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The non-uniqueness of the limit solutions of the scalar Chern-Simons equations with signed measures

Adilson Eduardo Presoto (2021)

Mathematica Bohemica

We investigate the effect of admitting signed measures as a datum at the scalar Chern-Simons equation - Δ u + e u ( e u - 1 ) = μ in Ω with the Dirichlet boundary condition. Approximating μ by a sequence ( μ n ) n ∈ ℕ of L 1 functions or finite signed measures such that this equation has a solution u n for each n ∈ ℕ , we are interested in establishing the convergence of the sequence ( u n ) n ∈ ℕ to a function u # and describing the form of the measure which appears on the right-hand side of the scalar Chern-Simons equation solved by u # .

The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes

Jonathan Luk (2013)

Journal of the European Mathematical Society

We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region { r ≤ t 4 } .

The output least squares identifiability of the diffusion coefficient from an H 1 –observation in a 2–D elliptic equation

Guy Chavent, Karl Kunisch (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.

The Output Least Squares Identifiability of the Diffusion Coefficient from an H1–Observation in a 2–D Elliptic Equation

Guy Chavent, Karl Kunisch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.

The Regularization of the Second Order Lagrangians in Example

Dana Smetanová (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper is devoted to geometric formulation of the regular (resp. strongly regular) Hamiltonian system. The notion of the regularization of the second order Lagrangians is presented. The regularization procedure is applied to concrete example.

The relation between the porous medium and the eikonal equations in several space dimensions.

Pierre-Louis Lions, Panagiotis E. Souganidis, Juan Luis Vázquez (1987)

Revista Matemática Iberoamericana

We study the relation between the porous medium equation ut = Δ(um), m > 1, and the eikonal equation vt = |Dv|2. Under quite general assumtions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as m↓1 to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.

The solution operator for a partial differential equation with delay

Gabriella Di Blasio, Karl Kunisch, Eugenio Sinestrari (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Viene dimostrata l’esistenza e l’unicità globale della soluzione di un’equazione funzionale in uno spazio di Hilbert e si caratterizza il generatore infinitesimale del semigruppo ad essa associato. Il risultato è applicato ad equazioni integrodifferenziali a derivate parziali di tipo parabolico in cui compaiono argomenti con ritardo (discreto e continuo) nelle derivate spaziali di ordine massimo.

The spectrum of Schrödinger operators with random δ magnetic fields

Takuya Mine, Yuji Nomura (2009)

Annales de l’institut Fourier

We shall consider the Schrödinger operators on ℝ 2 with the magnetic field given by a nonnegative constant field plus random δ magnetic fields of the Anderson type or of the Poisson-Anderson type. We shall investigate the spectrum of these operators by the method of the admissible potentials by Kirsch-Martinelli. Moreover, we shall prove the lower Landau levels are infinitely degenerated eigenvalues when the constant field is sufficiently large, by estimating the growth order of the eigenfunctions...

Currently displaying 61 – 80 of 110