Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs.
Linearization and explicit solutions of the minimal surface equations.
We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Δh + 2h = 0 describing locally germs of minimal surfaces. Here Δ is the Laplace-Beltrami operator on the standard two-dimensional sphere. It explains the existence of the sum operator of minimal surfaces, introduced recently. In 4-dimensional space the equation Δ h + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero.
Linearization and normal form of the Navier-Stokes equations with potential forces
Linearization of nonlinear differential equations. IV: Nonlinear second order partial differential equations equivalent to linear base equation.
Linear-quadratic optimal control for the Oseen equations with stabilized finite elements
For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure...
Line-energy Ginzburg-Landau models : zero-energy states
We consider a class of two-dimensional Ginzburg-Landau problems which are characterized by energy density concentrations on a one-dimensional set. In this paper, we investigate the states of vanishing energy. We classify these zero-energy states in the whole space: They are either constant or a vortex. A bounded domain can sustain a zero-energy state only if the domain is a disk and the state a vortex. Our proof is based on specific entropies which lead to a kinetic formulation, and on a careful...
Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity
L'intégrale de Riemann-Liouville et le problème de Cauchy pour l'équation des ondes
Liouville integrability of geometric variational problems.
Liouville results for -Laplace equations of Lane-Emden-Fowler type
Liouville theorem for Dunkl polyharmonic functions.
Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems
In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.
Liouville theorems and blow up behaviour in semilinear reaction diffusion systems
Liouville theorems based on symmetric diffusions
Liouville theorems for a class of linear second-order operators with nonnegative characteristic form.
Liouville Theorems for Nondiagonal Elliptic Systems in Arbitrary Dimensions.
Liouville Theorems for Nonlinear Elliptic Equations and Systems.
Liouville theorems for self-similar solutions of heat flows
Let be a compact Riemannian manifold. A quasi-harmonic sphere on is a harmonic map from to () with finite energy ([LnW]). Here is the Euclidean metric in . Such maps arise from the blow-up analysis of the heat flow at a singular point. In this paper, we prove some kinds of Liouville theorems for the quasi-harmonic spheres. It is clear that the Liouville theorems imply the existence of the heat flow to the target . We also derive gradient estimates and Liouville theorems for positive...
Liouville theorems for semilinear equations on the Heisenberg group
Liouville type condition and the interior regularity of quasilinear parabolic system (the case of BMO-solutions)