DIRICHLET PROBLEMS FOR STATIONARY VON NEUMANN-LANDAU WAVE EQUATIONS

In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation：{（-△x＋△y）φ（x,y）=0,x,y∈Ωφ｜δΩxδΩ=fwhere Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods.

Transition-layer Solutions of Quasilinear Elliptic Boundary Blow-up Problems and Dirichlet Problems

We study profiles of positive solutions for quasilinear elliptic boundary blow-up problems and Dirichlet problems with the same equation：where ω 〉 0, a（x） is a continuous function satisfying 0 〈 a（x） 〈 1 for x ∈Ω, Ω is a bounded smooth domain in R^N. We will see that the profile of a minimal positive boundary blow-up solution of the equation shares some similarities to the profile of a positive minimizer solution of the equation with homogeneous Dirichlet boundary condition.

Dirichlet Problems for the Quasilinear Second Order Subelliptic Equations

In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.

EXISTENCE OF SOLUTIONS OF NONLOCAL PERTURBATIONS OF DIRICHLET DISCRETE NONLINEAR PROBLEMS

This paper is devoted to the study of second order nonlinear difference equations. A Nonlocal Perturbation of a Dirichlet Boundary Value Problem is considered. An exhaustive study of the related Green＇s function to the linear part is done. The exact expression of the function is given, moreover the range of parameter for which it has constant sign is obtained. Using this, some existence results for the nonlinear problem are deduced from monotone iterative techniques, the classical Krasnoselski fixed point theorem or by application of recent fixed point theorems that combine both theories.

SPLITTING EXTRAPOLATIONS FOR SOLVING BOUNDARY INTEGRAL EQUATIONS OF LINEAR ELASTICITY DIRICHLET PROBLEMS ON POLYGONS BY MECHANICAL QUADRATURE METHODS

Taking hm as the mesh width of a curved edge Гm （m = 1, ..., d ） of polygons and using quadrature rules for weakly singular integrals, this paper presents mechanical quadrature methods for solving BIES of the first kind of plane elasticity Dirichlet problems on curved polygons, which possess high accuracy O（h0^3） and low computing complexities. Since multivariate asymptotic expansions of approximate errors with power hi^3 （i = 1, 2, ..., d） are shown, by means of the splitting extrapolations high precision approximations and a posteriori estimate are obtained.

ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER

In the paper we study questions about solvability of some boundary value prob- lems for a non-homogenous poly-harmonic equation. As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense. The considered problem is a generalization of well-known Dirichlet and Neumann problems.

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION

In this paper,for a bounded C2 domain,we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive C2 boundary data.We have a non-existence result,which is the justification for taking into account the restricted boundary data.There is a smooth positive boundary datum that precludes the existence of the positive classical solution.

WEIGHTED A PRIORI ESTIMATES FOR SOLUTION OF （-△）^mu = f WITH HOMOGENEOUS DIRICHLET CONDITIONS

Let u be a weak solution of （-△）mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω C Rn. Then, the main goal of this paper is to prove the following a priori estimate：｜｜u｜｜w2m/ω·p（Ω）≤C｜｜f｜｜L^pω（Ω）,where ω is a weight in the Muckenhoupt class Ap.

AN IMPROVEMENT FOR LOWER－ORDER PANEL METHOD BASED ON THE DIRICHLET BOUNDARY CONDITION

An improved algorithm for velocity field of general configurations ispresentd for low-order panel method based on the internal Dirichlet boundary condi-tion. A direct calculating method for the velocity distribution by means of a limit pro-cess combining with analytic evaluation of higher-order singular integrals instead of theconventional method of doublet strength gradient is devised in order to avoid the diffi-culty of edge extrapolation of doublet strength. The problem of substantialunderpredictions of the induced drag coefficient obtained from the VSAERO analysisdisappears for the present improved algorithm. Illustrative calculations for several testcases such as swept back wing, swept forward wing and wing-body combination showthat the accuracy of results may be improved and is competitive with high-order panelmethod. In addition, the present direct integral method can be used to evaluate the ve-locity distribution for external flow field correctly, where the method of gradient cannot be used at all.

