Bessel Potential Research Articles

Complex interpolation with Dirichlet boundary conditions on the half line

AbstractWe prove results on complex interpolation of vector‐valued Sobolev spaces over the half‐line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space‐valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space H^{s-2}(Omega) or widetilde{H}^{s-2}( Omega), frac{1}{2}< s<frac{3}{2}, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.

Fundamental and dressed annular solitons in saturable nonlinearity with parity–time symmetric Bessel potential**Project supported by the National Natural Science Foundation of China (Grant No. 61308019), the Guangdong Provincial Natural Science Foundation, China (Grant Nos. 2015A030313650 and 2014A030310262), and the Guangdong Provincial Science and Technology Planning Program, China (Grant No. 2017A010102019).

We theoretically study the existence and stability of optical solitons in saturable nonlinearity with a two-dimensional parity–time (PT) symmetric Bessel potential. Besides the fundamental solitons, a novel type of dressed soliton, whose intensity looks like a ring dressed on an intensity hump, are presented. It is found that both the fundamental solitons and dressed solitons can exist when the propagation constant is beyond a certain critical value. The propagation stability is investigated with a linear stability analysis corroborated by a beam propagation method. All the fundamental solitons are stable, while dressed solitons are unstable for low values of saturable parameter. As the value of saturable parameter increases, the dressed solitons tend to be stable at high powers.

Some generalizations of Bessel and Flett potentials associated to the Laplace–Bessel differential operator

ABSTRACTWe introduce the notion of bi-parametric potential-type operators, which generalize the Bessel and Flett potentials associated to singular Laplace–Bessel differential operator. Explicit inversion formulas for them are obtained by using a special wavelet-like transform, generated by the so-called ‘modified beta-semigroup’.

Second order optimality conditions for optimal control of quasilinear parabolic equations

We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy.Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

Cones of Functions with Monotonicity Conditions for Generalized Bessel and Riesz Potentials

Рассматриваются вопросы порядкового накрывания и порядковой эквивалентности для конусов функций со свойствами монотонности, связанных с убывающими перестановками обобщенных потенциалов Бесселя и Рисса. Библиография: 8 названий.

Дифференциальные свойства обобщённых потенциаловтипа Бесселя и типа Рисса

In this paper we study differential properties of convolutions of functions with kernels thatgeneralize the classical Bessel-Macdonald kernels ... The theory ofclassical Bessel potentials is an important section of the general theory of spaces of differentiablefunctions of fractional smoothness and its applications in the theory of partial differentialequations. The properties of the classical Bessel-Macdonald kernels are studied in detail in thebooks of Bennett and Sharpley, S. M. Nikolskii, I. M. Stein, V. G. Mazya. The local behavior ofthe Bessel-Macdonald kernels in the neighborhood of the origin is characterized by the presenceof a power-type singularity ||-. At infinity, they tend to zero at an exponential rate. Therecent work of M. L. Goldman, A. V. Malysheva, and D. Haroske was devoted to the investigationof the differential properties of generalized Bessel-Riesz potentials.In this paper we study the differential properties of potentials that generalize the classicalBessel-Riesz potentials. Potential kernels can have nonpower singularities in the neighborhoodof the origin. Their behavior at infinity is related only to the integrability condition, so thatkernels with a compact support are included. In this connection, the spaces of generalized Besselpotentials generated by them belong to the so-called spaces of generalized smoothness. The casewith the satisfied criterion for embedding potentials in the space of continuous bounded functionsis considered. In this case, the differential properties of the potentials are expressed in termsof the behavior of their module of continuity in the uniform metric. Criteria for embedding ofpotentials in Calderon spaces are established and explicit descriptions of the module of continuityof potentials and optimal spaces for such embeddings are obtained in the case when the basespace for potentials is the Lorentz weight space. These results specify the general constructionsestablished in previous works.

Analytic dependence of a periodic analog of a fundamental solution upon the periodicity parameters

We prove an analyticity result in Sobolev–Bessel potential spaces for the periodic analog of the fundamental solution of a general elliptic partial differential operator upon the parameters which determine the periodicity cell. Then we show concrete applications to the Helmholtz and the Laplace operators. In particular, we show that the periodic analogs of the fundamental solution of the Helmholtz and of the Laplace operator are jointly analytic in the spatial variable and in the parameters which determine the size of the periodicity cell. The analysis of the present paper is motivated by the application of the potential theoretic method to periodic anisotropic boundary value problems in which the “degree of anisotropy” is a parameter of the problem.

Multipliers in spaces of Bessel potentials: The case of indices of nonnegative smoothness

The aim of the paper is to study spaces of multipliers acting from the Bessel potential space H (ℝ n ) to the other Bessel potential space H (ℝ n ). We obtain conditions ensuring the equivalence of uniform and standard multiplier norms on the space of multipliers $$M\left[ {H_p^s({\mathbb{R}^n}) \to H_q^t({\mathbb{R}^n})} \right]fors,t \in \mathbb{R},p,q > 1.$$ In the case $$p,q > 1,p \leqslant q,s > \frac{n}{p},t \geqslant 0,s - \frac{n}{p} \geqslant t - \frac{n}{q}$$ , the space M[H (ℝ n ) → H (ℝ n ) can be described explicitly. Namely, we prove in this paper that the latter space coincides with the space H , unif (ℝ n ) of uniformly localized Bessel potentials introduced by Strichartz. It is also proved that if both smoothness indices s and t are nonnegative, then such a description is possible only for the given values of the indices.

