Przemysław Górka

One-dimensional Klein-Gordon equation with logarithmic nonlinearities

In this paper, we consider the relativistic version of the logarithmic Schrodinger equation, namely the logarithmic Klein–Gordon equation. We show the existence of classical solutions, and we investigate weak solutions. Finally, we examine the traveling waves of this model.

The dual modified Korteweg-de Vries–Fokas–Qiao equation: Geometry and local analysis

We study a bi-hamiltonian equation with cubic nonlinearity shown to appear in the theory of water waves by Fokas, derived by Qiao using the two-dimensional Euler equation, and also known to arise as the dual of the modified Korteweg-de Vries equation thanks to work by Fokas, Fuchssteiner, Olver, and Rosenau. We present a quadratic pseudo-potential, we compute infinite sequences of local and nonlocal conservation laws, and we construct an infinite-dimensional Lie algebra of symmetries which contains a semi-direct sum of the sl(2,R)-loop algebra and the centerless Virasoro algebra. As an application we prove a theorem on the existence of smooth solutions, and we construct some explicit examples. Moreover, we consider the Cauchy problem and we prove existence and uniqueness of weak solutions in the Sobolev space Hq+2(R), q > 1/2.

Functional calculus via Laplace transform and equations with infinitely many derivatives

We study nonlocal linear equations of the form f(∂t)ϕ=J(t), t≥0, in which f is an entire function. We develop an appropriate functional calculus via Laplace transform, we solve the aforementioned equation completely in the space of exponentially bounded functions, and we analyze the delicate issue of the formulation of initial value problems for nonlocal equations.

A comparison principle for Hamilton-Jacobi equations with discontinuous Hamiltonians

We show a comparison principle for viscosity super- and subsolutions to HamiltonJacobi equations with discontinuous Hamiltonians. The key point is that the Hamiltonian depends upon u and it has a special structure. The supersolution must enjoy some additional regularity.

The initial value problem for ordinary differential equations with infinitely many derivatives

We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We develop a Lorentzian functional calculus via Laplace transform which allows us to interpret rigorously operators of the form f(∂t) on the half line, in which f is an analytic function. We find the most general solution to the equation in the space of exponentially bounded functions, and we also analyze in full detail the delicate issue of the initial value problem. In particular, we state conditions under which the solution ϕ admits a finite number of derivatives, and we prove rigorously that if a finite set of a priori data directly connected with our Lorentzian calculus is specified, then the initial value problem is well posed and it requires only a finite number of initial conditions.

Generalized Euclidean Bosonic String Equations

We consider nonlinear equations of the form p(Δ)u = U(x, u(x)), in which p is a real-valued function satisfying some suitable technical conditions, and Δ stands for the Laplacian operator. We formulate a functional calculus appropriate for the study of such equations, and we establish results on the existence and regularity of solutions to the Euclidean bosonic string equation Δexp(-c Δ) u = U(x, u(x)), and we introduce a functional calculus appropriate for the study of very general nonlinear equations depending on functions of the Laplace operator. We also prove that under some further technical conditions, these “nonlocal” equations admit smooth, and even realanalytic, solutions. Our motivation comes from recent developments in string theory and nonlocal cosmology.

Harmonic functions on metric measure spaces

We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipschitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. Finally, we discuss and prove the Liouville type theorems. Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

Brézis-Wainger Inequality on Riemannian Manifolds

The Brézis-Wainger inequality on a compact Riemannian manifold without boundary is shown. For this purpose, the Moser-Trudinger inequality and the Sobolev embedding theorem are applied.

Sobolev spaces on locally compact abelian groups: Compact embeddings and local spaces

We continue our research on Sobolev spaces on locally compact abelian (LCA) groups motivated by our work on equations with infinitely many derivatives of interest for string theory and cosmology. In this paper, we focus on compact embedding results and we prove an analog for LCA groups of the classical Rellich lemma and of the Rellich-Kondrachov compactness theorem. Furthermore, we introduce Sobolev spaces on subsets of LCA groups and study its main properties, including the existence of compact embeddings into -spaces.

Multipole matrix elements of Green function of Laplace equation

Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation.