Search results for "Elliptic partial differential equation"

Kleine periodische L�sungen bei nichtlinearen stark-elliptischen Systemen von partiellen Differentialgleichungen I

1971

Strongly elliptic systems of nonlinear partial differential equations are considered in the case when the derivatives of the solutions occuring in the nonlinear terms have the same order as those in the linear principal part. The existence of periodic solutions for such systems is investigated. It is shown that this problem can be reduced to the study of algebraic bifurcation equations, whose small solutions correspond to the classical solutions of the given problem. A discussion of the bifurcation equations will be given in a forthcoming paper.

Removability of a Level Set for Solutions of Quasilinear Equations

2005

In this paper, we study the removability of a level set for the solutions of quasilinear elliptic and parabolic equations of the second order. We show, under rather general assumptions on the coeff...

Regularity and polar sets for supersolutions of certain degenerate elliptic equations

1988

On considere l'equation ⊇•⊇ h F(x,⊇u(x))=0. Cette equation est non lineaire et degeneree avec des coefficients mesurables. On etudie la regularite des supersolutions

Scheduled Relaxation Jacobi method: improvements and applications

2016

Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficien…

Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions

2006

We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a(x, Du)+γ(u) ∋ φ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x, Du)·η+β(u) ∋ ψ. The nonlinear elliptic operator div a(x, Du) is modeled on the p-Laplacian operator Δp(u) = div (|Du|p−2Du), with p > 1, γ and β are maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) and 0 ∈ β(0), and the data φ ∈ L1 (Ω) and ψ ∈ L1 (∂Ω). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with c…

Nonlinear Eigenvalue Problems of Schrödinger Type Admitting Eigenfunctions with Given Spectral Characteristics

2002

The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form in a real Hilbert space ℋ with a semi-bounded self-adjoint operator A0, while for every y from a dense subspace X of ℋ, B(y ) is a symmetric operator. The left-hand side is assumed to be related to a certain auxiliary functional ψ, and the associated linear problems are supposed to have non-empty discrete spectrum (y ∈ X). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (∗) on a sphere SR ≔ {y ∈ X | ∥y∥ℋ = R} whose ψ-value is the n-th Ljusternik-Schnirelman level of ψ| and whose corresponding eigenvalue is t…

Removability theorems for solutions of degenerate elliptic partial differential equations

1993

Sur une classe d’equations du type parabolique lineaires

1996

The application of the variational method for the existence theorem, developped by J. L. Lions, for the evolution equations in Hilbert spaces to a considerably large class of systems of linear partial differential equations of parabolic type is studied by defining Hilbert spaces in relation to the elliptic operator of the system, and an example insired by the system of equations for a viscous gas is examined.

Global integrability of the gradients of solutions to partial differential equations

1994

Holder continuity of solutions for a class of nonlinear elliptic variational inequalities of high order

2001