Symmetries of Fundamental Solutions and Their Application in Continuum Mechanics

This paper is a brief survey of the recent results in problems of approximating functions by solutions of homogeneous elliptic systems of PDEs on compact sets in the plane in the norms of C^m spaces, m ≥ 0. We focus on general second-order systems. For such systems the paper complements the recent survey by M. Mazalov, P. Paramonov, and K. Fedorovskiy (2012), where the problems of C^m approximation of functions by holomorphic, harmonic, and polyanalytic functions as well as by solutions of homogeneous elliptic PDEs with constant complex coefficients were considered.     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

On the curve of critical exponents for nonlinear elliptic problems in the case of a zero mass

For semilinear elliptic equations ?Δu = λ|u| ^p?2 u?|u| ^q?2 u, boundary value problems in bounded and unbounded domains are considered. In the plane of exponents p × q, the so-called curves of critical exponents are defined that divide this plane into domains with qualitatively different properties of the boundary value problems and the corresponding parabolic equations. New solvability conditions for boundary value problems, conditions for the stability and instability of stationary solutions, and conditions for the existence of global solutions to parabolic equations are found.     

To the theory of the Dirichlet and Neumann problems for strongly elliptic systems in Lipschitz domains

For strongly elliptic Systems with Douglis-Nirenberg structure, we investigate the regularity of variational solutions to the Dirichlet and Neumann problems in a bounded Lipschitz domain. The solutions of the problems with homogeneous boundary conditions are originally defined in the simplest L ₂-Sobolev spaces H ^σ . The regularity results are obtained in the potential spaces H _p ^σ and Besov spaces B _p ^σ . In the case of second-order Systems, the author’s results obtained a year ago are strengthened. The Dirichlet problem with nonhomogeneous boundary conditions is considered with the use of Whitney arrays.     

The convolution method for fourier expansions. The case of generalized transmission conditions of crack (screen) type in piecewise inhomogeneous media

We consider boundary value problems for the equation ? _x (K ? _x ?) + KL[?] = 0 in the space R ⁿ with generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen on the surface x = 0. (Here L is an arbitrary linear differential operator with respect to the variables y ₁, …, y _n?1.) The coefficients K(x) > 0 are monotone functions of certain classes in the regions separated by the screen x = 0. The desired solution has arbitrary given singular points and satisfies a sufficiently weak condition at infinity.We derive formulas expressing the solutions of the above-mentioned problems in the form of simple quadratures via the solutions of the considered equation with a constant coefficient K for given singular points in the absence of a crack or a screen. In particular, the obtained formulas permit one to solve boundary value problems with generalized transmission conditions for equations with functional piecewise continuous coefficients in the framework of the theory of harmonic functions.     

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions.     

Grid analogs of two functions harmonic on the plane with a cut

For the Laplace equation in an unbounded domain (in the first quadrant, upper half-plane, plane with a cut), the Dirichlet and Neumann problems whose solutions are the imaginary and real parts of the complex function z ²lnz, respectively, are considered. Both problems are approximated on a square grid using the classic five-point difference scheme. The grid Fourier transform is applied to represent the solutions to the aforementioned grid problems in an integral form and obtain their asymptotic decompositions. It follows from the results that the accuracy of these grid solutions in the L _∞ ^h norm is O(h ²), where h is the grid step.     

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class C ^μ, where μ ∈ (0, 1) is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the L _∞ ^h norm. The convergence rate is O(N ^?2ln² N), where N is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.     

Complex vector measure and integral over manifolds with locally finite variations

It is well known that any compactly supported continuous complex differential n-form can be integrated over real n-dimensional C¹ manifolds in C^m (m ≥ n). For n = 1, the integral along any locally rectifiable curve is defined. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞ differential forms). The topic of the article is the integration of measurable complex differential (n, 0)-forms (containing no \(d{\bar z_j}\)) over real n-dimensional C⁰ manifolds in C^m with locally finite n-dimensional variations (a generalization of locally rectifiable curves to dimensions n > 1). The last result is that a real n-dimensional manifold C¹ embedded in C^m has locally finite variations, and the integral of a measurable complex differential (n, 0)-form defined in the article can be calculated by a well-known formula.     

