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We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.
Projection methods based on wavelet functions combine optimal convergence rates with algorithmic efficiency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguelette decompositions enable the realization of especially fast algorithms for certain operators.
Contents: 1 Introduction 1.1 Tikhanov-Phillips Regularization of Ill-Posed Problems 1.2 A Compact Course to Wavelets 2 A Multilevel Iteration for Tikhonov-Phillips Regularization 2.1 Multilevel Splitting 2.2 The Multilevel Iteration 2.3 Multilevel Approach to Cone Beam Reconstuction 3 The use of approximating operators 3.1 Computing approximating families {Ah}
Parabolic equations on manifolds with singularities require a new calculus of anisotropic pseudo-differential operators with operator-valued symbols. The paper develops this theory along the lines of sn abstract wedge calculus with strongly continuous groups of isomorphisms on the involved Banach spaces. The corresponding pseodo-diferential operators are continuous in anisotropic wedge Sobolev spaces, and they form an alegbra. There is then introduced the concept of anisotropic parameter-dependent ellipticity, based on an order reduction variant of the pseudo-differential calculus. The theory is appled to a class of parabolic differential operators, and it is proved the invertibility in Sobolev spaces with exponential weights at infinity in time direction.
Preclinical work indicates that calcitriol restores vascular function by normalizing the endothelial expression of cyclooxygenase-2 and thromboxane-prostanoid receptors in conditions of estrogen deficiency and thus prevents the thromboxane-prostanoid receptor activation-induced inhibition of nitric oxide synthase. Since endothelial dysfunction is a key factor in the pathogenesis of cardiovascular diseases, this finding may have an important translational impact. It provides a clear rationale to use endothelial function in clinical trials aiming to find the optimal dose of vitamin D for the prevention of cardiovascular events in postmenopausal women.
In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
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(2011)