Cristian Enache

Blow-up phenomena for a class of quasilinear parabolic problems under Robin boundary condition

Abstract This note deals with a class of heat emission processes in a medium with a non-negative source, a nonlinear decreasing thermal conductivity and a linear radiation (Robin) boundary condition. For such heat emission problems, we make use of a first-order differential inequality technique to establish conditions on the data sufficient to guarantee that the blow-up of the solutions does occur or does not occur. In addition, the same technique is used to determine a lower bound for the blow-up time when blow-up occurs.

New carbocyclic N(6)-substituted adenine and pyrimidine nucleoside analogues with a bicyclo[2.2.1]heptane fragment as sugar moiety; synthesis, antiviral, anticancer activity and X-ray crystallography.

New nucleoside analogues with an optically active bicyclo[2.2.1]heptane skeleton as sugar moiety and 6-substituted adenine were synthesized by alkylation of 6-chloropurine intermediate. Thymine and uracil analogs were synthesized by building the pyrimidine ring on amine 1. X-ray crystallography confirmed an exo-coupling of the thymine to the ring and an L configuration of the nucleoside analogue. The library of compounds was tested for their inhibitory activity against influenza virus A∖California/07/09 (H1N1)pdm09 and coxsackievirus B4 in cell culture. Compounds 13a and 13d are the most promising for their antiviral activity against influenza, and compound 3c against coxsackievirus B4. Compounds 3b and 3g were tested for anticancer activity.

A maximum principle for some fully nonlinear elliptic equations with applications to Weingarten hypersurfaces

In this article, we investigate a general class of fully nonlinear elliptic equations, including Weingarten equations. Our first aim is to construct a general elliptic inequality for an appropriate functional combination of u(x) and |∇u(x)|, i.e. a kind of P-function P(x), in the sense of L.E. Payne (see the book of Sperb [Sperb, Maximum Principles and Their Applications, Academic Press, New York, 1981]), where u(x) is a given solution of our class of fully nonlinear equations. From this inequality, making use of Hopfs first maximum principle, we derive a maximum principle for P(x), which extend some similar results obtained by Philippin and Safoui [Philippin and Safoui, Some maximum principles and symmetry results for a class of boundary value problems involving the Monge-Ampère equation, Math. Models Methods Appl. Sci. 11 (2001), pp. 1073–1080; Philippin and Safoui, Some applications of the maximum principle to a variety of fully nonlinear elliptic PDEs, Z. Angew. Math. Phys. 54 (2003), pp. 739–755], Porru et al. [Porru, Safoui and Vernier-Piro, Best possible maximum principles for fully nonlinear elliptic partial differential equations, Zeit. Anal. Anwend. 25 (2006), pp. 421–434] and Enache [Enache, Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs, Nonlinear Differ. Eqns Appl. 17 (2010), pp. 591–600]. This maximum principle is then used to obtain various a priori estimates with applications to some class of Weingarten hypersurfaces.

Maximum principles, Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations

Abstract This paper is concerned with a general class of quasilinear anisotropic equations. We first derive some maximum principles for two appropriate P-functions, in the sense of Payne (see the book of Sperb [18]). These maximum principles are then employed to obtain a Liouville-type result and a Serrin–Weinberger-type symmetry result.

New carbocyclic nucleoside analogues with a bicyclo[2.2.1]heptane fragment as sugar moiety; synthesis, X-ray crystallography and anticancer activity.

An amine group was synthesized starting from an optically active bicyclo[2.2.1]heptane compound, which was then used to build the 5 atoms ring of a key 6-chloropurine intermediate. This was then reacted with ammonia and selected amines obtaining new adenine- and 6-substituted adenine conformationally constrained carbocyclic nucleoside analogues with a bicyclo[2.2.1]heptane skeleton in the sugar moiety. X-ray crystallography confirmed an exo-coupling of base to the ring and a L configuration of the nucleoside analogues. The compounds were tested for anticancer activity.

Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane

This note is mainly devoted to the solvability of Monge-Ampere equation det ( D2u ) = 1, in a C2 bounded strictly convex domain Ω ⊂ R2, subject to a contact angle boundary condition. A necessary condition for the solvability of this problem, involving the maximal value of the curvature k (s) of ∂Ω and the contact angle, was derived by X.-N. Ma in [10], making use of a maximum principle for an appropriate P-function (see, also, J. Urbas [20], for a different proof). Our main goal here is to prove a complementary result. More precisely, we will derive a new necessary condition of solvability, involving the minimal value of the curvature k (s) of ∂Ω and the contact angle. The main ingredients of our proof are the derivation of a minimum principle for the P-function employed by X.-N. Ma in his proof, respectively the use of some computations in normal coordinates with respect to the boundary ∂Ω. Finally, a similar minimum principle will be employed to derive some isoperimetric estimates for the classical convex solution of Monge-Ampere equation, subject to the homogeneous Dirichlet boundary condition.

On some isoperimetric inequalities involving eigenvalues of fixed membranes

Abstract In 1956, Hersch (1965) derived some isoperimetric inequalities for eigenvalues of a fixed membrane Ω , simply connected, with a center of symmetry O . In this note we are going to derive some sharper versions of Hersch’s results. More precisely, if Ω is symmetric of order 2 we show that we have λ 2 ( Ω ) + λ 3 ( Ω ) ≤ 2 λ 2 ( D r o ) , where D r o is a disc of radius r o ( Ω ) (the conformal radius of Ω at O ). Also, if Ω is symmetric of order 4, we have λ 4 ( Ω ) + λ 5 ( Ω ) ≤ 2 λ 4 ( D r o ) .

MAXIMUM PRINCIPLES FOR A CLASS OF NONLINEAR SECOND-ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS IN DIVERGENCE FORM

For a class of nonlinear elliptic boundary value problems in divergence form, we construct some general elliptic inequalities for appropriate combinations of and, where are the solutions of our problems. From these inequalities, we derive, using Hopfs maximum principles, some maximum principles for the appropriate combinations of and, and we list a few examples of problems to which these maximum principles may be applied.

On some inequalities for low eigenvalues of closed surfaces in

This paper is concerned with the low eigenvalues of closed surfaces in , of given measure, which are topologically equivalent to a sphere. Our aim is to obtain an isoprimetric inequality giving an upper bound for the product of the first three non-trivial eigenvalues of a convex closed surface topologically equivalent to a sphere. Moreover, we will also derive some lower bounds for the first non-trivial eigenvalue of the regular octahedron and icosahedron.

Synthesis and properties of 4-(3-substituted azulen-1-yl)-2,6-diphenylpyridines

The syntheses of 4-azulen-1-yl-2,6-diphenylpyridines substituted at the azulene C-3 moiety with electron donating or withdrawing groups, are reported. When electron-donating groups (EDG) are present, the reaction of the corresponding pyranylium perchlorates and ammonium acetate takes place. Because of the difficulties in the generation of pyranylium salts with electronwithdrawing groups (EWG), the corresponding pyridines are obtained by a PdCl2-promoted substitution of halogen in 4-chloro-2,6-diphenylpyranylium salt, with an azulene derivative followed by an in situ replacement of oxygen with nitrogen. The physical and chemical properties of the pyridines obtained are discussed.