A priori bounds for elliptic equations

In this paper we study the relationship existing between the maximum principle and the principal eigenvalue of second order elliptic operators and the expected exit times of the corresponding diffusions. A probabilistic approach is discussed that provides a good understanding of classical results established analytically. We also obtain an Alexandrov-Bakelman-Pucci type of estimate using similar methods.     

Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.     

Lower bounds for matrices on block weighted sequence spaces I

In this paper we consider some matrix operators on block weighted sequence spaces l _p (w, F). The problem is to find the lower bound of some matrix operators such as Hausdorff and Hilbert matrices on l _p (w, F). This study is an extension of papers by G. Bennett, G.J.O. Jameson and R. Lashkaripour.      

A priori bounds for approximations of Markov programs

This note determines a priori bounds for B. L. Fox's [J. Math. Anal. Appl., 34 (1971), 665–670] scheme of approximating discounted Markov programs, thus refining bounds recently obtained by D. J. White (Notes in Descision Theory No. 43, University of Manchester, 1977). The approximation scheme focuses careful attention on only a subset of the state space and uses a fixed function to characterize future returns outside the designated subset. The a priori bounds are useful to design the specific approximation, that is, to select the appropriate subset on which the approximation is based.     

A convolution operator in weighted spaces

Convolution operators in R ^m with a homogeneous function of an arbitrary complex order are considered in spaces with weighted norms. The role of the weight is played by some power of the distance to a k-dimensional plane. The fundamental content of the paper consists in the investigation of the transversal operators (TO), which arise from convolutions as a result of a Fourier transform along the mentioned plane and a change of variables. Formulas for the composition of TO are derived and the (finite-dimensional) kernels and cokernels of elliptic TO are described. Problems for TO are indicated, containing additional boundary conditions or potentials. The properties of the convolution operators are obtained from the corresponding properties of the TO.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 208–237, 1990.     

A priori bounds in the cauchy problem for coupled elliptic systems

By means of the logarithmic convexity of a suitable functional, an a priori inequality is developed for the sum of the squares of the solutions of the following improperly posed Cauchy problem. Consider the coupled elliptic system Lu = aν+f,Lν= bu+g, where L is a uniformly elliptic differential operator, a,b,f and g are bounded integrable functions with |b(x)|≧b₀>0 and ν satisfies a stabilizing condition, and where upper bounds for the error in measurement of the Cauchy data on the initial surface are prescribed. From the a priori estimate uniqueness, stability, and pointwise bounds for the solutions u and n are simultaneously deduced. The bounds are improvable by the Ritz technique. Moreover, the method presented here can be extended to the nonlinear system Lu = f(x, ν), Lν =g(x,u)provided g is a suitable form     

A priori bounds for periodic solutions of a Duffing equation

In this paper a well known Duffing type equation is considered. By means of a Liapunov function and careful estimation, we establish a priori bounds for periodic solutions and their derived functions. Coincidence theorems can then be applied to yield sufficient conditions for the existence of periodic solutions. Our conclusion improve several well known results in the literature.     

A priori bounds for superlinear problems involving the N-Laplacian

In this paper we establish a priori bounds for positive solutions of the equation

     

A priori bounds for a class of semi-linear degenerate elliptic equations

In this paper,we mainly discuss a priori bounds of the following degenerate elliptic equation,a ij(x)■ij u+b i(x)■iu+f(x,u)=0,in ΩRn,(*)where aij■iφ■jφ=0 on■Ω,andφis the defining function of ■Ω.Imposing suitable conditions on the coefficients and f(x,u),one can get the L∞-estimates of(*)via blow up method.     

A priori bounds for degenerate and singular evolutionary partial integro-differential equations

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional p-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi’s iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle is valid for such equations.     

A priori bounds and necessary conditions for solvability of prescribed curvature equations

We prove an estimate for the magnitude of solutions of the prescribed higher order mean curvature equations and examine the necessity of our conditions. Our results include well known sharp estimates for the mean and Gauss curvature and our previous estimate for scalar curvature as special cases.     

A priori bounds for an indefinite superlinear elliptic equation with exponential growth

This paper considers positive solutions to problem

     

A Schur test for weighted mixed-norm spaces

Summary We prove a Schur test for mixed-norm spaces L^p,q, 1 < p,q < ∞. Also we prove another version of the Schur test for discrete weighted mixed-norm spaces l^p,q _w, 1 < p,q < ∞, and wis a weight. We show that if w ₁, and w ₂are two weight functions on the index sets Jx Iand K x Lrespectively, and A =(a _{ji, kl} ) _{j∈J, i∈I, k∈K, l∈L} is an infinite matrix, then under certain conditions, Ais a bounded operator from l^p,q _w1, 1 < p,q < ∞ to l^p,q _w2. This will be a key result in proving boundedness of important operators in our work in time-frequency analysis.</o:p>     

A priori error bounds on invariant subspace approximations by block Krylov subspaces

We derive a priori error bounds for the block Krylov subspace methods in terms of “the sine” between the desired invariant subspace and the block Krylov subspace. The obtained results can be seen as the block analogue of the classical a priori estimates for standard projection methods.