We consider the Hamilton–Jacobi–Bellman system ∂tu−Δu=H(u,∇u)+ffor u∈RN , where the Hamiltonian H(u,∇u) satisfies a super‐quadratic growth condition with respect to |∇u| . Such a non‐linear parabolic system corresponds to a stochastic differential game with N players. We obtain the existence of bounded weak solutions and prove regularity results in Sobolev spaces for the Dirichlet problem.

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.

The objective of this paper is to study nonlinear partial differential systems likepartial derivative(t)u - Delta u + H(x, t, u, del u) = 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 H of the nonlinear system satisfies a quadratic growth condition in del u and has a positive definite Jacobian 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.

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

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 C 0 ‐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

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.

Parabolic systems with p-structure are considered on convex polyhedral domains under Dirichlet boundary conditions. A fully discrete scheme is studied using C-0-piecewise linear finite elements in space and the backward Euler difference scheme in time. A priori error estimates in quasi norms are proved, and optimal convergence rates are obtained.

The system $${-{\rm{div}}\,{\rm\bf{S}}({\rm\bf{D}}({\rm\bf{u}}))+({\rm\bf{u}}\cdot\nabla){\rm\bf{u}}+\nabla\pi\,{=}\,{\rm\bf{f}}, \quad{\rm{div}}\,{\rm\bf{u}}\,{=}\,0}$$ is considered on a bounded three‐dimensional domain under no‐stick boundary value conditions, where S has p‐structure for some p<2 and D(u) is the symmetrized gradient of u. Various regularity results for the velocity u and the pressure π in fractional order Sobolev and Nikolskii spaces are obtained. Copyright © 2006 John Wiley & Sons, Ltd.

The p-Laplace equation is considered for p > 2 on a n-dimensional convex polyhedral domain under a Dirichlet boundary value condition. Global regularity of weak solutions in weighted Sobolev spaces and in fractional order Nikolskij and Sobolev spaces are proven.

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-Laplace equation. The result is obtained via regularization and a comparison theorem.

The degenerate parabolic equation ut=Δ(|u|m−1u),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−1u in weighted Sobolev spaces and in fractional order Nikolskii spaces are proved.

Nonlinear elliptic equations with p-structure on non-convex polyhedral domains under homogeneous Dirichlet boundary values are considered. Global regularity in fractional order Nikolskij and Sobolev spaces is proved.

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.

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 Dirichlet problem for strongly nonlinear elliptic equations on a non-Lipschitz domain is studied, It is assumed that the domain is two-dimensional and has a piecewise smooth boundary with a cuspidal point. The global regularity of a weak solution in fractional-order Sobolev spaces is investigated. Therefore, a difference quotient technique is applied, which provides regularity results in Nikolskii spaces. Utilizing the imbedding theorem of Nikolskii spaces into Sobolev spaces it follows that weak solutions are W-s,W-2(Omega)-functions for all s < 3/2. This result cannot be 2 improved. In fact, there is a counterexample in the case that s = 3/2.

The nonlinear elliptic system-Sigma(i=1)(n)partial derivative(i)F(i)(x,delu) = f(x)+Sigma(i=1)(n)partial derivative(i)f(i)(x)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 < infinity). Global regularity results of u and \delu\(p/2) in fractional order Sobolev spaces are proven.

%\noindent Strongly nonlinear elliptic equations and systems are investigated under mixed boundary value conditions. It is supposed that the domain is a multidimensional polyhedral domain. Global regularity results of $|\nabla u|^{\sigma}$ $(\sigma\geq 1)$ are proven, in particular, $W^{s,p}(\Omega)$-regularity $(s<\frac{1}{2})$ of $|\nabla u|$ and $|\nabla u|^2$.