Related papers: Two-Component Reduction of Nahm's Equations and Hy…
In this study, we devote our attention to the question of clarifying the existence of a weak solution to a class of quasilinear double-phase elliptic equations with logarithmic convection terms under some appropriate assumptions on data.…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
Considered herein is a multi-component Novikov equation, which admits bi-Hamiltonian structure, infinitely many conserved quantities and peaked solutions. In this paper, we deduce two blow-up criteria for this system and global existence…
We announce a new bi-Hamiltonian integrable two-component system admitting the scalar 3rd-order Burgers equation as a reduction.
We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence)…
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably…
This paper deals with the existence of solutions for an elliptic system of partial differential equations. The solution method is based on the sub- and super-solutions approach. An application to a stochastic control problem is presented.…
On a compact Riemannian manifold, we prove the existence of multiple solutions for an elliptic equation with critical Sobolev growth and critical Hardy potential.
In this paper, we establish the modified concavity inequality for complex Hessian equations under the semi-convexity assumption inspired by Lu \cite{Lu23} and Zhang \cite{Z24} for real case. Then second order estimates for admissible…
We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the…
In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a…
We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and…
The full ADHM-Nahm formalism is employed to find exact higher charge caloron solutions with non-trivial holonomy, extended beyond the axially symmetric solutions found earlier. Particularly interesting is the case where the constituent…
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial…
We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.
This paper considers a class of noncoercive nonlinear elliptic problems with coefficients defined in Marcinkiewicz and Lorentz spaces. We prove the existence of a solution for the corresponding Dirichlet problem and investigate the higher…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators…
We derive second order estimates for $\chi$-plurisubharmonic solutions of complex Hessian equations with right hand sides depending on gradients on compact Hermitian manifolds.