A new representation for second order stochastic integral-differential operators and its applications

In this paper, we’ll prove new representation theorems for a kind of second order stochastic integraldifferential operator by stochastic differential equations(SDEs) and backward stochastic differential equations(BSDEs) with jumps, and give some applications.     

Near operators theory and fully nonlinear elliptic equations

We give a short survey of the Campanato near operators theory and of its applications to fully nonlinear elliptic equations.     

Comparison theorems for nonlinear elliptic systems of second order with applications

In this paper comparison theorems concerning Dirichlet problem for a class of weakly coupled nonlinear second order elliptic systems have been developed. Some applications of these results to deflection or vibration of membranes and plates are also given.     

On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications

Dedicated to Professor Shmuel Agmon     

Existence and uniqueness of global strong solutions to fully nonlinear second order elliptic systems

We consider the problem of existence and uniqueness of strong a.e. solutions \({u: \mathbb{R}^n \longrightarrow \mathbb{R}^N}\) to the fully nonlinear PDE system

$$\label{equ1}F(\cdot,D^2u ) = f, \quad \text{ a.e. on }\mathbb{R}^n,\quad \quad(1)$$

when \({ f\in L^2(\mathbb{R}^n)^N}\) and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1).

     

On nonlinear higher order elliptic systems positive in the first derivatives

Elliptic systems of a divergent type with a natural energy spaceW _p ^m W _q ¹ are considered. Under certain restrictions imposed on the modulus of ellipticity of the lower part of the system, the Hölder property of the generalized solutions to this system and the Liouville theorem are established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 942–947, July, 1993.     

Essential self-adjointness of semibounded elliptic operators of second order

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 710–716, May, 1989.     

Spectral stability of Dirichlet second order uniformly elliptic operators

We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation.     

Existence of solutions of Two-Phase free boundary problems for fully nonlinear elliptic equations of second order

In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hⁿ⁻¹ Hausdorff measure and a normal in a measuretheoretic sense Hⁿ⁻¹ almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact that the free boundary is locally a C^1,α surface near Hⁿ⁻¹-a.e. point.     

On surjective second order non-linear Markov operators and associated nonlinear integral equations

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.     

On the range of elliptic, second order, nonvariational operators in Sobolev spaces

The subfamilyR(p, Ω) of the class of second order, uniformly elliptic, non variational operators L with bounded measurable coefficients, such that L(W^{2, p} (Ω)) is dense in L^p (Ω), is studied. Sufficient and necessary and sufficient conditions are given, for L to belong toR(p, Ω). An operator L of the class above and not inR(p, Ω) is constructed. Entrata in Redazione il 5 ottobre 1998.     

Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators

We study uniformly elliptic fully nonlinear equations of the type F(D²u,Du,u,x)=f(x). We show that convex positively 1-homogeneous operators possess two principal eigenvalues and eigenfunctions, and study these objects; we obtain existence and uniqueness results for nonproper operators whose principal eigenvalues (in some cases, only one of them) are positive; finally, we obtain an existence result for nonproper Isaac's equations.     

Schauder estimates for a class of second order elliptic operators on a cube

We consider a class of second order elliptic operators on a d-dimensional cube S_d. We prove that if the coefficients are of class C^k+δ(S_d), with k=0,1 and δ∈(0,1), then the corresponding elliptic problem admits a unique solution u belonging to C^k+2+δ(S_d) and satisfying non-standard boundary conditions involving only second order derivatives.