The Dirichlet problem of higher order quasilinear elliptic equation

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

     

On solvability of the second boundary value problem in a half-space for one elliptic equation

We consider the second boundary value problem for one elliptic equation with a lower term in a half-space. We prove theorems of solvability of the problem in the Sobolev space.     

On solvability of the Dirichlet problem for the third-order hyperbolic equation

We consider a Dirichlet problem for the third-order hyperbolic equation and show the existence and uniqueness of its classical solution. For the proof of unique solvability, we use the methods of Riemann’s function and integral equations.     

On the solvability of a boundary-value problem for an elliptic equation

Consider a boundary-value problem for a second-order linear elliptic equation in a bounded domain. The coefficient of the required function is nonpositive everywhere in the domain, except for a small neighborhood of an interior point. The following question arises: Under what constraints on this coefficient in the given small domain do the statements on the existence and uniqueness of the solution of the first boundary-value problem remain valid?     

Probabilistic approach to the Dirichlet problem of second order elliptic PDE

In this paper we provide a probabilistic approach to the following Dirichlet Problem{(∑x~4(α~(ij) x~j) ∑b~ix~i ξ)u=0, iD u=g, on D,without assuming that the eigenvalues of the operator∑x~i(α~(ij)x~j) ∑b~ix~i ξwith Dirichlet boundary conditions are all strictly negative. The results of this paper generalizedthose of Ma.     

On boundary values of solutions of the dirichlet problem for second order elliptic equation

The paper suggests some conditions on the lower order terms, which provide that the solution of the Dirichlet problem for the general elliptic equation of the second order

General non local boundary value problem for second order elliptic equation

In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper 1 . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem.     

The mixed problem for a nonlinear degenerate elliptic equation of second order

The present paper deals with the mixed boundary value problem for a nonlinear elliptic equation with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the uniqueness and existence of solutions of the above problem for the nonlinear elliptic equation by the extremum principle and the method of parameter extension. The complex method is used to discuss the corresponding problem for degenerate elliptic complex equation of first order and then that of second order.     

On the solvability of the higher order elliptic boundary value problem

In this paper, by employing a non-variational version of a max–min principle, new results on existence and uniqueness of generalized solution to the boundary value problem of higher order elliptic partial differential equations at resonance are given.     

On the oblique derivative problem for second order elliptic equations

A calculus of polyhomogeneous paired Lagrangian distributions, associated to any two cleanly intersecting Lagrangain submanifolds, is constructed. The class is given an intrinsic characterisation using radial operators and a symbol calculus is developed. A class of pseudo—differential operators with singular symbols is developed within the calculus. This is used to give symbolic constructions of parametrices for operators of real principal type and paired Lagrangian distributions. The calculus is then applied to give a symbolic construction of the forward fundamental solution of the wave operator.     

A Riemann-Hilbert type problem for second order non-regular elliptic equation in weighted spaces

The paper investigates a Riemann-Hilbert type problem for second order nonregular elliptic equation in weighted spaces. It is established that the number of linearly independent solutions of the homogeneous problem and the number of conditions on the boundary functions depend not only on the order of singularity of the weight function and coefficient indices of the considered problem, but also on the behavior of these functions at singular points.     

On the solvability of the first nonlocal boundary value problem for an elliptic equation

We consider problems of the existence, uniqueness, and sign-definiteness of the classical solutions of the problem

$(Lu)(x) = f(x)(x \in D),u(x) - \beta (x)u(\sigma x) = \psi (x)(x \in S),$

where L is a linear second-order operator elliptic in the closure of a domain D ? R ⁿ and σ is a single-valued continuous mapping of S ≡ ?D into \(\bar D\).

We show that, under natural assumptions on the smoothness of β, σ, and the coefficients of L, this problem is Fredholm provided that either σ has no attractors on S or σ generates an attractor Θ on S and the spectral radius of the operator A defined on η(x) ∈ C(Θ) by the formula (Aη)(x) = |β(x)|η(σx) is less than unity.We obtain semieffective (in terms of a test function) conditions for the unique solvability of the problem.     

Spectral stability of Dirichlet second order uniformly elliptic operators

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation.     

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.