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В статье рассматривается метод определения нелинейных дифракционных сил, действующих на судно в условиях мелководья на основании трехмерной потенциальной теории. Производится оценка влияния относительной глубины водоема Н/Т на данные нелинейные силы. Для нахождения нелинейных сил, требуется определение потенциалов второго порядка малости. Решение основано на методах малого параметра, интегральных уравнений Фредгольма и функции Грина для случая мелководья. Данное решение является новым для отечественной практики. При определении потенциалов второго порядка учитываются нелинейные граничные условия на свободной поверхности жидкости и на смоченной поверхности судна. На основании изложенного метода разработана программа расчета нелинейных сил при различных Н/Т. Приводятся результаты расчетов сил и моментов для трех различных типов судов. Приведено сравнение с расчетами, основанными на двумерной теории. Особое внимание уделяется учету потенциала набегающего волнения второго порядка. Учитывается его вклад в образовании нелинейных дифракционных сил. Приводится сравнение расчетов с учетом влияния данного потенциала и без него. The article discusses a method for determining nonlinear diffraction forces acting on a ship in shallow water conditions on the basis of a three-dimensional potential theory. The influence of the relative depth of the fluid Н/Т on these nonlinear forces is assessed. To determine them, it is necessary to calculate the potentials of the second order of smallness. The solution of the problem is carried out on the basis of small parameter methods, Fredholm’s integral equations and Green’s function for a fluid of limited depth. The presented solution in national practice is new. During the determination of the second order potentials, nonlinear boundary conditions on the free surface of the liquid and on the wetted surface of the ship are taken into account on the basis of methods program was developed, considering various Н/Т. The results of calculations of the forces and moments for three types of different ships are presented. Comparison with calculations based on two-dimensional theory is given. Special attention is paid to taking into account the potential of the second-order of incoming wave. Its contribution to the formation of nonlinear diffraction forces is taken into account. A comparison of calculations with and without the influence of this potential is given

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid., Comment: This is a corrected version of the published paper https://doi.org/10.1007/s11118-021-09947-8. Proofs of Lemmas 9.3 and 9.4 had minor gaps (estimates of integrals I_2 and IV in Lemma 9.3 and integral IV in Lemma 9.4 were incorrect). These are now fixed. We also corrected some typos

Let $$\mathcal {E}$$ be a symmetrically $$\Delta $$ -normed ideal in B(H). For $$1\le p

We prove a Korovkin-type theorem for sequences of positive linear operators on C([0, 1]). This theorem allows us to discuss some general results concerning positive linear operators. In particular, we give simple and efficient convergence criteria for the iterates of positive linear operators preserving the affine functions. By way of examples, we consider the Bernstein, the Meyer-Konig and Zeller, the q-Bernstein operators and their iterates.

Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak convergence with the unbounded absolutely weak convergence (uaw-convergence) in the definition of a continuous operator between Banach lattices. We characterize order continuous Banach lattices and reflexive Banach lattices in terms of these spaces of operators. Moreover, motivated by characterizing of a reflexive Banach lattice in terms of unbounded absolutely weakly Cauchy sequences, we consider pre-unbounded continuous operators between Banach lattices which maps uaw-Cauchy sequences to weakly (uaw- or norm) convergent sequences. This allows us to characterize KB-spaces and reflexive spaces in terms of these operators, too. Furthermore, we consider the unbounded Banach–Saks property as an unbounded version of the weak Banach–Saks property. There are many relations between spaces possessing the unbounded Banach–Saks property with spaces fulfilled by different types of the known Banach–Saks property. In particular, we characterize order continuous Banach lattices in terms of these relations, as well.

This paper introduces a ‘new’ class of operators. An operator T from a Banach lattice E into a Banach space Y will be called an LW-compact if T carries L-weakly compact sets of E into relatively compact sets of Y. Various characterizations of LW-compact operators are given. Also, we study the domination problem in this class of operators.

