Sobolev Type Research Articles

Existence Results for Some Partial Integro-Differential Equations

In this note, we deal with semilinear integro-differential equations subject to homogeneous Dirichlet boundary conditions given on the boundaries of the sections. Even if the differentiation will be taken only in some directions, it is not possible to see the main problem parameterized by the other coordinates because of the non-local terms which also obliged the problem to be degenerate. We establish the existence of solutions by employing the singular perturbations method as a natural tool. The perturbed problems are classical, non-local, semilinear elliptic problems and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. Some improvement, concerning the existence of the solutions and the convergence results depending on the weights, will be established. The paper also gives an idea about the study of the anisotropic singular perturbations in the framework of weighted spaces.

Semilinear pseudodifferential equations in spaces of tempered ultradistributions

We study a class of semilinear elliptic equations on spaces of tempered ultradistributions of Beurling and Roumieu type. Assuming that the linear part of the equation is a pseudodifferential operator of infinite order satisfying a suitable ellipticity condition we prove a regularity result in the functional setting above for weak Sobolev type solutions.

Global well-posedness of the three dimensional incompressible anisotropic Navier–Stokes system

In this paper, we study the global well-posed problem for the three dimensional incompressible anisotropic Navier–Stokes system (ANS) with initial data in the scaling invariant Besov–Sobolev type spaces. We prove that (ANS) has a unique global solution provided that the initial vertical velocity is large while initial horizontal data are sufficiently small compared with the horizontal viscosity. In particular, our result implies the global well-posedness of (ANS) with highly oscillating initial data.

Sharp affine Sobolev type inequalities via the Lp Busemann–Petty centroid inequality

We show that the Lp Busemann–Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo–Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases.

Non-autonomous functionals, borderline cases and related function classes

We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p, q)-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima. 1. Almost fifty years of degenerate operators in Russia In 1967 a seminal paper [60] of Ural’tseva appeared, featuring the proof of the C-nature of energy solutions to the degenerate equation (1.1) − div(|Du|p−2Du) = 0 . The one on the left hand side is nowadays very well known as the p-Laplacean operator. This operator is relevant in a large number of situations as for instance in the Calculus of Variations, in Geometric Analysis, in the theory of quasiconformal mappings, in the modelling of non-Newtonian fluids. Ural’tseva herself, by exhibiting a counterexample, showed that the regularity of solutions does not go beyond Holder continuity of the gradient, for some exponent β ∈ (0, 1). The proof the Holder gradient continuity result also appears in the second, yet untranslated, edition of the classical book [41]. Ural’tseva’s fundamental result is at the origin of a huge literature, up to the point that it is nowadays hard to find another single nonlinear operator that has attracted so much attention as long as the elliptic regularity theory is concerned. We quote here the important paper of Uhlenbeck [59], where Ural’tseva’s result has been extended to the vectorial case, and the papers [23, 29, 42, 47], where different proofs and extensions to equations with coefficients have been given. The equation appearing in (1.1) is the Euler-Lagrange equation of the functional (1.2) w 7→ ∫ |Dw| dx , p > 1 and, in fact, several of the results and techniques coming from the analysis of (1.1) have been eventually found to be useful in the Calculus of Variations. Starting from the eighties, new models and functionals related to the one in (1.2) were developed by Zhikov [62, 63, 64, 65, 66], together with a group of Russian mathematicians, in order to describe the behaviour of strongly anisotropic materials in the context of homogenisation and nonlinear elasticity. These functionals revealed to be important 1 2 BARONI, COLOMBO, AND MINGIONE also in the study of duality theory and in the context of the Lavrentiev phenomenon. They are non-autonomous functionals of the form

On the completeness of root functions of a holomorphic family of non-coercive mixed problem

We consider a non-coercive mixed boundary value problem in a bounded domain D of complex space for a second order parameter-dependent elliptic differential operator with complex-valued essentially bounded measured coefficients and complex parameter . The differential operator is assumed to be of divergent form in D, the boundary operator is of Robin type. The boundary of D is assumed to be a Lipschitz surface. Under reasonable assumptions the pair (A, B) induces a family of non-coercive mixed problems and a holomorphic family of Fredholm operators in suitable Hilbert spaces , of Sobolev type (here are the Sobolev-Slobodetskii spaces over D). If there is a Lipschitz function close enough to the (possibly discontinuous) argument of the complex-valued multiplier of the parameter in then we prove that the operators are continuously invertible for all with sufficiently large modulus on each angle on the complex plane where the operator is parameter-dependent elliptic. We also describe reasonable conditions for the system of root functions related to the family to be (doubly) complete in the spaces , and the Lebesgue space .

Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in [formula omitted

We investigate the rate of convergence of linear sampling numbers of the embedding Hα,β(Td)↪Hγ(Td). Here α governs the mixed smoothness and β the isotropic smoothness in the space Hα,β(Td) of hybrid smoothness, whereas Hγ(Td) denotes the isotropic Sobolev space. If γ>β we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on “energy-norm based sparse grids” for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding Hmixα(Td)↪Hmixγ(Td) and achieve optimality for Smolyak’s algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding Hmixα(Td)↪Lq(Td) for 2<q≤∞ where again Smolyak’s algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation β=γ (including the embedding into L2(Td)).

