On the exterior Dirichlet problem for Hessian quotient equations

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge–Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.     

The exterior Dirichlet problem for a class of fourth order elliptic equations

In some exterior domain G of the Euclidian p-space $\text{R}$^p the Dirichlet boundary value problem is considered for the equation (L + κ²)²u = f, where L is a uniformly elliptic operator and κ is a real number different from 0. It can be shown that each solution u of this equation splits into u = x_l?_lu₁ + u₂, where u₁ and u₂ satisfy Heimholte equations. Asymptotic conditions for u are formulated by imposing Sommerfeld radiation conditions on u₁ and u₂. If u₁ and u₂ are assumed to satisfy the same radiation condition, we prove a “Fredholm alternative theorem.” If u₁ and u₂ satisfy different radiation conditions, existence and uniqueness of the solution can be shown, provided the space dimension p is greater than 2.     

The exterior Dirichlet problem for quasi-linear elliptic equations with small boundary data

It is shown that if the order of non-uniformity of a quasi-linear elliptic equation is h,10,2(h–1)/h norm. For 0h1,existence of a bounded solution is guaranteed without any smallness assumption on the given boundary data.More precise information is given for the special case of the minimal surface equation.     

Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse

The present paper considers the Dirichlet problem for properly elliptic equations of fourth order in the exterior of an ellipse. No restrictions on the multiplicities of the roots of the characteristic polynomial are assumed.     

Radiation conditions and the exterior Dirichlet problem for a class of higher order elliptic equations

Denote by L a second order strongly elliptic operator in the Euclidian p-space $\text{R}$^p, and by P some real polynomial in one variable. First the wholespace-problem for the equation P(L)u = f is considered and asymptotic conditions are derived which yield an existence and uniqueness theorem. Then for the Dirichlet problem in some exterior domain G ? $\text{R}$^p a “Fredholm alternative theorem” is proved.     

The exterior Dirichlet problem for fully nonlinear elliptic equations related to the eigenvalues of the Hessian

In this paper, we establish the existence theorem for the exterior Dirichlet problems for a class of fully nonlinear elliptic equations, which are related to the eigenvalues of the Hessian matrix, with prescribed asymptotic behavior at infinity. This extends the previous results on Monge–Ampère equation and k-Hessian equation to more general cases, in particular, including the special Lagrangian equation.     

On the exterior dirichlet problem for linear elliptic equations in the plane

Summary In this paper the exterior Dirichlet problem for linear elliptic equations in two independent variables with bounded measurable coefficients is investigated. An existence-uniqueness theorem is established in a suitably weighted Sobolev class. Some a-priori estimates are derived. Entrata in Redazione il 6 agosto 1977. This research was supported by GNAFA-CNR.     

Solvability of the Dirichlet problem for second-order elliptic equations

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n?1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ? ? _n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W ₂ ¹ (Q) under the additional requirement that the boundary function belongs to the space W ₂ ^1/2 (?Q).     

On the existence of solutions of the Dirichlet problem for nonlinear elliptic equations

In this paper we investigate the Dirichlet problem (1), (2) with the boundary data ?∈L ^∞(Q) and the nonlinearityb(x, u, p) having a quadratic growh inp. Using the method of sub- and supersolutions we prove the existence of a solution in a weighted Sobolev spaceW ^1,2(Q).     

一类奇异半线性椭圆型方程的Dirichlet问题

得到了一类奇异半线性椭圆型方程 Ｄｉｒｉｃｈｌｅｔ问题解的存在性．     

The Dirichlet problem for Hessian quotient equations in exterior domains

In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian quotient equations in exterior domains.     

A spectral method for elliptic equations: the Dirichlet problem

Let Ω be an open, simply connected, and bounded region in ? ^d , d?≥?2, and assume its boundary \(\partial\Omega\) is smooth. Consider solving an elliptic partial differential equation Lu?=?f over Ω with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials u _n of degree ≤?n that is convergent to u. The transformation from Ω to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For \(u\in C^{\infty}( \overline{\Omega})\) and assuming \(\partial\Omega\) is a C ^?∞? boundary, the convergence of \(\left\Vert u-u_{n}\right\Vert _{H^{1}}\) to zero is faster than any power of 1/n. Numerical examples in ?² and ?³ show experimentally an exponential rate of convergence.     

Solvability of the Dirichlet problem for degenerate quasilinear elliptic equations

In a bounded domain of the n -dimensional (n?2) space one considers a class of degenerate quasilinear elliptic equations, whose model is the equation $$\sum\limits_{i = 1}^n {\frac{{\partial F}}{{\partial x_i }}} (a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i - 2} u_{x_i } ) = f(x),$$ where x =(x₁,..., x_r), l_i?0, m_i>1, the function f is summable with some power, the nonnegative continuous function a(u) vanishes at a finite number of points and satisfies \(\frac{{lim}}{{\left| u \right| \to \infty }}a(u) > 0\) . One proves the existence of bounded generalized solutions with a finite integral $$\int\limits_\Omega {\sum\limits_{i = 1}^n {a^{\ell _i } (u)\left| {u_{x_i } } \right|^{m_i } dx} }$$ of the Dirichlet problem with zero boundary conditions.     

Some existence theorems for the Dirichlet problem for quasilinear elliptic equations

Summary The purpose of this paper is to establish the existence of a solution of the Dirichlet problem for a quasilinear elliptic equation. We prove that the problem (1), (2) is solvable for all _j (j=1,...), where _j are eigenvalues of a linear elliptic operator that is associated with a given quasilinear elliptic operator by a limiting process. We also discuss this problem with a boundary data in L². This leads in a natural way to the Dirichlet problem in a weighted Sobolev space.