On the maximum principle for complete second-order elliptic operators in general domains

This paper is concerned with the maximum principle for second-order linear elliptic equations in a wide generality. By means of a geometric condition previously stressed by Berestycki-Nirenberg-Varadhan, Cabré was very able to improve the classical ABP estimate obtaining the maximum principle also in unbounded domains, such as infinite strips and open connected cones with closure different from the whole space. Now we introduce a new geometric condition that extends the result to a more general class of domains including the complements of hypersurfaces, as for instance the cut plane. The methods developed here allow us to deal with complete second-order equations, where the admissible first-order term, forced to be zero in a preceding result with Cafagna, depends on the geometry of the domain.     

Oscillation theorems for differential equations with a nonlinear damping term

In this paper, we are concerned with a class of nonlinear second-order differential equations with a nonlinear damping term. Passage to more general class of equations allows us to remove a restrictive condition usually imposed on the nonlinearity, and, as a consequence, our results apply to wider classes of nonlinear differential equations. Two illustrative examples are considered.     

Diffraction by a wedge at skew incidence: integral representations of Cauchy-Carleman for the electromagnetic fields

An electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional difference equations by means of Sommerfeld integrals and Malyuzhinets theorem. By introducing an integral operator it is shown that the solutions of this system of functional equations can be defined in terms of integral representations whose kernels are solutions of a singular integral equation of Cauchy-Carleman type for which an explicit solution is given.     

Existence for nonoscillatory solutions of higher-order nonlinear neutral difference equations

In this paper, we consider the following forced higher-order nonlinear neutral difference equation

     

Viscosity solutions of fully nonlinear parabolic systems

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 Proc. London Math. Soc. 63 (1991) 212-240 and Comm. Partial Differential Equations 16 (1991) 1095-1128.     

Stability of a system of Volterra integro-differential equations

Using new and known forms of Lyapunov functionals, this paper proposes new stability criteria for a system of Volterra integro-differential equations.     

Some noncoercive parabolic equations with lower order terms in divergence form

This paper deals with existence and regularity results for the problem $ \cases{u_t-\mathrm{div}(a(x,t,u )\nabla u)=-\mathrm{div}(u\,E) \qquad in \Omega\times (0,T),\cr u=0 \qquad on \partial \Omega\times (0,T), \cr u (0)= u_0 \qquad in \Omega ,\cr} $ under various assumptions on E and $ u_0 $. The main difculty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the "energy space" $ L^2 (0, T; H_0^1 (\Omega)) $.AMS Subject Classification: 35K10, 35K15, 35K65.     

Existence of solutions to an anisotropic phase‐field model

We consider an anisotropic phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger ( Acta. Metall. Mater. 1995; 43 (2):689). Existence of weak solutions is established under a certain convexity condition on the strongly non‐linear second‐order anisotropic operator and Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. The qualitative properties of the solutions are illustrated by a numerical example. Copyright © 2003 John Wiley & Sons, Ltd.     

Two Dimensional Interface Problems for Elliptic Equations

We study the structure of solutions to the interface problems for second order quasi-linear elliptic partial differential equations in two dimensional space. We prove that each weak solution can be decomposed into two parts near singular points, a finite sum of functions in the form of cr^α log^m rφ(θ) and a regular one w. The coefficients c and the C^{1,α} norm of w depend on the H¹-norm and the C^{º,α}-norm of the solution, and the equation only.