Abstract We consider the Dirichlet problem for partial trace operators which include smallest and largest eigenvalue of Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result known solutions, our knowledge. If some missing, such are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior estimate under...

The classical theory of reduction, initiated by Weierstrass, has found modern applications to the fields of Integrable Systems and Number Theory, to name but two. In this short note we only address specific cases, providing minimal historical references, listing the steps that we devised, and exhibiting some new explicit formulas. A full-length discussion of original motivation, theoretical adv...

Since the theory of spectral properties of non-self-accession differential operators on Sobolev spaces is an important field in mathematics, therefore, different techniques are used to study them. In this paper, two types of non-self-accession differential operators on Sobolev spaces are considered and their spectral properties are investigated with two different and new techniques.

We formulate and prove a non-local “maximum principle for semicontinuous functions” in the setting of fully nonlinear and degenerate elliptic integro-partial differential equations with integro operators of second order. Similar results have been used implicitly by several researchers to obtain comparison/uniqueness results for integro-partial differential equations, but proofs have so far been...

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The ell...

Spectral points of positive and negative type, and type π+ and type π− for closed linear operators and relations in Krein spaces are introduced with the help of approximative eigensequences. The main objective of the paper is to study these sign type properties in the non-selfadjoint case under various kinds of perturbations, e.g. compact perturbations and perturbations small in the gap metric....

[Last revised: 18 February, 2007. Includes corrections noted by Tim Nguyen] What do I want to do:(1) Variable coefficient, elliptic PDE. (2) Monopoles (3) Something else if time permits I have pretty much decided not to use one book but to take bits and pieces from several. For one thing there does not seem to be an approptriate treatment of monopoles. The elliptic part of the course is intende...

We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special elliptic curves defined over Q which do not deform to generic elliptic curves. We also study algebraically integrable operators of higher order with several...