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Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Oana Ivanovici (2012)

Journal of the European Mathematical Society

In this paper we consider a smooth and bounded domain Ω ⊂ ℝ d of dimension d ≥ 2 with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...

Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

For a principal type pseudodifferential operator, we prove that condition  ( ψ ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ + 3 / 2 for any ϵ > 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesnotimply local solvability with a loss of 1 derivative,...

Decay estimates of solutions of a nonlinearly damped semilinear wave equation

Aissa Guesmia, Salim A. Messaoudi (2005)

Annales Polonici Mathematici

We consider an initial boundary value problem for the equation u t t - Δ u - ∇ ϕ · ∇ u + f ( u ) + g ( u t ) = 0 . We first prove local and global existence results under suitable conditions on f and g. Then we show that weak solutions decay either algebraically or exponentially depending on the rate of growth of g. This result improves and includes earlier decay results established by the authors.

Dispersive and Strichartz estimates for the wave equation in domains with boundary

Oana Ivanovici (2010)

Journées Équations aux dérivées partielles

In this note we consider a strictly convex domain Ω ⊂ ℝ d of dimension d ≥ 2 with smooth boundary ∂ Ω ≠ ∅ and we describe the dispersive and Strichartz estimates for the wave equation with the Dirichlet boundary condition. We obtain counterexamples to the optimal Strichartz estimates of the flat case; we also discuss the some results concerning the dispersive estimates.

Dispersive estimates and absence of embedded eigenvalues

Herbert Koch, Daniel Tataru (2005)

Journées Équations aux dérivées partielles

In [2] Kenig, Ruiz and Sogge proved ∥ u ∥ L 2 n n - 2 ( ℝ n ) ≲ ∥ L u ∥ L 2 n n + 2 ( ℝ n ) provided n ≥ 3 , u ∈ C 0 ∞ ( ℝ n ) and L is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with C 2 coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues for potentials in L n + 1 2 and variants thereof.

Currently displaying 181 – 200 of 920