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The waiting time property for parabolic problems trough the nondiffusion of support for the stationary problems.

Luis Alvarez, Jesús Ildefonso Díaz (2003)

RACSAM

In this note we study the waiting time phenomenon for local solutions of the nonlinear diffusion equation through its connection with the nondiffusion of the support property for local solutions of the family of discretized elliptic problems. We show that, under a suitable growth condition on the initial datum near the boundary of its support, a finite part of the family of solutions of the discretized problem maintain unchanged its support.

The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

Alexander Khapalov (2013)

International Journal of Applied Mathematics and Computer Science

We introduce and investigate the well-posedness of a model describing the self-propelled motion of a small abstract swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, typically associated with low Reynolds numbers. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by rotational and elastic Hooke forces. Models like this are of interest in biological and engineering applications...

The Wiener test for degenerate elliptic equations

E. B. Fabes, D. S. Jerison, C. E. Kenig (1982)

Annales de l'institut Fourier

We consider degenerated elliptic equations of the form ∑ i , j D x i ( a i j ( x ) D x j ) , where λ w ( x ) | ξ | 2 ≤ ∑ i , j a i j ( x ) ξ i ξ j ≤ Λ w ( x ) | ξ | 2 . Under suitable assumptions on w , we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on w . The main tool we use is an estimate for the Green function in terms of w .

The Wolff gradient bound for degenerate parabolic equations

Tuomo Kuusi, Giuseppe Mingione (2014)

Journal of the European Mathematical Society

The spatial gradient of solutions to non-homogeneous and degenerate parabolic equations of p -Laplacean type can be pointwise estimated by natural Wolff potentials of the right hand side measure.

Thin vortex tubes in the stationary Euler equation

Alberto Enciso, Daniel Peralta-Salas (2013)

Journées Équations aux dérivées partielles

In this paper we outline some recent results concerning the existence of steady solutions to the Euler equation in ℝ 3 with a prescribed set of (possibly knotted and linked) thin vortex tubes.

Three cylinder inequalities and unique continuation properties for parabolic equations

Sergio Vessella (2002)

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

We prove the following unique continuation property. Let u be a solution of a second order linear parabolic equation and S a segment parallel to the t -axis. If u has a zero of order faster than any non constant and time independent polynomial at each point of S then u vanishes in each point, x , t ′ , such that the plane t = t ′ has a non empty intersection with S .

Three solutions for a nonlinear Neumann boundary value problem

Najib Tsouli, Omar Chakrone, Omar Darhouche, Mostafa Rahmani (2014)

Applicationes Mathematicae

The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form - Δ p ( x ) u + a ( x ) | u | p ( x ) - 2 u = μ g ( x , u ) in Ω, | ∇ u | p ( x ) - 2 ∂ u / ∂ ν = λ f ( x , u ) on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

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