{gathered} Lu = - D_i (a_{ij} (x)D_j u) = b(x)u^p + f(x,u) in\Omega , \hfill \\ p = (n + 2)/(n - 2) \hfill \\ u > 0 in\Omega , u = 0 \partial \Omega , \hfill \\ \end{gathered} \right.$$ whereL is a uniformly elliptic operator,b(x)≥0,f(x,u) is a lower order perturbation ofu p at infinity. The existence of solutions to (A) is strongly dependent on the behaviour ofa ij (x), b(x) andf(x, u). For example, for any bounded smooth domain Ω, we have \(a_{ij} \left( x \right) \in C\left( {\bar \Omega } \right)\) such thatLu=u p possesses a positive solution inH 0 1 (Ω). We also prove the existence of nonradial solutions to the problem ?Δu=f(|x|, u) in Ω,u>0 in Ωu=0 on ?Ω, Ω=B(0,1). for a class off(r, u).  相似文献   

10.
We consider the problems of dientifying the parametersa ij (x), b i (x), c(x) in a 2nd order, linear, uniformly elliptic equation, $$\begin{gathered} - \partial _i (a_{ij} (x)\partial _j u) + b_i (x)\partial _i u + c(x)u = f(x),in\Omega , \hfill \\ \partial _v u|_{\partial \Omega } = \phi (s),s \in \partial \Omega , \hfill \\ \end{gathered} $$ on the basis of measurement data $$u(s) = z(s),s \in B \subset \partial \Omega ,$$ with an equality constraint and inequality constraints on the parameters. The cost functionals are one-sided Gâteaux differentiable with respect to the state variables and the parameters. Using the Duboviskii-Milyutin lemma, we get maximum principles for the identification problems, which are necessary conditions for the existence of optimal parameters.  相似文献   

11.
In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for BVPs $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u(0) - B(u\prime (\eta )) = 0, u\prime (1) = 0 \hfill \\ \end{gathered} \right.$$ and $$\left\{ \begin{gathered} (\phi _p (u\prime ))\prime + q(t)f(t, u) = 0,0< t< 1, \hfill \\ u\prime (0) = 0, u(1) + B(u\prime (\eta )) = 0 \hfill \\ \end{gathered} \right.$$ The main tool is the monotone iterative technique. Here, the coefficientq(t) may be singular att = 0,1.  相似文献   

12.
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.  相似文献   

13.
We establish sufficient conditions for the solvability of boundary-value problems of the form $$\begin{gathered} u'' = f(t,u,u'); \hfill \\ \begin{array}{*{20}c} {(u(0),} & {u'(0)) \in S_0 ,} & {(u(1),} & {u'(1)) \in S_1 .} \\ \end{array} \hfill \\ \end{gathered} $$   相似文献   

14.
In this paper,we consider the following chemotaxis model with ratio-dependent logistic reaction term u/t=D▽(▽u-u▽ω/ω)+u(α-bu/ω),(x,t)∈QT,ω/t=βu-δω,(x,t)∈QT,u▽㏑(u/w)·=0,x ∈Ω,0tT,u(x,0)=u0(x)0,x ∈,w(x,0)=w0(x)0,x ∈,It is shown that the solution to the problem exists globally if b+β≥0 and will blow up or quench if b+β0 by means of function transformation and comparison method.Various asymptotic behavior related to different coefficients and initial data is also discussed.  相似文献   

15.
Let Ω be some open subset of ?N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L loc (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?+ and above on ??, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L loc (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .  相似文献   

16.
The main purpose of this paper is to use variational methods in the study of problems of following type: $$\left\{ {\begin{array}{*{20}c} {Lu = \lambda f(x,u) in \Omega } \\ {u = 0 on \partial \Omega } \\ \end{array} } \right.$$ Here Ω is supposed to be a bounded domain with smooth boundary ? Ω,L an elliptic operator, λ∈IR andf(x, t) a real function defined on \(\tilde \Omega \times IR\) having one or several simple discontinuities ont. Mainly we are interested in solutions which satisfy (.) a. e., which are most meaningful in physical problems, and we prove various existence theorems for several choices ofL, f and λ. The main difficulty consists in the fact that the functionals related to (.) are not Fréchet differentiable in every point, sincef is discontinuous.  相似文献   

17.
In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition $$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$ where Ω is a bounded smooth domain in ${\mathbb{R}^N}$ for ${N \ge 1, p > 1}$ , and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.  相似文献   

18.
The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞  相似文献   

19.
We study regularity results for solutions uHW 1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? n . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.  相似文献   

20.
In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu 0; finduK, whereK is a closed convex subset of the Sobolev spaceW 0 1,p (Ω), such that for anyvK $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.  相似文献   

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