Geometry and a priori estimates for free boundary problems of the Euler's equation

In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh‐Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh‐Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc.     

A priori estimates for optimal Dirichlet boundary control problems

We consider variational discretization of control constrained elliptic Dirichlet boundary control problems on smooth twoand three-dimensional domains, where we take into account the domain approximation. The state is discretized by linear finite elements, while the control variable is not discretized. We obtain optimal error bounds for the optimal control in two and three space dimensions. Furthermore we prove a superconvergence result in two space dimensions under the assumption that the underlying finite element meshes satisfy certain regularity requirements. We confirm our findings by a numerical experiment. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A priori estimates and existence of positive solutions for strongly nonlinear problems

In this paper we study the existence of positive solutions for a nonlinear Dirichlet problem involving the m-Laplacian. The nonlinearity considered depends on the first derivatives; in such case, variational methods cannot be applied. So, we make use of topological methods to prove the existence of solutions. We combine a blow-up argument and a Liouville-type theorem to obtain a priori estimates. Some Harnack-type inequalities which are needed in our reasonings are also proved.     

A priori estimates for solutions of boundary value problems for fractional-order equations

We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.     

A priori estimates for fluid interface problems

We consider the regularity of an interface between two incompressible and inviscid fluid flows in the presence of surface tension. We obtain local‐in‐time estimates on the interface in H^{(3/2)k + 1} and the velocity fields in H^(3/2)k. These estimates are obtained using geometric considerations which show that the Kelvin‐Helmholtz instabilities are a consequence of a curvature calculation. © 2007 Wiley Periodicals, Inc.     

A priori error estimates for elliptic optimal control problems with a bilinear state equation

In this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered. The control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints. We derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples.     

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.     

A priori estimates for the vector p-Laplacian with potential boundary conditions

This paper deals with the solvability of the boundary value problem

A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities

A priori bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to p-Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.     

On the asymptotic properties of a priori minimax estimates

Properties are studied of the a priori minimax estimates /1,2/ of an unknown initial state of a linear dynamic object under the condition that the measurements are taken at discrete instants, while noise exists only in the measurement channel and is modelled as an independent random variable. The paper continues the investigations begun in /3/.     

Optimal a priori estimates for interface problems

Summary. We consider a priori estimates in weighted norms for interface problems with piecewise constant diffusion constants which do not depend on the ratio between the constants. Our result generalizes an estimate of Lemrabet to arbitrary dimensions and includes curved boundaries. Furthermore, we discuss criteria for the existence of a uniform Poincaré estimate in weighted norms. In the affirmative case we obtain a robust finite element error bound in weighted norms. Finally, we present numerical experiments including a case with no uniform Poincaré constant. Mathematics Subject Classification (1991):65N15, 65N20Supported in part by the Deutsche Forschungsgemeinschaft, SFB 404     

Dirichlet operators: A priori estimates and uniqueness problems II

We study some basic properties of the Dirichlet operator in weighted L^p$L^p$ spaces. We work under rather general assumptions on weights that destroy minimal local Sobolev regularity of the form domain.     

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.     

Global existence,asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.     

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x ₀; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u₀ (x) on . Here is a regular and bounded domain in Rⁿ, with n 1, and₀ and ₁ endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n – 1) / (n – 2) whenn 3, r = when n = 1,2, and u₀ H^1/2 , u₀ = 0 on ₀. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary).     

A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case

We consider here solutions of a nonlinear Neumann elliptic equation Δu +?f (x, u) =?0 in Ω, ?u/?ν =?0 on ?Ω, where Ω is a bounded open smooth domain in ${\mathbb{R}^N, N\geq2}$ and f satisfies super-linear and subcritical growth conditions. We prove that L ^∞?bounds on solutions are equivalent to bounds on their Morse indices.     

L estimates and existence theorems for on Lipschitz boundaries

Mathematische Zeitschrift -