Mixed boundary value problems for the Laplace–Beltrami equation

We investigate the mixed Dirichlet–Neumann boundary value problems for the Laplace–Beltrami equation on a smooth bounded surface with a smooth boundary in non-classical setting in the Bessel potential space for , . To the initial BVP we apply a quasi-localization and obtain a model BVP for the Laplacian. The model mixed BVP on the half plane is investigated by the potential method and is reduced to an equivalent system of Mellin convolution equations in Sobolev–Slobodečkii space. Boundary integral equations are ivestigated in both Bessel potential and Sobolev–Slobodečkii spaces. The symbol of the obtained system is written explicitly and is responsible for the Fredholm properties and the index of the system. An explicit criterion for the unique solvability of the initial BVP in the non-classical setting is derived as well.

H-distributions with unbounded multipliers

We investigate H-distributions for sequences in the dual pairs of Bessel spaces, \((H^q_s , H^{p}_{-s}), s\in \mathbb {R}, q>1\) and \(q=p/(p-1),\) by the use of unbounded multipliers, with the finite regularity, as test functions. The results relating weak convergence, H-distributions and strong convergence are applied in the analysis of strong convergence for a sequence of approximated solutions to a class of differential equations \(P(x,D)u_n=f_n\), where P(x, D) is a differential operator of order k with coefficients in the Schwartz class and \((f_n)\) is a strongly convergent sequence in an appropriate Bessel potential space.

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L p -L q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L p -L q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L p -L q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Napoli, Drelichman and Duran, and Duoandikoetxea.

Sobolev Spaces on Non-Lipschitz Subsets of $${\mathbb {R}}^n$$ R n with Application to Boundary Integral Equations on Fractal Screens

We study properties of the classical fractional Sobolev spaces (or Bessel potential spaces) on non-Lipschitz subsets of $\mathbb{R}^n$. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschitz, including cases where the screen is fractal or has fractal boundary.

Multipliers in Spaces of Bessel Potentials: The Case of Indices of Nonnegative Smoothness

Цель работы - изучить пространства мультипликаторов, действующих из одного пространства бесселевых потенциалов $H^s_p(\mathbb{R}^n)$ в другое пространство бесселевых потенциалов $H^t_q(\mathbb{R}^n)$. Найдены условия, обеспечивающие эквивалентность равномерной и стандартной мультипликаторных норм на пространстве мультипликаторов $$ M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]\qquad при\quad s,t \in \mathbb{R},\quad p,q &gt; 1. $$ В случае $$ p,q &gt; 1,\qquad p \leqslant q,\qquad s &gt; \frac np,\qquad t \geqslant 0,\qquad s-\frac np \geqslant t-\frac nq $$ пространство $M[H^s_p(\mathbb{R}^n) \to H^t_q(\mathbb{R}^n)]$ удается описать явно. А именно, в работе доказано, что оно совпадает с введенным Р. С. Стрихартцем пространством $H^t_{q,\mathrm{unif}}(\mathbb{R}^n)$ равномерно локализованных бесселевых потенциалов. Доказано также, что если оба показателя гладкости $s$ и $t$ неотрицательны, то такое описание возможно только при указанных значениях индексов. Библиография: 16 названий.

Radial symmetry results for Bessel potential integral equations in exterior domains and in annular domains

The purpose of this paper is to investigate positive solutions of integral equations involving Bessel potential. Exploiting the moving plane method in integral form, we give the radial symmetry of both the domain and solutions of our integral equations in exterior domains and in annular domains respectively.

On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

This paper concerns the following question: given a subset [Formula: see text] of [Formula: see text] with empty interior and an integrability parameter [Formula: see text], what is the maximal regularity [Formula: see text] for which there exists a non-zero distribution in the Bessel potential Sobolev space [Formula: see text] that is supported in [Formula: see text]? For sets of zero Lebesgue measure, we apply well-known results on set capacities from potential theory to characterize the maximal regularity in terms of the Hausdorff dimension of [Formula: see text], sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of [Formula: see text], together with the sets of values of [Formula: see text] for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterizing the regularity that can be achieved on certain special classes of sets, such as [Formula: see text]-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.

Difference norms for vector-valued Bessel potential spaces with an application to pointwise multipliers

In this paper we prove a randomized difference norm characterization for Bessel potential spaces with values in UMD Banach spaces. The main ingredients are R-boundedness results for Fourier multiplier operators, which are of independent interest. As an application we characterize the pointwise multiplier property of the indicator function of the half-space on these spaces. All results are proved in the setting of weighted spaces.

Embedding of Sobolev spaces with limit exponent revisited

An embedding of the Sobolev spaces Wps (ℝn) in Lizorkin-type spaces of locally integrable functions of smoothness zero is obtained; a similar assertion for Riesz and Bessel potentials is presented. The embedding theorem is extended to Sobolev spaces on irregular domains in n-dimensional Euclidean space. The statement of the theorem depends on geometric parameters of the domain of functions.

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley &amp; Sons, Ltd.

Fredholm theory connected with a Douglis–Nirenberg system of differential equations over an exterior region in

We consider a boundary problem over an exterior subregion of for a Douglis–Nirenberg system of differential operators under limited smoothness asumptions and under the assumption of parameter‐ellipticity in a closed sector in the complex plane with vertex at the origin. We pose the problem in an Sobolev–Bessel potential space setting, , and denote by the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for for values of the spectral parameter λ lying in as well as results pertaining to the invariance of the Fredholm domain of with p.