The Cauchy problem for the well-known Benjamin–Bona–Mahoney–Burgers equation in the class of Holder initial functions from $${{\mathbb{C}}^{{2 + \alpha }}}({{\mathbb{R}}^{3}})$$ with $$\alpha \in (0,1]$$ is considered. For such initial functions, it is proved that the Cauchy problem has a unique time-unextendable classical solution in the class $${{\mathbb{C}}^{{(1)}}}([0,T];{{\mathbb{C}}^{{2 + \lambda }}}({{\mathbb{R}}^{3}}))$$ for any $$T \in (0,{{T}_{0}});$$ moreover, either $${{T}_{0}} = + \infty $$ or $${{T}_{0}} < + \infty $$ and, in the latter case, $${{T}_{0}}$$ is the solution blow-up time. To prove the solvability of the Cauchy problem, we examine volume and surface potentials associated with the Cauchy problem in Holder spaces. Finally, a Schauder estimate is obtained.

A numerical wave tank (NWT) is developed by employing the local radial point interpolation collocation method, mixed Eulerian–Lagrangian approach, and the fourth-order Runge–Kutta method. The potential theory is used for a mathematical explanation of the wave-propagation problem. The Laplace equation in the Eulerian manner and the free surface conditions in the Lagrangian manner are used to simulate the fully nonlinear water waves. The incident waves are generated by imposing analytic forms of the potential on the upstream boundary. To avoid any reflection of the waves at the end of the tank, a damping zone is placed on the free surface before the downstream wall boundary. In order to demonstrate the efficiency and accuracy of the meshless NWT, the first-, second-, third-, and fifth-order Stokes waves are simulated and the numerical results are compared with the analytical solutions. In addition, the wave propagation of water waves in uneven bottom NWT is investigated. Fairly good agreements between the numerical results, the analytical solutions, and experimental data are observed.

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a Caratheodory reaction f(z, x) which is ( $$p-1$$ )-superlinear in $$x\in {\mathbb {R}}$$ but without satisfying the Ambrosetti–Rabinowitz condition. Using variational tools and critical groups, we prove a multiplicity theorem producing three nontrivial smooth solutions all with sign information (one positive, one negative and one nodal).

Since 2001, rough convergence has been discussed in research where the core factor degree of roughness has been treated only as a real number. Instead of analyzing different results based on the same definition, we start with the new flow of ideas to enlarge the spectrum of degree of roughness. As a consequence we invent a new approach to degree of roughness with approximation of neighborhood conception over locally solid Riesz spaces. While we start working with this new definition and scrutinize existing results, many of the earlier results are violated. Naturally our curiosity begins to know that under which condition(s) we may get back either some of the previous results or not? Apart from these, we consider various non-trivial results which enhance and signify the motivation of our paper.

We prove a uniform large deviations principle for the law of the solutions to a class of Burgers-type stochastic partial differential equations in any space dimension. The equation has nonlinearities of polynomial growth of any order, the driving noise is a finite dimensional Wiener process, and the proof is based on variational principle methods. We prove the uniform large deviations principle for the law of the solutions in two different topologies. First, in the C([0, T] : Lρ(D)) topology where the uniformity is over Lρ(D)-bounded sets of initial conditions, and secondly in the $C([0,T]\times \bar D)$ topology with uniformity being over bounded subsets in the $C_{0}(\bar D)$ norm.

This paper proves two-sided estimates for the Dirichlet heat kernel on inner uniform domains in metric measure Dirichlet spaces satisfying the volume doubling condition, the Poincaré inequality, and a cutoff Sobolev inequality. More generally, we obtain local upper and lower bounds for the Dirichlet heat kernel on locally inner uniform domains under local geometric assumptions on the underlying space.

