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.     

On the exterior two-dimensional Dirichlet problem for elliptic equations

A necessary and sufficient condition is given on the boundary datum in order to the Dirichlet problem for an elliptic equation in a two-dimensional exterior Lipschitz domain has a unique solution with a finite Dirichlet integral which converges uniformly at infinity to an assigned constant value.     

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.     

Dirichlet problem for a class of linear second order elliptic partial differential equations with discontinuous coefficients

Summary I give a sufficient condition in order that a Dirichlet problem is solvable in H ² (Ω) for a class of linear second order elliptic partial differential equations. Such a class includes some particular cases for which the result is known.

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Sunto Si prova una condizione sufficiente affinchè un problema di Dirichlet sia risolubile in H ² (Ω) per una classe di equazioni differenziali alle derivate parziali lineari ellittiche del secondo ordine. Tale classe comprende alcuni casi particolari per i quali il risultato è noto.

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The present work was written while the author was a member of the ? Centro di Matematica e Fisica Teorica del C.N.R. ? at the University of Genova, directed by professorJ. Cecconi.

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Entrata in Redazione il 25 febbraio 1971.     

The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation:

     

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 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.     

The Dirichlet problem for second order elliptic equations with coefficients in the Stummel class in unbounded domains

In this note we prowe existence and unicity of solution of a Dirichlet problem for second order elliptic operator in the divergence form, with the coefficients of the lower order terms belonging to a variant of the Stummel-Kato class, in an unbounded domain, extending the works [6] and [2].

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Sunto In questa nota proviamo un Teorema di esistenza e unicità per la soluzione di un problema di Dirichlet relativo ad un operatore ellittico del secondo ordine in forma di divergenza, con i coefficienti dei termini di ordine inferiore appartenenti ad una variante dello spazio di Stummel-Kato, in un dominio non limitato, estendendo i lavori [6] e [2].

     

Dirichlet problem for elliptic equations with nonlinear first order terms: a comparison result

Summary We deal with the Dirichlet problem for elliptic equations with a nonlinearity involving the gradient of the solution. By symmetrization techniques, we reduce the problem of finding sharp estimates of solutions to an analogous problem for ordinary differential equations.Lavoro svolto nell'ambito del G.N.A.F.A. con parziale contributo del M.P.I.     

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 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.     

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).     

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.