The superconvergence property of a certain external method for solving two point boundary value problems is established. In the case when piecewise polynomial spaces are applied, it is proved that the convergence rate of the approximate solution at the knot points can exceed the global one.

In this paper we consider a nonlinear parabolic equation of the following type:(P)      ∂u/∂t - div(|∇p|p-2 ∇u) = h(x,u)with Dirichlet boundary conditions and initial data in the case when 1 < p < 2.We construct supersolutions of (P), and by use of them, we prove that for tn → +∞, the solution of (P) converges to some solution of the elliptic equation associated with (P).

On considère le problème de Dirichlet :$$(d{d}^{c}u{)}^n=0\mathrm{dans}B$$et$$u{|}_{\partial B}=\varphi$$où $B$ désigne la boule unité de ${ℂ}^n.$Nous donnons une démonstration simple du fait que si $\varphi \in C^{1,1}\left(\partial B\right)$, alors $u\in C^{1,1}\left(B\right)$; de plus la croissance du coefficient de Lipschitz de la différentielle de $u$ est contrôlée par l’inverse de la distance au bord.

We study uniformly elliptic fully nonlinear equations $F(D^{2}u,Du,u,x)=0$, and prove results of Gidas–Ni–Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.

We establish absolute continuity of the elliptic measure associated to certain second order elliptic equations in either divergence or nondivergence form, with drift terms, under minimal smoothness assumptions on the coefficients.

We study the Dirichlet problems for elliptic partial differential systems with nonuniform growth. By means of the Musielak-Orlicz space theory, we obtain the existence of weak solutions, which generalizes the result of Acerbi and Fusco [1].

For certain Fréchet spaces of entire functions of several variables satisfying some specified growth conditions, we define a constant coefficient differential operator $\check{\alpha }$ as the transpose of a convolution operation in the dual space of continuous linear functionals and show that for $f\left(z\right)$ in one of these spaces, their always exists a solution of the differential equation $\check{\alpha }\left(x\right)=f$ in the same space.

We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t=[sign\left(u_x\right)log|u_x{\left|\right]}_x$ on $Q_T=[0,T]\times [0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us...

We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the $3d$ formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences...

We establish two new formulations of the membrane problem by working in the space of $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-Young measures and $W_{{\Gamma }_0}^{1,p}(\Omega ,{𝐑}^3)$-varifolds. The energy functional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. The interest of the first formulation is to encode the oscillation informations on the gradients minimizing...