Let T be a Calderon-Zygmund operator of type $$\omega $$ with $$\omega (t)$$ being nondecreasing and satisfying a kind of Dini’s type condition and let $$T_{\vec {b}}$$ be the multilinear commutators of T with $$BMO^m$$ functions. In this paper, we study the boundedness of the operators T and $$T_{\vec {b}}$$ on generalized weighted variable exponent Morrey spaces $$M^{p(\cdot ),\varphi }(w)$$ with the weight function w belonging to variable Muckenhoupt’s class $$A_{p(\cdot )}({{\mathbb {R}}^n})$$ . We find the sufficient conditions on the pair $$(\varphi _1,\varphi _2)$$ with $$\vec {b} \in BMO^m({{\mathbb {R}}^n})$$ which ensures the boundedness of the operators T and $$T_{\vec {b}}$$ from $$M^{p(\cdot ),\varphi _1}(w)$$ to $$M^{p(\cdot ),\varphi _2}(w)$$ .

This paper deals with (1, 1; r)-convexity of sequences. First, we prove several results on the sets of (1, 1; r)-convex sequences for various values of r. Then we show that the Cesaro matrix does not preserve (1, 1; r)-convexity of sequences for any $$r\ge 2$$ .

In this paper, primal and dual necessary/sufficient optimality conditions are investigated for vector optimization problem with set, cone and equality constraints. We establish some necessary optimality conditions for strict local Pareto minima in terms of contingent derivatives, contingent epiderivatives and contingent hypoderivatives with steady functions in Banach spaces to such problem. Under suitable assumptions, necessary optimality conditions become sufficient optimality conditions. Examples illustrate the applicability of the obtained results.

In this paper, we study sufficient conditions for the (strict) inverse-positivity of the linear fourth order operator corresponding to the one-dimensional beam equation with a spatially variable coefficient. We use a modification of results obtained by the operator reduction technique introduced by Schroder and show that the extrema of the coefficient can go beyond the originally derived bounds significantly.

Let $\mathcal {X}$ be a separable Hilbert space with norm $\left \|{\cdot }\right \|$ and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on $\mathcal {X}$ , let $F:\mathcal {X}\rightarrow \mathcal {X}$ be a (smooth enough) function and let W(t) be a $\mathcal {X}$ -valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator $A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}$ . We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation $$ \left\{\begin{array}{ll} dX(t,x)=\left( AX(t,x)+F(X(t,x))\right)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. $$ and in its associated transition semigroup $$ P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_{b}(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; $$ where $B_{b}(\mathcal {X})$ is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of $\mathcal {X}$ . More precisely there exists K := K(F, T) > 0 such that for every $\varphi \in B_{b}(\mathcal {X})$ , $x\in \mathcal {X}$ , t ∈ (0, T] and $h\in Q^{\alpha }(\mathcal {X})$ it holds $$ |P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|. $$

The article deals with the one-group approximation to the problem of parametric identification of the distribution of isotropic sources, providing the required configuration of the neutron field in vacuum.

In this paper we consider nonlocal energies defined on probability measures in the plane, given by a convolution interaction term plus a quadratic confinement. The interaction kernel is $-\log|z|+\alpha\, x^2/|z|^2, \; z=x+iy,$ with $-1 < \alpha< 1.$ This kernel is anisotropic except for the Coulombic case $\alpha=0.$ We present a short compact proof of the known surprising fact that the unique minimiser of the energy is the normalised characteristic function of the domain enclosed by an ellipse with horizontal semi-axis $\sqrt{1-\alpha}$ and vertical semi-axis $\sqrt{1+\alpha}.$ Letting $\alpha \to 1^-$ we find that the semicircle law on the vertical axis is the unique minimiser of the corresponding energy, a result related to interacting dislocations, and previously obtained by some of the authors. We devote the first sections of this paper to presenting some well-known background material in the simplest way possible, so that readers unfamiliar with the subject find the proofs accessible, Comment: 17 pages, minor modifications after referee's report

The characteristics of how graphite particles adhere to the walls of high-temperature gas-cooled reactors are very important for analyzing reactor source terms. In the present study, atomic force microscopy (AFM) is used to measure the particle adhesion force and the wall morphology, then Lennard-Jones potential theory and the finite element method (FEM) are coupled to calculate the adhesion force between different particles and rough walls. The obvious deviations between the AFM measurements and the theoretical models are due to the contact hypothesis of a sphere with a smooth flat plane in the latter. The FEM results reveal the formation of maximum adhesion at the pull-off point of the particle and the corresponding stress distribution. For aspherical particles, the local curvature of the particle contact interface is the main factor affecting the adhesion force. In addition, for rough walls, different contact regimes correspond to different adhesion trends. When discrete wall roughness with hemispheres and truncated cones is used, the FEM predictions are closer to the AFM measurements, indicating that roughness dominates the adhesion weakening.

