A bifurcation result for a Dirichlet problem with discontinuous nonlinearity

In questo lavoro si considera un problema di Direchlet con nonlinearità discontinua; risultati di esistenza e molteplicità di soluzione per problemi di questo tipo sono stati dati in [A-B], [A-T] utilizzando il principio di azione duale di Clarke; in questo lavoro, invece, si introduce un problema multivoco associato al problma in modo naturale e si dimostra, utilizzando una variante del teorema di biforcazione globale di Rabinowitz che poggia sul grado multivoco ([C-L]), che per esso esiste un ramo globale di biforcazione.     

Iterative reduction of the Dirichlet problem to the Neumann problem

An iteration procedure for reduction of the Dirichlet boundary-value problem to the Neumann problem is suggested. Estimates of the rate of convergence are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 362–369, March, 1995.This paper was supported by the Foundation for Fundamental Research of the Ukrainian State Committee on Science and Technology and by the American Mathematical Society.     

Remarks on multiplicity result for the Dirichlet problem for nonlinear elliptic equations

In this paper we prove the multiplicity result for the Dirichlet problems (A _s ) and (B _t ) with a boundary data inL ² (ϖQ) and with the nonlinearity interacting with the spectrum of the elliptic operatorL. The fact that the boundary data is inL ² leads in a natural way to the Dirichlet problem in a weighted Sobolev space. We follow methods and arguments from the recent papers of Walter and McKenna [11] and [12].     

A uniqueness result for a Neumann problem involving the critical Sobolev exponent

 In this paper we consider the problem

On Dirichlet and Neumann problems for harmonic functions

The aim of the paper is to examine some aspects of the boundary value problems for harmonic functions in half-spaces related to approximation theory. M. V. Keldyshmentioned curious fact on richness in some sense of the solutions of Dirichlet problem in upper half-plane for a fixed continuous boundary data on the real axis. This can be considered as a model version for the Dirichlet problem with continuous boundary data, defined except a single boundary point, with no restrictions imposed on solutions near that point.Some extensions and multi-dimensional versions of Keldysh’s richness are obtained and related questions on existence, representation and richness of solutions for the Dirichlet and Neumann problems discussed.     

Global stability result for the generalized quasivariational inequality problem

This paper studies some stability properties for the generalized quasivariational inequality problem. The study of this topic is motivated by the work of Harker and Pang (Ref. 1). A global stability result is obtained for problems satisfying certain conditions.The author would like to express his gratitude to the referees for helpful suggestions for the revision of the paper.     

Three solutions for some Dirichlet and Neumann nonlocal problems

This article is concerned with a class of nonlocal Dirichlet and Neumann boundary-value problems depending on two real parameters. Our approach relies on variational methods: we establish the existence of three weak solutions via a recent abstract multiplicity result by Ricceri about nonlocal problems.     

A complete classification of bifurcation diagrams of a p -Laplacian Dirichlet problem

We study the bifurcation diagrams of (classical) positive solutions u with |u |_∞ ∈ (0, ∞) of the p -Laplacian Dirichlet problem (φ_p (u ′(x)))′ + λf_q (u (x))) = 0, –1 ≤ x ≤ 1, u (–1) = 0 = u (1), where p > 1, φ_p (y) = |y |^{p –2} y, (φ_p (u ′))′ is the one-dimensional p -Laplacian, λ > 0 is a bifurcation parameter, and the nonlinearity f_q (u) = |1 – u |^q is defined on [0, ∞) with constant q > 0. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let Ω be an open connected subset of Rⁿ of finite measure for which the Poincaré-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rⁿ, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then, we show Lipschitz-type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then, we further assume that the imbedding of the Sobolev space W^1,2(Ω) into the space L²(Ω) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.     

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.     

Dirichlet and Neumann problems for Poisson equation in half lens

In this article, the Green and Neumann functions are given for a half lens and the Dirichlet and Neumann problems for Poisson equation are solved. All formulas are given in explicit form.     

Classification of bifurcation diagrams of a p-Laplacian Dirichlet problem with examples

We study bifurcation diagrams of positive solutions of the p-Laplacian Dirichlet problem

     

A complete classification of bifurcation diagrams of a p-Laplacian Dirichlet problem

We study the bifurcation diagrams of classical positive solutions u with ‖u_‖∞∈(0,∞) of the p-Laplacian Dirichlet problem

     

Dirichlet and Neumann eigenvalue problems on CR manifolds

We study the properties of Carnot–Carathéodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point \(x \in M\), versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form \(\theta \) on M. The weak Dirichlet problem for the sublaplacian \(\Delta _b\) on \((M, \theta )\) is solved on domains \(\Omega \subset M\) supporting the Poincaré inequality. The solution to Neumann problem for the sublaplacian \(\Delta _b\) on a \(C^{1,1}\) connected \((\epsilon , \delta )\)-domain \(\Omega \subset {{\mathbb {G}}}\) in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of \(\Delta _b\) on \(\Omega \subset M\) in a Carnot–Carthéodory complete pseudohermitian manifold \((M, \theta )\).     

Regularity of spectral fractional Dirichlet and Neumann problems

Consider the fractional powers and of the Dirichlet and Neumann realizations of a second‐order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of ‐Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on ‐calculus. We also include an overview of the various Dirichlet‐ and Neumann‐type boundary problems associated with the fractional Laplacian.