Dirichlet Problem for a Class of Quasilinear Elliptic Equations

In this paper, the Dirichlet problem for quasilinear elliptic equations is studied. New a priori estimates of the solution and its gradient are obtained. These estimates are derived without any assumptions on the smoothness of the coefficients and the right-hand side of the equation. Moreover, an arbitrary growth of the right-hand side with respect to the gradient of the solution is assumed. On the basis of the resulting estimates, existence theorems are proved.     

在无界区域上拟线性散度型椭圆型方程的Dirichlet问题

研究在无界区域上的二阶拟线性散度型椭圆型方程Dirichlet问题在无穷远处径向收敛的古典解存在性和唯一性。     

共振的半线性椭圆方程Dirichlet问题解的多重性

使用极小极大方法获得非线性非自治无界共振的半线性椭圆方程Dirichlet问题解的多重性结果。     

Dirichlet Problem for Degenerate Quasilinear Elliptic Equations Suggested by the Anisotropic Permeation

By Browder's pseudo-monotone operator theory and the techniques belonging to J. Leray and J. Lions, the existence theorem of the generalized solution of the Dirichlet problem for a strongly degenerate quasilincar elliptic equation has been proved in the anisotropic Sobolev space.     

The Dirichlet Problem for Quasilinear Elliptic Equations in Domains with Smooth Closed Edges

The solvability of the Dirichlet problem for quasilinear elliptic second-order equations of nondivergence form are studied in a domain whose boundary contains a conical point or an edge of an arbitrary codimension. Bibliography: 12 titles.     

多维区域上椭圆型方程Dirichlet问题的勒让德多小波多层扩充算法

用小波伽辽金方法求解多维区域上椭圆型方程齐次Dirichlet问题,构造了近似解空间的两个等价的勒让德多小波基,使得快速求解离散后的线性方程组的多层扩充算法得以实现.数值算例表明该算法是有效的.     

The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane

A solution of the Dirichlet problem for an elliptic systemof equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Hölder boundary, the differential properties of the solution are studied.     

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the L p Dirichlet Problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L_0 = div A~0(x)? + B~0(x) · ? is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the LpDirichlet problem for the operator L_0 is solvable in the upper half-space Rn+. In this paper we prove that the Lpsolvability is stable under small perturbations of L_0. That is if L_1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L_0 and L_1 are sufficiently close in the sense of Carleson measures, then the LpDirichlet problem for the operator L_1 is solvable for the same value of p. As a corollary we obtain a new result on Lpsolvability of the Dirichlet problem for operators of the form L = div A(x)? + B(x) · ? where the matrix A satisfies weaker Carleson condition(expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoˇs,Petermichl and Pipher.     

Summability of Solutions of the Dirichlet Problem for Nonlinear Elliptic Equations with Right-Hand Side in Classes Close to L1

Mathematical Notes -     

Extrapolation for the L p Dirichlet Problem in Lipschitz Domains

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the Lp Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in...     

Capacitary Estimates for Solutions of the Dirichlet Problem for Second Order Elliptic Equations in Divergence Form

We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equationdiv( in an n-dimensional domain , n 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain , which in turn lead to a sufficient condition for the Hölder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmén–Lindelöf type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div( in a domain and for the solutions of the nonhomogeneous equation -div with measure on the right-hand side.     

一类拟线性椭圆方程组的Dirichlet问题

The homogeneous Dirichlet problem(1) for quasilinear elliptic system in a bounded domain Ω is investigated in this paper. The existence of generalized solutions in [H₀¹(Ω)]^N is obtained by using the contructive Galerkin method. For the case of a_ij^lm=0 when i≠j, it is estatablished that such generalized solutions have bounded [L_∞(Ω)]N norm and possess Holeler continuity. Even in the particular case that f_i are independent of Du, our results have improved those of A. V. Lair [Ann. Mat. Pura Appl., 116(1978)], allowing b_i¹(x,u) and f_i(x,u) to have a growth in u arbitrarily close to 1.     

Boundedness of Solutions of Dirichlet Problem for a Class of Nonlinear Elliptic Equations with Weights

In this article, under some Hypotheses on weights and on coefficients, using a modification of Moser's method, we establish the boundedness of solutions of Dirichlet Problem for a class of nonlinear elliptic equations.