Interior estimates for some semilinear elliptic problem with critical nonlinearity

We study compactness properties for solutions of a semilinear elliptic equation with critical nonlinearity. For high dimensions, we are able to show that any solutions sequence with uniformly bounded energy is uniformly bounded in the interior of the domain. In particular, singularly perturbed Neumann equations admit pointwise concentration phenomena only at the boundary.     

Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

We consider the boundary value problem u +p|x2α||u|-1u = 0,-1 < α = 0, in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution up whose maxima and minima are2 located alternately near the origin and the other m points qll=(λ cosπ(l-1)2, λ-1)msinπ(m), l = 2,, m + 1,such that as p goes to +∞,m+1pα|x2||upp|-1up 8πe(1 + α)δ0 + 8πe(-1)l-1δl=2re λ∈(0, 1), m is an odd number with(1 + α)(m + 2)-1 > 0, or m is an even,ql whe number. The same techniques lead also to a more general result on general domains.     

Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries

In this article, we consider a distributed optimal control problem associated with the Laplacian in a domain with rapidly oscillating boundary. For simplicity, we consider a rectangular region in 2d with oscillations on one part of the boundary. We consider two types of functionals, namely a functional involving the L ²-norm of the state variable and another one involving its H ¹-norm. The homogenization of the optimality system is obtained and then we derive appropriate error estimates in both cases.     

Asymptotic approximation with Kantorovich polynomial

We present the complete asymptotic expansion for the Kantorovich polyno-mials K_n as n→∞.The result is in a form convenient for applications.All coefficients of n~(-k)(k=1,2,…)are calculated explicitly in terms ofStifling numbers of the first and second kind.Moreover,we treat the simultaneous approximation with Kantorovich poly-nomials and determine the rate of convergence of d/(dx)K_n(f;x)-f'(x).     

Solvability problem for nondiagonal elliptic systems with a quadratic nonlinearity in the gradient (The two-dimensional case)

The existence of a weak solution of the Dirichlet problem for nondiagonal elliptic systems with quadratic nonlinearity is proved in the two-dimensional case. The solution is established to be smooth in the closure of a given domain, with the exception of at most finitely many points. The result is essentially based upon the theorem on quasireverse Hölder inequalities previously proved by the author. Bibliography: 14 titles.Dedicated to Academician O. A. Ladyzhenskaya on the occasion of her jubilee__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 5–17.     

On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity

We study the existence of nodal solutions to the boundary valueproblem – u = |u|^{p – 1}u in a bounded, smooth domain in ², with homogeneous Dirichlet boundary condition, when pis a large exponent. We prove that, for p large enough, thereexist at least two pairs of solutions which change sign exactlyonce and whose nodal lines intersect the boundary of .     

On the existence of infinitely many nonperturbative solutions in a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity

The paper focuses on a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity which describes the propagation of transverse electric waves along a dielectric layer filled with nonlinear medium. It is proved that even if the nonlinearity coefficients are small, the nonlinear problem has infinitely many nonperturbative solutions, whereas the corresponding linear problem always has a finite number of solutions. This results in the theoretical existence of a novel type of nonlinear guided waves that exist only in nonlinear guided systems. Asymptotic distribution of the eigenvalues is found and a comparison theorem is proved; periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. The results found essentially extend the theory evolved earlier (particular cases for Kerr, cubic-quintic, septic nonlinearities, etc. are easily extracted from the general results found here). Numerical results are also presented.     

Decay estimates for oscillatory integrals with polynomial phase

We obtain estimates for certain oscillatory integrals with polynomial phase. These estimates are stated in terms of roots of various derivatives of the phase polynomial.     

Rate estimates of gradient blowup for a heat equation with exponential nonlinearity

We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.     

A bifurcation result for a Dirichlet problem with discontinuous nonlinearity

In questo lavoro si considera un problema di Direchlet con nonlinearità discontinua; risultati di esistenza e molteplicità di soluzione per problemi di questo tipo sono stati dati in [A-B], [A-T] utilizzando il principio di azione duale di Clarke; in questo lavoro, invece, si introduce un problema multivoco associato al problma in modo naturale e si dimostra, utilizzando una variante del teorema di biforcazione globale di Rabinowitz che poggia sul grado multivoco ([C-L]), che per esso esiste un ramo globale di biforcazione.     

Asymptotic estimates for Laguerre polynomials

We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsL _n ⁽⁾ (x). First we summarize the results of a paper of Frenzen and Wong in whichn and >–1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.     

Uniform estimates for monotonic polynomial approximation

The uniform estimate is established for a monotone polynomial approximation of functions whose smoothness decreases at the ends of a segment.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 38–43, January, 1993.