Related papers: Generalised second order vectorial $\infty$-eigenv…

We study minimisation problems in $L^\infty$ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear…

In this paper we study $2$nd order $L^\infty$ variational problems, through seeking to minimise a supremal functional involving the Hessian of admissible functions as well as lower-order terms. Specifically, given a bounded domain…

We study a minimisation problem in $L^p$ and $L^\infty$ for certain cost functionals, where the class of admissible mappings is constrained by the Navier-Stokes equations. Problems of this type are motivated by variational data assimilation…

In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the…

Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…

We study a vectorial $L^\infty$-variational problem of second order, where the supremal functional depends on the vector function $u$ through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser…

Consider the supremal functional \[ \tag{1} \label{1} E_\infty(u,A) \,:=\, \|L(\cdot,u,D u)\|_{L^\infty(A)},\quad A\subseteq \Omega, \] applied to $W^{1,\infty}$ maps $u:\Omega\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$, $N\geq 1$.…

Let $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ and $g\in C^1(\mathbb R^N)$, where $N,n \in \mathbb N$. We study the minimisation problem of finding $u \in W^{1,\infty}_0(\Omega;\mathbb R^N)$ that satisfies \[ \big\|…

We discuss two distinct minimality principles for general supremal first order functionals for maps and characterise them through solvability of associated second order PDE systems. Specifically, we consider Aronsson's standard notion of…

We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural…

This paper is concerned with the P1 finite element approximation of the eigenvalue problem of second-order elliptic operators subject to the Dirichlet boundary condition. The focus is on the preservation of basic properties of the principal…

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error…

This paper is concerned with the Dirichlet eigenvalue problem associated to the $\infty$-Laplacian in metric spaces. We establish a direct PDE approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without…

This paper gives a framework to produce the lower bound of eigenvalues defined in a Hilbert space by the eigenvalues defined in another Hilbert space. The method is based on using the max-min principle for the eigenvalue problems.

In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$ and relate it to a divergence-form PDE, similarly to what…

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a…

We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…

Motivated, in particular, by the entropy-regularized optimal transport problem, we consider convex optimization problems with linear equality constraints, where the dual objective has Lipschitz $p$-th order derivatives, and develop two…

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for $k$-Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to…

In this paper we investigate a class of $2$-Hessian equations and establish a minimum principle for a $P$-function in the sense of L.E. Payne (see R. Sperb \cite{Sp81}). The analysis is based on a sharp matrix inequality providing an…