Two Forms of An Inverse Operator to the Generalized Bessel Potential

In this paper, we treat a convolution-type operator called the generalized Bessel potential. Our main result is the derivation of two different forms of its inversion. The first inversion is provided in terms of an approximative inverse operator using the method of an improving multiplier. The second one employs the regularization technique for the divergent integrals in the form of the appropriate segments of the Taylor–Delsarte series.

CONVOLUTION EQUATIONS ON THE ABELIAN GROUP $\cA(-1,1)$

The interval $j=[-1,1]$ turns into an Abelian group $\cA(\cJ)$ under the group operation $x+_\cJ y:=(x+y)(1+xy)^{-1},\qquad x,y\in\cJ$. This enables definition of the invariant measure $d_\cJ x=(1-x^2)^{-1}dx$ and the Fourier transform $\cF_\cJ$ on the interval $\cJ$ and, as a consequence, we can consider Fourier convolution operators $W^0_{\cJ,\cA}:=\cF_\cJ^{-1}\cA\cF_\cJ$ on $\cJ$. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative $\fD_\cJ u(x)=-(1-x^2)u’(x)$, $t\in\cJ$. Equations are solved in the scale of Bessel potential $\bH^s_p(\cJ,d_\cJ x)$, $1\leqslant p\leqslant\infty$, and H\”older-Zygmound $\bZ^\nu(\cJ,(1-x^2)^\mu)$, $0<\mu,\nu<\infty$ spaces, adapted to the group $\cA(\cJ)$. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol $\cA(\xi)$, $\xi\in\bR$, of a convolution equation $W^0_{\cJ,\cA}u=f$ defines solvability: the equation is uniquely solvable if and only if the symbol $\cA$ is elliptic. The solution is written explicitely with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Abelian group $\cA(\cJ^n)$.

Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order α for α > 1. Based on the properties of the convolution operator, we apply a global bifurcation technique to show the existence of a highest, even, 2 π-periodic traveling-wave solution. The regularity of this wave is proved to be exactly Lipschitz.

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{1}{p}, 1<p<\infty. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain \Omega_{\alpha}\subset\mathbb{R}^{2} of magnitude 𝛼. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J.27 (2020), 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \mathbb{R}^{+} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys.60 (2013), 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl.443 (2016), 2, 707–731].

Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain

The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces {\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, {s>\frac{1}{p}}, {1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.