On the total-variation convergence of regularizing algorithms for ill-posed problems

It is well known that ill-posed problems in the space V[a, b] of functions of bounded variation cannot generally be regularized and the approximate solutions do not converge to the exact one with respect to the variation. However, this convergence can be achieved on separable subspaces of V[a, b]. It is shown that the Sobolev spaces W ₁ ^m [a, b], m ∈ ? can be used as such subspaces. The classes of regularizing functionals are indicated that guarantee that the approximate solutions produced by the Tikhonov variational scheme for ill-posed problems converge with respect to the norm of W ₁ ^m [a, b]. In turn, this ensures the convergence of the approximate solutions with respect to the variation and the higher order total variations.     

Locally Lipschitz vector optimization with inequality and equality constraints

The present paper studies the following constrained vector optimization problem: \(\mathop {\min }\limits_C f(x),g(x) \in - K,h(x) = 0\), where f: ? ⁿ → ? ^m , g: ? ⁿ → ? ^p are locally Lipschitz functions, h: ? ⁿ → ? ^q is C ¹ function, and C ? ? ^m and K ? ? ^p are closed convex cones. Two types of solutions are important for the consideration, namely w-minimizers (weakly efficient points) and i-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x ⁰ to be a w-minimizer and first-order sufficient conditions for x ⁰ to be an i-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done.     

Maximal regularity of second order delay equations in Banach spaces

Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：
u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈R
to have maximal regularity in HSlder continuous function spaces C^α （R, X）, where X is a Banach space, A is a closed operator in X, F, G ∈L（C（[-r, 0], X）, X） are delay operators for some fixed r 〉 0.     

Composition limits and separating examples for some boolean function complexity measures

Block sensitivity (bs(f)), certificate complexity (C(f)) and fractional certificate complexity (C*(f)) are three fundamental combinatorial measures of complexity of a boolean function f. It has long been known that bs(f) ≤ C*(f) ≤ C(f) = O(bs(f)²). We provide an infinite family of examples for which C(f) grows quadratically in C*(f) (and also bs(f)) giving optimal separations between these measures. Previously the biggest separation known was \(C(f) = C*(f)^{\log _{4,5} 5}\). We also give a family of examples for which C*(f)= Ω (bs(f)^3/2).These examples are obtained by composing boolean functions in various ways. Here the composition fog of f with g is obtained by substituting for each variable of f a copy of g on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure s(f). The measures s(f), C(f) and C*(f) behave nicely under composition: they are submultiplicative (where measure m is submultiplicative if m(fog) ≤ m(f)m(g)) with equality holding under some fairly general conditions. The measure bs(f) is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure m at function f, m ^lim(f) to be the limit as k grows of m(f ^(k))^1/k , where f ^(k) is the iterated composition of f with itself k-times. For any function f we show that bs ^lim(f) = (C*)^lim(f) and characterize s ^lim(f); (C*)^lim(f), and C ^lim(f) in terms of the largest eigenvalue of a certain set of 2×2 matrices associated with f.     

Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Let (X,d,μ) be a RCD^?(K,N) space with \(K\in \mathbb {R}\) and N∈[1,∞). We derive the upper and lower bounds of the heat kernel on (X,d,μ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L^p boundedness of (local) Riesz transforms.     

Operator algebras associated with multiplicative convolutions of arithmetic functions

The action of N on l~2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l~2(N)), the set of all bounded operators on l~2(N).Moreover, the differential operator on l~2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) 1}. Interesting problems concerning are discussed.     

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^?1/4?ε; in the smoothness direction, a sufficient condition in Hölder classes isq∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^?1/4 and belong toC^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Denjoy–Carleman Differentiable Perturbation of Polynomials and Unbounded Operators

Let \({t\mapsto A(t)}\) for \({t\in T}\) be a C ^M -mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here C ^M stands for C ^ω (real analytic), a quasianalytic or non-quasianalytic Denjoy–Carleman class, C ^∞, or a Hölder continuity class C ^0,α . The parameter domain T is either \({\mathbb R}\) or \({\mathbb R^n}\) or an infinite dimensional convenient vector space. We prove and review results on C ^M -dependence on t of the eigenvalues and eigenvectors of A(t).     

Spaces of lower semicontinuous set-valued maps I

We introduce a lower semicontinuous analog, L ^?(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L ^?(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L ^?(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L ^?(X) and L ^?(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L ^?(X) and L ^?(Y) can be characterized by a unique factorization.