Extra-strong uncertainty principles in relation to phase derivative for signals in Euclidean spaces

This paper devotes to studying uncertainty principles of Heisenberg type for signals defined on Rn taking values in a Clifford algebra. For real-para-vector-valued signals possessing all first-order partial derivatives we obtain two uncertainty principles of which both correspond to the strongest form of the Heisenberg type uncertainty principles for the one-dimensional space. The lower-bounds of the new uncertainty principles are in terms of a scalar-valued phase derivative. Through Hardy spaces decomposition we also obtain two forms of uncertainty principles for real-valued signals of finite energy with the first order Sobolev type smoothness.

Optimal Sobolev trace embeddings

Optimal target spaces are exhibited in arbitrary-order Sobolev type embeddings for traces of n n -dimensional functions on lower dimensional subspaces. Sobolev spaces built upon any rearrangement-invariant norm are allowed. A key step in our approach consists of showing that any trace embedding can be reduced to a one-dimensional inequality for a Hardy type operator depending only on n n and on the dimension of the relevant subspace. This can be regarded as an analogue for trace embeddings of a well-known symmetrization principle for first-order Sobolev embeddings for compactly supported functions. The stability of the optimal target space under iterations of Sobolev trace embeddings is also established and is part of the proof of our reduction principle. As a consequence, we derive new trace embeddings, with improved (optimal) target spaces, for classical Sobolev, Lorentz-Sobolev and Orlicz-Sobolev spaces.

ОБОБЩЕННАЯ МОДЕЛЬ НЕСЖИМАЕМОЙ ВЯЗКОУПРУГОЙ ЖИДКОСТИ В МАГНИТНОМ ПОЛЕ ЗЕМЛИ

The initial boundary value problem for a system of partial differential equations modeling the dynamics of Kelvin-Voigt incompressible viscoelastic fluid of higher order in the Earth's magnetic field is studied. Problems of this type arise in the study of the process of rotation of a certain volume of fluid in the Earth's magnetic field. Research of the models of Kelvin–Voigt media has its source in the scientific works by A.P. Oskolkov, who summarizes the system of Navier–Stokes equations and theorems of unique existence of solutions to the corresponding initial boundary value problems. Subsequently, these models are studied by G.A. Sviridyuk and his followers. This model is studied for the first time and summarizes corresponding results for the model of magnetohydrodynamics of the nonzero order. The article deals with local unique solvability of chosen problem in the framework of the theory of autonomous semilinear Sobolev type equations. The main method is the method of phase space. The basic tool is the notion of p -sectorial operator and resolving singular semigroup of operators generated by it. In other words, the semigroup approach is used in the research. Besides the introduction, conclusion and reference list, the article includes three parts. In the first part of the article, the abstract Cauchy problem for semilinear autonomous equation of Sobolev type is presented. Here the concepts of Cauchy problem for Sobolev type equations, the phase space, quasi-stationary semitrajectory are introduced, and the theorems providing necessary and sufficient conditions for the existence of quasi-stationary semitrajectories are presented. In the second part, the Cauchy–Dirichlet problem is considered as a specific interpretation of the abstract problem. In the third part, the existence of a unique solution to the problem, which is a quasi-stationary semitrajectory is proved, and the description of its phase space is obtained. In conclusion, the possible ways of further research are outlined

НЕКЛАССИЧЕСКИЕ УРАВНЕНИЯ МАТЕМАТИЧЕСКОЙ ФИЗИКИ. ФАЗОВЫЕ ПРОСТРАНСТВА ПОЛУЛИНЕЙНЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА

The article surveys the results concerning the morphology of phase spaces for semilinear Sobolev type equations. The first three paragraphs present specific boundary value problems for Sobolev type partial differential equations whose phase spaces are simple smooth Banach manifolds. The last section contains the mathematical models whose phase spaces lie on a smooth Banach manifolds with singularities. The purpose of this article is the formation of a basis for future studies of the morphology of phase spaces for semilinear Sobolev type equations. In addition, the article provides an explanation of the phenomenon of nonexistence of solutions to the Cauchy problem and the phenomenon of nonuniqueness of solutions to the Showalter–Sidorov problem for the semilinear Sobolev type equations

Numerical Study of a Flow of Viscoelastic Fluid of Kelvin - Voigt Having Zero Order in a Magnetic Field

The article developed algorithms for the numerical solution of the initial - boundary problem of the flow of an incompressible viscoelastic Kelvin--Voigt fluid in the Earth's magnetic field. The theorem on an existence and uniqueness of this problem solution is proved using the theory of semilinear Sobolev type equations in the works written by T.G.~Sukachev, S.A.~Kondyukova. The original initial-boundary problem is transformed to the Cauchy problem for ordinary systems of nonlinear equations by sampling. Algorithms based on the explicit one-step schemes having Runge--Kutta type of seventh-order accuracy with a choice of integration step are used to find a numerical solution of the Cauchy problem. Evaluation of control of calculation accuracy at each time step is carried out by a scheme of the eighth order of accuracy. A time step is chosen according to the results of monitoring. Computational experiments show high computational efficiency of the developed algorithms for solving of the problem considered.

