Local analysis of a polynomial system and its perturbations

On the real (x, y)-plane, we consider an autonomous system of differential equations whose right-hand sides are polynomials of special form in x and y and a perturbed system obtained from the former by varying the coefficients in the class of functions of (x, y) satisfying the Lipschitz condition. We study the behavior of trajectories of the system in a neighborhood of the isolated equilibrium point O = (0, 0). For the main (polynomial) system, we find all possible types of arrangement of the trajectories in a neighborhood of O. For the case in which the system has TO-curves, we give coefficient criteria for each of the possible types of the point O and study conditions under which the type is preserved in the perturbed system.     

The space clcv(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>) with the Hausdorff-Bebutov metric and differential inclusions

The paper is devoted to studying the space of nonempty closed convex (but not necessarily compact) sets in ? ⁿ , a dynamical system of translations, and existence theorems for differential inclusions. We make this space complete by equipping it with the Hausdorff-Bebutov metric. The investigation of these issues is important for certain problems of optimal control of asymptotic characteristics of a control system. For example, the problem \(\dot x = A(t,u)x\), (u, x) ∈ ? ^m+n , λ _n (u(·))→ min, where λ _n (u(·)) is the largest Lyapunov exponent of the system {ie121-2} = A(t, u)x, leads to a differential inclusion with a noncompact right-hand side.     

Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

We consider a (generally, noncoercive) mixed boundary value problem in a bounded domain D of Rⁿ for a second order elliptic differential operator A(x, ?). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ?) is of Robin type on ?D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ? ?D and control the growth of solutions near Y. We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y. Moreover, we prove the completeness of root functions related to L.     

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

We first investigate the Lipschitz continuity of (K,K’)-quasiregular C ² mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of ρ-harmonic (K,K’)-quasiregular mappings, and as the other application, we study the Lipschitz continuity of (K,K’)- quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation Δw = g. These results generalize and extend several recently obtained results by Kalaj, Mateljevi? and Pavlovi?.     

Dirichlet and Neumann problems for Laplace and heat equations in domains with right angles

The Dirichlet and Neumann problems are considered in the n-dimensional cube and in a right angle. The right-hand side is assumed to be bounded, and the boundary conditions are assumed to be zero. The author obtains a priori bounds for solutions in the Zygmund space, which is wider than the Lipschitz space C ^1,1 but narrower than the Hölder space C ^{1, α}, 0 < α < 1. Also, the first and second boundary-value problems are considered for the heat equation with similar conditions. It is shown that the solutions belong to the corresponding Zygmund space.     

Random extremal solutions of differential inclusions

Given a Lipschitz continuous multifunction F on \({\mathbb{R}^{n}}\), we construct a probability measure on the set of all solutions to the Cauchy problem \(\dot{x}\in F(x)\) with x(0) = 0. With probability one, the derivatives of these random solutions take values within the set extF(x) of extreme points for a.e. time t. This provides an alternative approach in the analysis of solutions to differential inclusions with non-convex right hand side.     

Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I

We consider (in general noncoercive) mixed problems in a bounded domain D in ? ⁿ for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.     

Two-dimensional homogeneous cubic systems: Classification and normal forms II

This work is the second in a series of papers concerning two-dimensional homogeneous cubic systems. In the first paper of the series, structural principles were developed to introduce a total order on the set of structural forms, i.e., vector polynomials with a fixed number of zero coefficients that are right-hand sides of two-dimensional homogeneous cubic systems of ODEs. Among them, structural forms normalized on the basis of normalization principles and canonical forms (CFs) that are linearly nonequivalent to each other and are the simplest in their class were sequentially distinguished. In this paper, for above-mentioned systems with proportional right-hand side components, all CFs with their canonical sets of permissible values are distinguished. For each CF, (a) conditions on the coefficients of the original system, (b) linear substitutions that reduce the right-hand side of a system under these conditions to the chosen CF, and (c) the resulting values of the CF’s coefficients are given.     

On Lie’s problem and differential invariants of ODEs <Emphasis Type="Italic">y</Emphasis>″ = <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis>, <Emphasis Type="Italic">y</Emphasis>)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.     

Averaging of systems of differential inclusions with slow and fast variables

We consider the system of differential inclusions

$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$

, where F,G: D → Kυ (\(R^{m_1 } \)), Kυ (\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces \(R^{m_1 } \) and \(R^{m_2 } \), respectively, D = R ₊ × \(R^{m_1 } \) × \(R^{m_2 } \) × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem \(\dot u\) ∈ µF ₀(u), u(0) = x ₀, F ₀(u) ∈ Kυ (\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ₀] and any solution x _µ(·), y _µ(·) of the original problem, there exists a solution u _µ(·) of the averaged problem such that ∥x _µ(t) ? y _µ(t) ∥ ≤ ? for t ∈ [0, 1/µ]. Furthermore, for each solution u _µ(·)of the averaged problem, there exists a solution x _µ(·), y _µ(·) of the original problem with the same estimate.

     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.     

