Related papers: A variational Approach to complex Hessian equation…
In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of…
In this paper we provide a different approach for existence of the variational solutions of the gradient flows associated to functionals on Sobolev spaces studied in \cite{BDDMS20}. The crucial condition is the convexity of the functional…
Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently,…
We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation $(-\Delta)^s u = h$ in $\Omega$ and $u=0$ in $\Omega^c$, where…
The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…
New necessary and sufficient conditions are given for the quantization of a class of periodic second order non-homogeneous ordinary differential equations in the complex plane in this paper. The problem is studied from the viewpoint of…
Although many programs have been published for fully numerical Hartree--Fock (HF) or density functional (DF) calculations on atoms, we are not aware of any that support hybrid DFs, which are popular within the quantum chemistry community…
We study linear and semi-linear wave, heat, and Schr\"odinger equations defined by Kre\u{\i}n-Feller operator $-\Delta_\mu$ on a complete Riemannian $n$-manifolds $M$, where $\mu$ is a finite positive Borel measure on a bounded open subset…
The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when…
In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=\mu\,\partial…
Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the…
For any $\alpha $ small, we construct infinitely many $C^{1,\alpha}$ very weak solutions to the 2-Hessian equation with prescribed boundary value. The proof relies on the convex integration method and cut-off technique.
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
We prove existence and uniqueness results of positive viscosity solutions of fully nonlinear degenerate elliptic equations with power-like zero order perturbations in bounded domains. The principal part of such equations is either…
We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which…
Post buckling problem of a large deformed beam is analyzed using canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy…
In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll}…
A Maxwell-Stefan system for fluid mixtures with driving forces depending on Cahn-Hilliard-type chemical potentials is analyzed. The corresponding parabolic cross-diffusion equations contain fourth-order derivatives and are considered in a…
We consider the vectorial system \[ \begin{cases} Du \in \mathcal{O}(2), & \mbox{a.e. in}\;\Omega, u=0, & \mbox{on} \;\partial \Omega, \end{cases} \] where $\Omega$ is a subset of $\R^2$, $u:\Omega\to \R^2$ and $\mathcal{O}(2)$ is the…
In this paper, we first study the comparison principle for the operator $H_{\chi,m}$. This result is used to solve certain weighted complex $m-$ Hessian equations.