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It is known that difference equations generated as the Newton-Raphson iteration for quadratic equations are solvable in closed form, and the solution can be constructed from linear three-term recurrence relations with constant coefficients.…
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
We prove existence, uniqueness and regularity results for mixed boundary value problems associated with fully nonlinear, possibly singular or degenerate elliptic equations. Our main result is a global H\"older estimate for solutions,…
The aim of this paper is to implement some new techniques, based on conjugate duality in convex optimization, for proving the existence of global error bounds for convex inequality systems. We deal first of all with systems described via…
In this paper we prove a general uniqueness result in the inverse boundary value problem for the weighted p-Laplace equation in the plane, with smooth weights. We also prove a uniqueness result in dimension 3 and higher, for real analytic…
This work is devoted to the studies of a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa--Holm equation. It is given by a quadratic form with a singular…
In this paper we present some very recent results regarding existence, uniqueness, and multiplicity of solutions for quasilinear elliptic equations and systems, exhibiting both singular and convective reaction terms. The importance of…
The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main…
Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness…
The aim of this thesis is to derive new gradient estimates for parabolic equations. The gradient estimates found are independent of the regularity of the initial data. This allows us to prove the existence of solutions to problems that have…
We establish near-optimal quantitative uniqueness of continuation for solutions of evolution equations vanishing on the lateral boundary. These results were obtained simply by combining existing observability inequalities and energy…
We establish new results of first-order necessary conditions of optimality for finite-dimensional problems with inequality constraints and for problems with equality and inequality constraints, in the form of John's theorem and in the form…
The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on…
We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a…
We study the homogenization of first-order Hamilton-Jacobi equations on an infinite-dimensional Hilbert space, motivated by systems of infinitely many indistinguishable particles on the torus. A central difficulty is that the analysis takes…
The direct or algorithmic approach for the Jacobian problem, consisting of the direct construction of the inverse polynomials is proposed. The so called principle and derived Jacobi conditions are proposed and discussed. The algorithmic…
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
In this paper we present some basic uniqueness results for evolutive equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field…
We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second…