Related papers: Interplay between nonlinear potential theory and f…
In this paper, we show how the motion of physical fields, in particular the electromagnetic potential, is connected with the choice of a space and time decomposition of the background spacetime manifold. The relation of the field dynamics…
Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations…
The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All…
The properties of nonlinear PDEs that generate filtered solutions are explored with particular attention given to the constraints on the residual term. The analysis is carried out for nonlinear PDEs with an emphasis on evolution problems…
In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This…
We discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin. The main feature of these new free boundary problems is that…
The subject of PT-symmetry and its areas of application have been blossoming over the past decade. Here, we consider a nonlinear Schr\"odinger model with a complex potential that can be tuned controllably away from being PT-symmetric, as it…
We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a…
The paper concerns the theory of parabolic equations on a broad class of closed subsets of Euclidean space possessing a kind of tangent structure. A necessary framework for considering evolutionary problems is developed, and fundamental…
The main result establishes the existence of a solution in a generalized sense for a nonlinear Dirichlet problem driven by a competing operator and exhibiting a convection term composed with an intrinsic operator. A finite dimensional…
We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The nonlinear SUSY for complex potentials is considered and the theorems…
Motivated by the famous Waddington's epigenetic landscape metaphor in developmental biology, biophysicists and applied mathematicians made different proposals to realize this metaphor in a rationalized way. We adopt comprehensive…
We propose that the behavior of asymmetric binary fluid mixtures with a large class of attractive or repulsive interparticle interactions can be understood by mapping onto effective non-additive hard-sphere models. The latter are best…
Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2x2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of…
Dyson's model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of particles with strength proportional to the inverse of distances with proportionality constant $\beta/2$. We give…
The theory of complete generalized Jordan sets is employed to reduce the PDE with the irreversible linear operator $B$ of finite index to the regular problems. It is demonstrated how the question of the choice of boundary conditions is…
The double-layer potential plays an important r$\hat{\rm o}$le in solving boundary value problems of elliptic equations. Here, in this paper, we aim at introducing and investigating double layer potentials for a generalized bi-axially…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
We prove the existence and uniqueness of weak solution of a Neumann boundary problem for an elliptic partial differential equation (PDE for short) with a singular divergence term which can only be understood in a weak sense. A probabilistic…
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem…