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Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Catherine Bandle, Joachim von Below, Wolfgang Reichel (2008)

Journal of the European Mathematical Society

We consider linear elliptic equations - Δ u + q ( x ) u = λ u + f in bounded Lipschitz domains D ⊂ ℝ N with mixed boundary conditions ∂ u / ∂ n = σ ( x ) λ u + g on ∂ D . The main feature of this boundary value problem is the appearance of λ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient σ ( x ) . We study positivity principles and anti-maximum principles. One of our main results states that if σ is somewhere negative, q ≥ 0 and ∫ D q ( x ) d x > 0 then there exist two eigenvalues λ - 1 , λ 1 such the positivity principle...

Problème de Dirichlet pour les fonctions α -harmoniques sur les domaines coniques

Krzysztof Bogdan, Tomasz Jakubowski (2005)

Annales mathématiques Blaise Pascal

On considère le noyau de Poisson du processus α -stable symétrique pour un domaine conique. Puis on considère le problème d’intégrabilité du noyau de Poisson à la puissance p . On donne des conditions sur q pour qu’il existe une solution au problème de Dirichlet pour les fonctions α -harmoniques sur les domaines coniques, avec une condition au bord donnée par une fonction de L q .

Pseudo-Differential Operators in a Wave Diffraction Problem with Impedance Conditions

Castro, L.P., Kapanadze, D. (2008)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30We consider an impedance boundary-value problem for the Helmholtz equation which models a wave diffraction problem with imperfect conductivity on a strip. Pseudo-differential operators are used to deal with this wave diffraction problem. Therefore, single and double layer potentials allow a reformulation of the problem into a system of integral equations. By using operator theoretical methods, the well-posedness of the problem...

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