Key wordsIn this paper we prove a comparison principle between the semicontinuous viscosity sub-and supersolutions of the tangential oblique derivative problem and the mixed Dirichlet-Neumann problem for fully nonlinear elliptic equations. By means of the comparison principle, the existence of a unique viscosity solution is obtained.

Let P be a linear partial differential operator with coefficients in the Gevrey class G s (T n ), where T n is the n-dimensional torus and s 1. We prove a necessary condition for the s-global solvability of P on T n . We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.  2004 Elsevier Inc. All rights reserved.In the last years many papers are concerned with the study of the global solvability and hypoellipticity of linear partial differential operators on compact manifolds, e.g., torus, in large scales of functional spaces (see, e.g., [3,[7][8][9][10][18][19][20]23] and references listed therein). It is well known that the theory of global properties of differential operators is not well developed in comparison with the one of local properties. In particular, the global properties are open problems except for certain classes of operators. On the other hand, the local and global solvability/hypoellipticity are rather different in general. In fact, there are (A.A. Albanese), popivano@banmatpc.math.bas.bg (P. Popivanov).

This book deals with equations of mathematical physics as the different modifications of the KdV equation, the Camassa-Holm type equations, several modifications of Burger's equation, the Hunter-Saxton equation, conservation laws equations and others. The equations originate from physics but are proposed here for their investigation via purely mathematical methods in the frames of university courses. More precisely, we propose classification theorems for the traveling wave solutions for a sufficiently large class of third order nonlinear PDE when the corresponding profiles develop different kind of singularities (cusps, peaks), existence and uniqueness results, etc. The orbital stability of the periodic solutions of traveling type for mKdV equations are also studied. Of great interest too is the interaction of peakon type solutions of the Camassa-Holm equation and the solvability of the classical and generalized Cauchy problem for the Hunter-Saxton equation. The Riemann problem for special systems of conservation laws and the corresponding δ-shocks are also considered. As it concerns numerical methods we apply the CNN approach.The book is addressed to a broader audience including graduate students, Ph.D. students, mathematicians, physicist, engineers and specialists in the domain of PDE.

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