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On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes

Mathias Rousset (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the...

On a quasi-variational inequality arising in semiconductor theory.

José-Francisco Rodrigues (1992)

Revista Matemática de la Universidad Complutense de Madrid

Some new mathematical results of existence and uniqueness of solutions are obtained for a class of quasi-variational inequalities modeling the free boundary problem for the determination of the depletion zone in reverse biased semiconductor diodes. The corresponding one (or two) obstacle implicit problems are solved by direct methods with weak regularity estimates for mixed boundary value elliptic problems of second order.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, u | ∂ Ω = 0 , where f(x) and h(u) satisfy minimal regularity assumptions.

On a semilinear elliptic equation in ℍ n

Gianni Mancini, Kunnath Sandeep (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

On a semilinear variational problem

Bernd Schmidt (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We provide a detailed analysis of the minimizers of the functional u ↦ ∫ ℝ n | ∇ u | 2 + D ∫ ℝ n | u | γ , γ ∈ ( 0 , 2 ) , subject to the constraint ∥ u ∥ L 2 = 1 . This problem,e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties...

On a semilinear variational problem

Bernd Schmidt (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We provide a detailed analysis of the minimizers of the functional u ↦ ∫ ℝ n | ∇ u | 2 + D ∫ ℝ n | u | γ , γ ∈ ( 0 , 2 ) , subject to the constraint ∥ u ∥ L 2 = 1 . This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties...

Currently displaying 221 – 240 of 2166