Rachid Regbaoui

On the heat flow for harmonic maps with potential

Let (M, g) and (N, h) be twoconnected Riemannian manifolds without boundary (M compact,N complete). Let G ε C∞(N): ifu: M → N is a smooth map, we consider the functional EG(u) = (1/2) ∫M [|du|2− 2G(u)]dVM and we study its associated heat equation. Inthe compact case, we recover a version of the Eells–Sampson theorem,while for noncompact target manifold N, we establishsuitable hypotheses and ensure global existence and convergence atinfinity. In the second part of the paper, we study phenomena of blowingup solutions.

Strong unique continuation for stokes equations

We prove a strong unique continuation theorem for the stationary Stokrs system where the potential A satisfies .

Unique Continuation from Sets of Positive Measure

Let Ω be a connected open subset of ℝ n and let V, W be functions on Ω. We say that the differential inequality

Strong Uniqueness for Second Order Differential Operators

\left| {\Delta u} \right| \leqslant \left| {Vu} \right| + \left| {W\nabla u} \right|

Q-curvature flow on 4-manifolds

(1.1) has the weak unique continuation property (w.u.c.p) if any solution u of (1.1) which vanishes on an open subset of Ω is identically zero. And we say that (1.1) has the strong unique continuation property (s.u.c.p) if any solution u is identically zero whenever it vanishes of infinite order at a point of Ω. We recall that a function \(u \in L_{{loc}}^{p}\) is said to vanish of infinite order at a point x 0 (or that u is flat at x 0) if for all N > 0,

Heat flow for p-harmonic maps between compact Riemannian manifolds

\int_{{\left| {x - {{x}_{0}}} \right|}} {{{{\left| {u\left( x \right)} \right|}}^{p}}dx = O\left( {{{R}^{N}}} \right)asR \to 0} .