Системы функций, ортогональных относительно скалярных произведений типа Соболева с дискретными массами, порожденных классическими ортогональными системами

Системы функций, ортогональных относительно скалярных произведений типа Соболева с дискретными массами, порожденных классическими ортогональными системами

The Convergence of Approximate Solutions of the Cauchy Problem for the Model of Quasi-Steady Process in Conducting Nondispersive Medium with Relaxation

This article deals with numerical method for solving of the Dirichlet--Cauchy problem for equation modeling the quasi-steady process in conducting nondispersive medium with relaxation. This problem describes a complex electrodynamic process, allows to consider and predict its flow in time. The study of electrodynamic models is necessary for the development of electrical engineering and new energy saving technologies. The main equation of the model is considered as a quasi-linear Sobolev type equation. The convergence of approximate solutions obtained from the use of the method of straight lines with $\varepsilon$-embedding method and complex Rosenbrock method is proven in the article. The lemmas on the local error and on the distribution of error are proven. Estimates of a global error of the method are obtained.

НЕКЛАССИЧЕСКИЕ УРАВНЕНИЯ МАТЕМАТИЧЕСКОЙ ФИЗИКИ. ЛИНЕЙНЫЕ УРАВНЕНИЯ СОБОЛЕВСКОГО ТИПА ВЫСОКОГО ПОРЯДКА

The article presents the review of authors’ results in the field of non-classical equations of mathematical physics. The theory of Sobolev-type equations of higher order is introduced. The idea is based on generalization of degenerate operator semigroups theory in case of the following equations: decomposition of spaces, splitting of operators' actions, the construction of propagators and phase spaces for a homogeneous equation, as well as the set of valid initial values for the inhomogeneous equation. The author uses a proven phase space technology for solving Sobolev type equations consisting of reduction of a singular equation to a regular one defined on some subspace of initial space. However, unlike the first order equations, there is an extra condition that guarantees the existence of the phase space. There are some examples where the initial conditions should match together if the extra condition can’t be fulfilled to solve the Cauchy problem. The reduction of nonclassical equations of mathematical physics to the initial problems for abstract Sobolev type equations of high order is conducted and justified

Incomplete Sobolev type equations of higher order with "white noise" in quasi-Banach spaces

Incomplete Sobolev type equations of higher order with "white noise" in quasi-Banach spaces

Sobolev type fractional stochastic integro-differential evolution

In this paper, we prove the existence of $\alpha$-mild solutions for a class of fractional stochastic integrodifferential evolution equations of sobolev type with fractional sobolev stochastic nonlocal conditions in a real separable Hilbert space. To establish our main results, we use the Banach contraction mapping principle, fractional calculus, stochastic analysis and an analytic semigroup of linear operators. An example is given to illustrate the feasibility of our abstract result.

Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and entropy numbers of the embedding ${id}:\ell_{\|\cdot\|}^d \to \ell_\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identically to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schrodinger equation, where mixed regularity is present.

On Modified Method of Multistep Coordinate Descent for Optimal Control Problem for Semilinear Sobolev-Type Model

The paper describes a numerical method for solving the optimal control problem for a semilinear model of Sobolev-type. The method is based on both the modified projection Galerkin method and the method of multistep coordinate descent with memory. New numerical methods for solving nonlinear optimal control problems are need, because there exists a large number of applications and it is difficult to find their analytical solutions. We consider mathematical model of regulating potential distribution of speed of the filtered liquid free surface motion. In order to numerically investigate the mathematical model, we use the sufficient conditions for the existence of an optimal control by solutions of Showalter-Sidorov problem for semilinear Sobolev type equation with $ s $-monotone and $ p $-coercive operator. We present the results of computational experiment that demonstrate the work of the proposed numerical method.

A Linearized Model of Vibrations in the DNA Molecule in the Quasi-Banach Spaces

The system of Boussinesq equations, which simulates the vibrations in the DNA molecule, is researched in this paper. Earlier this model with a nonlinear right side was considered in a Banach space. In this paper, the model with a linear one is considered, but in quasi-Banach spaces of sequences. In the article the method of phase space and the theory of $(L,~p)$-bounded operators, developed by G. A. Sviridyuk and T. G. Sukacheva for first order equations, are used. It is also based on the theory of the Cauchy problem for linear equations of Sobolev type of high order in quasi-Banach spaces. A mathematical model of vibrations in the DNA molecule is reduced to an equation of Sobolev type in a quasi-Sobolev space. We construct the phase space of a system of linear Boussinesq equations. Sufficient conditions for the solvability of the initial boundary value problem for the Boussinesq equations in terms of the theory of $(L,~p)$-bounded operators are obtained.