Strict comparison theorems under sublinear expectations

In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class \(Lip(\Omega )\). Then we prove generalized strict comparison theorems on the enlarged space \(L_G^1(\Omega )\), which is the Banach completion of \(Lip(\Omega )\) under the G-expectation.     

Directional Derivative Problem for the Telegraph Equation with a Dirac Potential

In the domain Q = [0,∞)×[0,∞) of the variables (x, t), for the telegraph equation with a Dirac potential concentrated at a point (x₀, t₀) ∈ Q, we consider a mixed problem with initial (at t = 0) conditions on the solution and its derivative with respect to t and a condition on the boundary x = 0 which is a linear combination with coefficients depending on t of the solution and its first derivatives with respect to x and t (a directional derivative). We obtain formulas for the classical solution of this problem under certain conditions on the point (x₀, t₀), the coefficient of the Dirac potential, and the conditions of consistency of the initial and boundary data and the right-hand side of the equation at the point (0, 0). We study the behavior of the solution as the direction of the directional derivative in the boundary condition tends to a characteristic of the equation and obtain estimates of the difference between the corresponding solutions.     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Gradient continuity estimates

We consider degenerate elliptic equations of p-Laplacean type

$$-{\rm{div}}\, (\gamma(x)|Du|^{p-2}Du)=\mu\,,$$

and give a sufficient condition for the continuity of Du in terms of a natural non-linear Wolff potential of the right-hand side measure μ. As a corollary we identify borderline condition for the continuity of Du in terms of the data: namely μ belongs to the Lorentz space L(n, 1/(p ? 1)), and γ(x) is a Dini continuous elliptic coefficient. This last result, together with pointwise gradient bounds via non-linear potentials, extends to the non homogeneous p-Laplacean system, thereby giving a positive answer in the vectorial case to a conjecture of Verbitsky. Continuity conditions related to the density of μ, or to the decay rate of its L ⁿ -norm on small balls, are identified as well as corollaries of the main non-linear potential criterium.

     

Hölder continuity of solutions to the complex Monge–Ampère equation with the right-hand side in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>: the case of compact Kähler manifolds

We prove that on compact Kähler manifolds solutions to the complex Monge–Ampère equation, with the right-hand side in L ^p , p > 1, are Hölder continuous.     

Low distortion embeddings of some metric graphs into Banach spaces

We give a simple example of a countable metric graph M such that M Lipschitz embeds with distortion strictly less than 2 into a Banach space X only if X contains an isomorphic copy of l ₁. Further we show that, for each ordinal α < ω ₁, the space C([0, ω ^α ]) does not Lipschitz embed into C(K) with distortion strictly less than 2 unless K ^(α) ≠ 0. Also \(C\left( {\left[ {0,{\omega ^{{\omega ^\alpha }}}} \right]} \right)\) does not Lipschitz embed into a Banach space X with distortion strictly less than 2 unless Sz(X) ≥ ω ^α+1.     

Value functions for certain class of Hamilton Jacobi equations

We consider a class of Hamilton Jacobi equations (in short, HJE) of type

$ u_t + \frac{1}{2}\big(|u_{x_1}|^2+ \cdots +|u_{x_{n-1}}|^2\big) + \frac{\mathrm{e}^u}{m}|u_{x_n}|^m=0, $

in ? ⁿ ×?_?+? and m?>?1, with bounded, Lipschitz continuous initial data. We give a Hopf-Lax type representation for the value function and also characterize the set of minimizing paths. It is shown that the minimizing paths in the representation of value function need not be straight lines. Then we consider HJE with Hamiltonian decreasing in u of type

$ u_t + H_1\big(u_{x_1},\ldots,u_{x_i}\big) + \mathrm{e}^{-u}H_2\big(u_{x_{i+1}},\ldots, u_{x_n}\big)=0 $

where H ₁,H ₂ are convex, homogeneous of degree n,m?>?1 respectively and the initial data is bounded, Lipschitz continuous. We prove that there exists a unique viscosity solution for this HJE in Lipschitz continuous class. We also give a representation formula for the value function.

     

Boundary value problems for a higher-order parabolic equation with growing coefficients

We establish the unique solvability of boundary value problems in Hölder function classes for a linear parabolic equation of order 2m in noncylindrical domains of the class C ^{2m ? 1,α} , possibly unbounded (with respect to x as well as t), with nonsmooth (with respect to t) lateral boundary under the condition that the lower-order coefficients and the right-hand side of the equation can grow in a certain way when approaching the parabolic boundary of the domain and the leading coefficients may fail to satisfy the Dini condition near this boundary.     

The Lipschitzianity of convex vector and set-valued functions

It is well known that every scalar convex function is locally Lipschitz on the interior of its domain in finite dimensional spaces. The aim of this paper is to extend this result for both vector functions and set-valued mappings acting between infinite dimensional spaces with an order generated by a proper convex cone C. Under the additional assumption that the ordering cone C is normal, we prove that a locally C-bounded C-convex vector function is Lipschitz on the interior of its domain by two different ways. Moreover, we derive necessary conditions for Pareto minimal points of vector-valued optimization problems where the objective function is C-convex and C-bounded. Corresponding results are derived for set-valued optimization problems.