Search

A new forward–backward anisotropic diffusion model is introduced. The two limit cases are the Perona-Malik equation and the Total Variation flow model. A fully discrete finite element scheme is studied using C0-piecewise linear elements in space and the backward Euler difference scheme in time. A priori estimates are proven. Numerical results in image denoising and form generalization are presented.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

1The fast diffusion equation $u_t=\Delta(|u|^{m-1}u)(0

The quasi–steady power–law Stokes flow of a mixture of incompressible fluids with shear–dependent viscosity is studied. The fluids are immiscible and have constant densities. Existence results are presented for both the no–slip and the no–stick boundary value conditions. Use is made of Schauder’s fixed–point theorem, compactness arguments, and DiPerna-Lions renormalized solutions. CMUC/FCT; Project POCI/MAT/57546/2004

In this work, new interpolation error estimates have been derived for some well-known interpolators in the quasi-norms. The estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations. In particular, by using these estimates, we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2-d and 3-d p-Laplacian.

We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well-known feature in some particular cases such as the porous medium equation or the parabolic p p -Laplace equation. The result is obtained via regularization and a comparison theorem.

The nonlinear elliptic system is investigated on a non-smooth domain. Mixed boundary value conditions are given. The left-hand side of the system has p-structure (e.g., it is the p-Laplacian and 1 < p < ∞). Global regularity results of u and |∇u|p/2 in fractional order Sobolev spaces are proven.

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ⊂ IRn (n ≥ 3) has a piecewise smooth boundary. Ws,2 — regularity (s < 3/2) of u and Lp — properties of the first and the second derivatives of u are proven.

The degenerate parabolic equation \[ u_t=\Delta\beta(u) \] is considered in the cylinder $\Omega\times(0,T)$, where $\Omega\subset{\R}^n$ is bounded, $\beta(0)=\beta'(0)=0$, and $\beta'\geq0$ (e.g., $\beta(u)=u|u|^{m-1}$, $m>1$). A fully discrete scheme is given using C0-piecewise linear finite elements in space and a backward difference scheme in time. L2-error estimates for the approximation of u and of $\beta(u)$ are proven. For that purpose we employ the fact that the free boundary has a finite speed of propagation.

The objective of this paper is to study nonlinear partial differential systems like $$ \partial_t {\bf u}- \Delta {\bf u} +{\bf H}(x,t,{\bf u}, \nabla {\bf u})={\bf G}(x,t), $$ with applications to the solution of stochastic differential games with $N$ players, where $N$ is arbitrarily large. It is assumed that the Hamiltonian ${\bf H}$ of the nonlinear system satisfies a quadratic growth condition in $\nabla {\bf u}$ and has a positive definite Jacobian ${\bf H_u}$. An energy estimate and the uniqueness property for bounded weak solutions are proved. Moreover, applications to stochastic games and financial economics such as modern portfolio theory are discussed.

During the last century, the amount and the complexity of land surface data have enormously increased and have shown the need for efficient automated analysis methods. In this paper we focus our interest on methods helping to analyze Digital Elevation Models. A finite element method in shape generalization of Digital Elevation Models is presented and numerical results are given. The finite element scheme is a fully discrete approximation of a diffusion equation of forward-backward Perona-Malik type. C 0 -piecewise linear elements in space and the backward Euler difference scheme in time are used.

Nonlinear partial differential equations with $p$--structure for $1

Nonlinear elliptic systems with $p$-structure for $p \! \geq \! 2$, such as \[ -\mbox{div}\left((1+|\nabla u|^{p-2})\nabla u\right) =f(x)+\sum_{i=1}^n\partial_if_i(x) ,\] are considered under mixed boundary value conditions on nonsmooth domains. Regularity results in fractional-order Sobolev spaces are proven, e.g., $u\in W^{r,p}(\Omega)$ for all $r 1$.

The doubly nonlinear parabolic equation u t = div [| ∇ (|u| m−1 u)| p−2 ∇ (|u| m−1 u)] (m>1,m(p−1)>1) is considered in several dimensions and regularity results in fractional order Sobolev spaces are obtained. The main tools in the proof are a difference quotient technique and the imbedding theorem of Nikolskii spaces into Sobolev spaces.

The degenerate parabolic equation u t = Δ ( | u | m − 1 u ) , m > 0 is considered in a cylinder Ω × ( 0 , T ) under homogeneous Dirichlet boundary values. The regularity of weak solutions for the fast diffusion equation ( m 1 ) and the porous medium equation ( m > 1 ) are investigated. Regularity of u and | u | m − 1 u in weighted Sobolev spaces and in fractional order Nikolskii spaces are proved.