We construct a solution of the Poisson equation in exterior domains $\Omega \subset \mathbb R^n,\;n \ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(\Omega),;1 < q

178 p. This thesis is devoted to the spectral study of two types of perturbation of theDirac operator, which are singular from the point of view of scaling.In the first part of this thesis, we consider the coupling of the Dirac operator with a combinationof delta-shell interactions of electrostatic, Lorentz scalar, and magnetic type supportedeither on regular compact surfaces or locally deformed hyperplanes. We develop an approachbased on regularization techniques that will allow us to describe the self-adjoint realizationof the perturbed Dirac operator for any combination of the coupling constants. We then investigatethe qualitative spectral properties of the various models using a Birman-Schwingerprinciple and a Krein-type formula relating the resolvent of the perturbed operator to thatof the free Dirac operator, and we pay special attention to the case of critical combinationsof coupling constants and those that give rise to the phenomenon of confinement.In the second part, we study the coupling of the Dirac operator with non-critical combinationsof delta interactions supported on non-regular compact surfaces. We first generalizethe results obtained in the context of regular surfaces to the case of surfaces locally coincidentwith the graph of a Lipschitz function whose gradient is bounded and has vanishing meanoscillations. For this we use some techniques from harmonic analysis, potential theory andFredholm¿s theory. Moreover, in the case of Hölder surfaces, we show how the smoothnessof the surface supporting the delta interactions affects the Sobolev regularity of the domainof the operator under consideration. In a second step, we investigate delta-interactions supportedon surfaces satisfying certain weak topological conditions. We first study the Diracoperator coupled with the electrostatic and Lorentz scalar delta-shell interactions supportedon uniformly rectifiable surfaces. Under certain conditions on the coupling constants, weprove the self-adjointness fo the perturbed operator and we establish several spectral propertiesin the Lipschitz case. In particular, we determine the essential spectrum of the perturbedoperator and we show that at most a finite number of eigenvalues can appear in the gap.Moreover, we fit these results to other delta-shell interactions and derive several models ofDirac operators that give rise to the confinement phenomenon.In the third part of this thesis, we are concerned the study of the pseudodifferentialproperties of Poincaré-Steklov (PS) operators associated with the Dirac operator with theMIT bag boundary condition. First, we show that the PS operators fit well into the frameworkof classical pseudodifferential operators. Then, we study the PS operators from the pointof view of semiclassical pseudodifferential operators, where the semiclassical parameter isgiven by the inverse of the mass. In particular, using some regularity properties of the MITbag operator, we show that the PS operators are zero-order semiclassical pseudodifferentialoperators, and we determine their semiclassical principal symbols. In a second step, we studythe Dirac operator coupled with a potential depending on an additional mass and supportedoutside a regular domain. When the additional mass is large enough, using the symboliccalculus and the properties of the PS operators, we establish a Krein-type formula relatingthe resolvent of the perturbed operator to that of the MIT bag operator. With its help, weshow that the perturbed operator converges in the norm resolvent sense towards the MITbag operator and give a sharp estimate of the convergence rate.

In this paper, we introduce polynomial versions of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property for Banach lattices. By using Fremlin projective Banach lattice tensor products, we obtain several characterizations of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property in terms of regular homogeneous polynomials on Banach lattices.

In the paper, we introduce higher-order tangent epiderivatives for set-valued maps. Then, we study some basic properties of these concepts. Finally, we establish some results on duality in set-valued optimization. Several examples are given to illustrate the obtained results.

We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric mean of (infinite or finite) nonnegative matrices that define positive operators on Banach sequence spaces. Some of these inequalities complement the known inequalities and some of them refine known inequalities. Several inequalities appear to be new even in the finite dimensional case.