A Priori Estimates and Existence of Positive Solutions for a Quasilinear Elliptic Equation

On the basis of some new Liouville theorems, under suitableconditions, a priori estimates are obtained of positive solutionsof the problem where R^N (N 2) is a bounded smooth domain, p>1 and isa parameter, , q are given constants such that p–1<<p*–1, <q, p*=Np/(N–p) if N > p and p*= when N p, and a(x) is a continuous nonnegative function. Makinguse of the Leray–Schauder degree of a compact mappingand a priori estimates, the paper finds that the problem abovepossesses at least one positive solution. It also discussesthe corresponding perturbed problem, where a(x) is replacedby a(x)+, >0. The results are strikingly different from thoseobtained for the case =p–1.     

A Priori Estimates for Solutions of Ordinary Differential Equations of Emden--Fowler Type

Mathematical Notes -     

一类具有高阶转向点的二次问题的奇摄动

研究带有高阶转向点的二阶非线性微分方程的边值问题εy″=f(t)y′2+g(t,y)y(a,ε)=A,y(b,ε)=B的奇异摄动现象.在一定的条件下,得到了摄动解关于退化解的渐近性质及误差估计.     

A Priori Estimates for Quantum Dynamical Semigroups

We obtain a priori bounds for the solutions of master Markov quantum equations in the Heisenberg and Schrödinger pictures.     

Expansions for Singular Perturbations

The author applies the two-variable expansion procedure to anon-linear problem in singular perturbations, and compares theresults with those obtained using the composite expansion method.     

Two-Variable Expansions for Singular Perturbations

The author applies the two-variable expansion procedure to anexample in singular perturbations and shows how the validityof this procedure can be established in many cases (includingsome non-linear problems).     

A Priori Estimates and Existence of Positive Solutions to Quasilinear Elliptic Equations in General Form

In this paper we prove the existence of a positive solution to the following superlinear elliptic Dirichlet problem, - Σ^n_{i,j=1}aij(x, u, Du)D_{ij}u = f(x, u, Du) in Ω, \quad u = 0 on ∂Ω where f satisfies certain growth conditions.     

Refining Eigenvalue Estimates for a String with a Singular Weight

Differential Equations - We consider the linear differential pencil $$A(\lambda )\equiv L-\lambda V $$ , where $$L\equiv -y^{\prime {}\prime }+q(x)y$$ is the Sturm–Liouville operator with...     

Nonmonotone Interior Layer Solutions for Singularly Perturbed Semilinear Boundary Value Problems with a Turning Point

ＮｏｎｍｏｎｏｔｏｎｅＩｎｔｅｒｉｏｒＬａｙｅｒＳｏｌｕｔｉｏｎｓｆｏｒＳｉｎｇｕｌａｒｌｙＰｅｒｔｕｒｂｅｄＳｅｍｉｌｉｎｅａｒＢｏｕｎｄａｒｙＶａｌｕｅＰｒｏｂｌｅｍｓｗｉｔｈａＴｕｒｎｉｎｇＰｏｉｎｔＷｅｉＢａｏｓｈｅ（魏宝社）（Ｄｅｐａｒｔｍ...     

一类新的弱奇性Wendroff型积分不等式解的估计

在参数更广的分布下,讨论了一类新的弱奇性Wendroff型积分不等式解的估计,所得结果推广了已有的相关结果,并将结果应用到研究微分方程解的有界性中.     

On a Priori Energy Estimates for Characteristic Boundary Value Problems

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space \(H^1_{ tan}\) , can be transformed into an \(L^2\) a priori estimate of the same problem.     

A Priori Error Estimates for State-Constrained Semilinear Parabolic Optimal Control Problems

We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean values of the state variable. In order to control the feasibility violation induced by the discretization, error estimates for the semilinear partial differential equation are derived. Based upon these estimates, it can be shown that any local minimizer of the semilinear parabolic optimization problems satisfying a weak second-order sufficient condition can be approximated by the discretized problem. Rates for this convergence in terms of temporal and spatial discretization mesh sizes are provided. In contrast to other results in numerical analysis of optimization problems subject to semilinear parabolic equations, the analysis can work with a weak second-order condition, requiring growth of the Lagrangian in critical directions only. The analysis can then be conducted relying solely on the resulting quadratic growth condition of the continuous problem, without the need for similar assumptions on the discrete or time semidiscrete setting.