Search results for "Elliptic operator"

showing 10 items of 40 documents

Explicit solutions for second-order operator differential equations with two boundary-value conditions. II

1992

AbstractBoundary-value problems for second-order operator differential equations with two boundary-value conditions are studied for the case where the companion operator is similar to a block-diagonal operator. This case is strictly more general than the one treated in an earlier paper, and it provides explicit closed-form solutions of boundary-value problem in terms of data without increasing the dimension of the problem.

Numerical AnalysisAlgebra and Number TheoryMathematical analysisSemi-elliptic operatorp-LaplacianOrder operatorDiscrete Mathematics and CombinatoricsBoundary value problemGeometry and TopologyC0-semigroupDifferential algebraic geometryTrace operatorNumerical partial differential equationsMathematicsLinear Algebra and its Applications
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On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method

2017

The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method. …

Physics and Astronomy (miscellaneous)DiscretizationFOS: Physical sciencesJacobi method010103 numerical & computational mathematics01 natural sciencesMatemàtica aplicadasymbols.namesakeMatrix (mathematics)FOS: MathematicsMathematics - Numerical Analysis0101 mathematicsEigenvalues and eigenvectorsMathematicsHigh Energy Astrophysical Phenomena (astro-ph.HE)Numerical AnalysisApplied MathematicsLinear systemMathematical analysisNumerical Analysis (math.NA)Computational Physics (physics.comp-ph)Computer Science Applications010101 applied mathematicsComputational MathematicsElliptic operatorRate of convergenceModeling and SimulationsymbolsÀlgebra linealAstrophysics - High Energy Astrophysical PhenomenaPhysics - Computational PhysicsLaplace operatorJournal of Computational Physics
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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…

PhysicsElliptic operatorNonlinear systemPure mathematicsElliptic partial differential equationBounded functionStefan problemBoundary value problemUniquenessWeak formulation
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Principal eigenvalue of a very badly degenerate operator and applications

2007

Abstract In this paper, we define and investigate the properties of the principal eigenvalue of the singular infinity Laplace operator Δ ∞ u = ( D 2 u D u | D u | ) ⋅ D u | D u | . This operator arises from the optimal Lipschitz extension problem and it plays the same fundamental role in the calculus of variations of L ∞ functionals as the usual Laplacian does in the calculus of variations of L 2 functionals. Our approach to the eigenvalue problem is based on the maximum principle and follows the outline of the celebrated work of Berestycki, Nirenberg and Varadhan [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operator…

Pure mathematicsApplied MathematicsMathematical analysisMathematics::Analysis of PDEsLipschitz continuityElliptic operatorOperator (computer programming)Maximum principleInfinity LaplacianMaximum principleInfinity LaplacianPrincipal eigenvalueUniquenessLaplace operatorEigenvalues and eigenvectorsAnalysisMathematicsJournal of Differential Equations
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\( L^{1} \) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions

2007

Abstract In this paper we study the questions of existence and uniqueness of weak and entropy solutions for equations of type − div a ( x , D u ) + γ ( u ) ∋ ϕ , posed in an open bounded subset Ω of R N , with nonlinear boundary conditions of the form a ( x , D u ) ⋅ η + β ( u ) ∋ ψ . The nonlinear elliptic operator div a ( x , D u ) is modeled on the p-Laplacian operator Δ p ( u ) = div ( | D u | p − 2 D u ) , with p > 1 , γ and β are maximal monotone graphs in R 2 such that 0 ∈ γ ( 0 ) and 0 ∈ β ( 0 ) , and the data ϕ ∈ L 1 ( Ω ) and ψ ∈ L 1 ( ∂ Ω ) .

Pure mathematicsApplied MathematicsMathematical analysisSemi-elliptic operatorElliptic operatorHalf-period ratiop-LaplacianFree boundary problemBoundary value problemUniquenessLaplace operatorMathematical PhysicsAnalysisMathematicsAnnales de l'Institut Henri Poincaré C, Analyse non linéaire
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OBSTACLE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS WITH NONHOMOGENEOUS NONLINEAR BOUNDARY CONDITIONS

2008

In this paper we study the questions of existence and uniqueness of 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) modeled on the p-Laplacian operator Δp(u) = div (|Du|p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) ∩ β(0), [Formula: see text] and the data ϕ ∈ L1(Ω) and ψ ∈ L1(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solut…

Pure mathematicsElliptic operatorMonotone polygonApplied MathematicsModeling and SimulationWeak solutionBounded functionObstacle problemMathematical analysisBoundary value problemUniquenessType (model theory)MathematicsMathematical Models and Methods in Applied Sciences
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Removability theorems for solutions of degenerate elliptic partial differential equations

1993

Pure mathematicsParametrixGeneral Mathematics010102 general mathematicsFirst-order partial differential equation01 natural sciencesParabolic partial differential equation010101 applied mathematicsStochastic partial differential equationSemi-elliptic operatorElliptic partial differential equation0101 mathematicsSymbol of a differential operatorNumerical partial differential equationsMathematicsArkiv för Matematik
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Analysis of geometric operators on open manifolds: A groupoid approach

2001

The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras, results that are useful for the analysis of geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators on groupoids are in our algebras. This then leads to criteria for the Fredholmness of geometric operators on suitable non-compact manifolds, as well as to an inductive procedure to study their essentia…

Pure mathematicsSpectral theoryMathematics::Operator Algebras010102 general mathematicsMathematical analysisSpectral geometryFinite-rank operatorOperator theoryCompact operator01 natural sciencesQuasinormal operatorSemi-elliptic operatorElliptic operatorMathematics::K-Theory and Homology0103 physical sciences010307 mathematical physics0101 mathematicsMathematics::Symplectic GeometryMathematics
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SOME RELATIONS BETWEEN BOUNDED BELOW ELLIPTIC OPERATORS AND STOCHASTIC ANALYSIS

2019

International audience; We apply Malliavin Calculus tools to the case of a bounded below elliptic rightinvariant Pseudodifferential operators on a Lie group. We give examples of bounded below pseudodifferential elliptic operators on R d by using the theory of Poisson process and the Garding inequality. In the two cases, there is no stochastic processes besides because the considered semi-groups do not preserve positivity.

Pure mathematicsStochastic process010102 general mathematicsLie groupPoisson processMalliavin calculus01 natural sciences[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]010104 statistics & probabilityElliptic operatorsymbols.namesakeBounded functionsymbols0101 mathematics[MATH]Mathematics [math]Mathematics
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Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline

2019

In this chapter we will state a result asserting that for elliptic semi-classical (pseudo-)differential operators the eigenvalues are distributed according to Weyl’s law “most of the time” in a probabilistic sense. The first three sections are devoted to the formulation of the results and in the last section we give an outline of the proof that will be carried out in Chaps. 16 and 17.

Section (fiber bundle)Pure mathematicsElliptic operatorDistribution (number theory)Probabilistic logicState (functional analysis)Differential operatorEigenvalues and eigenvectorsMathematics
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