Existence of Multiple solutions for semilinear elliptic equations in the annulus

The existence of radial solutions of Δu ＋ λg（｜x｜）f（u） = 0 in annuli with Dirichlet（Dirichlet/Neumann） boundary conditions is investigated.It is proved that the problems have at least two positive radial solutions on any annulus if f is superlinear at 0 and sublinear at ∞.

Existence of <i>T</i>-<i>ν</i>-<i>p</i>(<i>x</i>)-Solution of a Nonhomogeneous Elliptic Problem with Right Hand Side Measure

Using the theory of weighted Sobolev spaces with variable exponent and the L1-version on Minty’s lemma, we investigate the existence of solutions for some nonhomogeneous Dirichlet problems generated by the Leray-Lions operator of divergence form, with right-hand side measure. Among the interest of this article is the given of a very important approach to ensure the existence of a weak solution of this type of problem and of generalization to a system with the minimum of conditions.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF NONNEGATIVE SOLUTIONS OF INHOMOGENEOUS p-LAPLACE EQUATION

Let Ω be a smooth bounded domain in R^n. In this article, we consider the homogeneous boundary Dirichlet problem of inhomogeneous p-Laplace equation --△pu = ｜u｜^q-1 u ＋ λf（x） on Ω, and identify necessary and sufficient conditions on Ω and f（x） which ensure the existence, or multiplicities of nonnegative solutions for the problem under consideration.

THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS

This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω , u = 0, x ∈δΩ, where Ω belong to R^n is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T. Yau et al.

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.

NONDEGENERACY OF POSITIVE SOLUTIONS TO HOMOGENEOUS SECOND-ORDER DIFFERENTIAL SYSTEMS AND ITS APPLICATIONS

This article considers the Dirichlet problem of homogeneous and inhomogeneous second-order ordinary differential systems. A nondegeneracy result is proven for positive solutions of homogeneous systems. Sufficient and necessary conditions for the existence of multiple positive solutions for inhomogeneous systems are obtained by making use of the nondegeneracy and uniqueness results of homogeneous systems.

Energy and Laplacian of fractal interpolation functions

Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions （FIFs） on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket （SG）. As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5 ：△=0 on SG ／ {q1, q2, q3}, and u（qi）=ai, i = 1, 2, 3, where qi, i=1, 2, 3, are boundary points of SG.

COMPLEX MONGE-AMPěRE EQUATIONS ON GENERAL DOMAINS

The smooth solutions of the Dirichlet problems for the complex Monge-Ampère equations on general smooth domains are found,provided that there exists a C 3 strictly plurisub harmonic subsolution with prescribed boundary value.It is the smooth version of an existence theorem given by Bedford and Taylor.

NONTRIVIAL SOLUTIONS FOR A CLASS OF NON-DIVERGENCE EQUATIONS ON POLARIZABLE CARNOT GROUP

Some new properties of polarizable Carnot group are given.By choosing a proper constant a nontrivial solution of a class of non-divergence Dirichlet problem on the polarizable Carnot group is constructed.Thus the multi-solution property of corresponding non-homogeneous Dirichlet problem is proved and the best possible of LQ norm in the famous Alexandrov-Bakelman-Pucci type estimate is discussed.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann oroblems with operators of a fractional order.

REGULAR SOLUTIONS FOR SCHRODINGER EQUATION ON UNBOUNDED DOMAINS

The authors study a class of solutions,namely, regular solutions of the Schrodinger equation (1/2 Delta + q)u = 0 on unbounded domains. They definite the regular solutions in terms of sample path properties of Brownian motion and then characterize them by analytic method. In Section 4, they discuss the regular solution to the stochastic Dirichlet problem for the equation (1/2 Delta + q)u = 0 having limit alpha at infinity.