Haim Brezis

Remarks on sublinear elliptic equations

On considere le probleme (1): −Δu=f(x,u) sur Ω, u≥0, u±0 sur Ω, u=0 sur ∂Ω, ou Ω⊂R N est un domaine borne a frontiere lisse et f[x,u):Ω×[0,∞)→R. On demontre que (1) a au plus une solution. De plus, une solution de (1) existe si et seulement si λ 1 (−Δ−a 0 (x)) 0. (a 0 (x)=lim u→0 f(x,u)/u; a ∞ (x)=lim u→∞ f(x,u)/u.)

Monotonicity Methods in Hilbert Spaces and Some Applications to Nonlinear Partial Differential Equations

Publisher Summary This chapter discusses the monotonicity methods in Hilbert spaces and presents some applications to nonlinear partial differential equations. It describes classical properties of maximal monotone operators in Hilbert spaces. It focuses on a particular class of monotone operators, namely those that are gradients of convex functions. The chapter also highlights their specific properties that do not hold for general monotone operators. Evolution equations associated with gradients of convex functions: smoothing effect on the initial data, behavior at infinity, and so on are discussed in the chapter along with some applications to nonlinear partial differential equations.

Asymptotics for the minimization of a Ginzburg-Landau functional

AbstractLetΩ ⊂ ℝ2 be a smooth bounded simply connected domain. Consider the functional

Remarks on the Euler equation

E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 }

Degree theory and BMO; part I: Compact manifolds without boundaries

on the classHg1={u εH1(Ω; ℂ);u=g on ∂Ω} whereg:∂Ω∂ → ℂ is a prescribed smooth map with ¦g¦=1 on ∂Ω∂ and deg(g, ∂Ω)=0. Let uuε be a minimizer for Eε onHg1. We prove that uε → u0 in

Produits infinis de resolvantes

C^{1,\alpha } (\bar \Omega )