Search ResultsShowing 1-20 of 43
-
On Second-Order $L^\infty$ Variational Problems with Lower-Order Terms
Comments: 21 pagesCategories: math.APIn 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 $\Omega\subseteq \mathbb R^n$ and $\mathrm H : \Omega\times\big(\mathbb R \times\mathbb R^n \times \mathbb R^{n^{\otimes2}}_s \big) \to \mathbb R$, we consider the functional \[ \mathrm{E}_\infty(u, \mathcal{O}) :=\underset{ \mathcal{O}}{\mathrm{ess}\sup}\hspace{1mm}\mathrm H (\cdot,u,\mathrm D u,\mathrm D^2u ) , \ \ u\in W^{2,\infty}(\Omega), \ \mathcal{O} \subseteq \Omega \text{ measurable}. \] We establish the existence of minimisers subject to (first-order) Dirichlet data on $\partial \Omega$ under natural assumptions, and, when $n=1$, we also show the existence of absolute minimisers. We further derive a necessary fully nonlinear PDE of third-order which arises as the analogue of the Euler-Lagrange equation for absolute minimisers, and is given by $$ \ \ \mathrm H_{\mathrm X}(\cdot,u,\mathrm D u,\mathrm D^2u): \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)\otimes \mathrm D\big(\mathrm H(\cdot,u,\mathrm D u,\mathrm D^2u)\big)=0\ \ \text{ in }\Omega. $$ We then rigorously derive this PDE from smooth absolute minimisers, and prove the existence of generalised D-solutions to the (first-order) Dirichlet problem. Our work generalises the key results obtained in [26] which first studied problems of this type with pure Hessian dependence only, providing at the same time considerably simpler streamlined proofs.
-
arXiv:2403.12625 (Published 2024-03-19)
Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$
Comments: 41 pagesCategories: math.APWe study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(\Omega)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is a positive matrix, and $\Omega \Subset \mathbb R^n$. Higher order problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for $\mathrm F_\infty$. We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for $\mathrm F_\infty$; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on $\partial \Omega$, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$.
-
arXiv:2403.06727 (Published 2024-03-11)
Minimisers of supremal functionals and mass-minimising 1-currents
Categories: math.APWe study vector-valued functions that minimise the $L^\infty$-norm of their derivatives for prescribed boundary data. We construct a vector-valued, mass minimising $1$-current (i.e., a generalised geodesic) in the domain such that all solutions of the problem coincide on its support. Furthermore, this current can be interpreted as a streamline of the solutions. It also has maximal support among all $1$-currents with certain properties. The construction relies on a $p$-harmonic approximation. In the case of scalar-valued functions, it is closely related to a construction of Evans and Yu. We therefore obtain an extension of their theory.
-
arXiv:2303.15982 (Published 2023-03-28)
Variational problems in $L^\infty$ involving semilinear second order differential operators
Categories: math.APFor an elliptic, semilinear differential operator of the form $S(u) = A : D^2 u + b(x, u , Du)$, consider the functional $E_\infty(u) = \mathop{\mathrm{ess \, sup}}_\Omega |S(u)|$. We study minimisers of $E_\infty$ for prescribed boundary data. Because the functional is not differentiable, this problem does not give rise to a conventional Euler-Lagrange equation. Under certain conditions, we can nevertheless give a system of partial differential equations that all minimisers must satisfy. Moreover, the condition is equivalent to a weaker version of the variational problem.
-
Generalised second order vectorial $\infty$-eigenvalue problems
Comments: 19 pages, Journal: Proceedings of the Royal Society of Edinburgh A, MathematicsCategories: math.APWe consider the problem of minimising the $L^\infty$ norm of a function of the hessian over a class of maps, subject to a mass constraint involving the $L^\infty$ norm of a function of the gradient and the map itself. We assume zeroth and first order Dirichlet boundary data, corresponding to the ``hinged" and the ``clamped" cases. By employing the method of $L^p$ approximations, we establish the existence of a special $L^\infty$ minimiser, which solves a divergence PDE system with measure coefficients as parameters. This is a counterpart of the Aronsson-Euler system corresponding to this constrained variational problem. Furthermore, we establish upper and lower bounds for the eigenvalue.
-
arXiv:2202.12005 (Published 2022-02-24)
On isosupremic vectorial minimisation problems in $L^\infty$ with general nonlinear constraints
Comments: 27 pages, 1 tableWe 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 operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, jacobian and null Lagrangian constraints. Via the method of $L^p$ approximations as $p\to \infty$, we illustrate the existence of a special $L^\infty$ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained $L^\infty$ variational problem.
-
arXiv:2103.15911 (Published 2021-03-29)
Generalised vectorial $\infty$-eigenvalue nonlinear problems for $L^\infty$ functionals
Comments: 30 pagesCategories: math.APLet $\Omega \Subset \mathbb R^n$, $f \in C^1(\mathbb R^{N\times n})$ 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 \[ \big\| f(\mathrm D u) \big\|_{L^\infty(\Omega)} \! = \inf \Big\{\big\| f(\mathrm D v) \big\|_{L^\infty(\Omega)} \! : \ v \! \in W^{1,\infty}_0(\Omega;\mathbb R^N), \, \| g(v) \|_{L^\infty(\Omega)}\! =1\Big\}, \] under natural assumptions on $f,g$. This includes the $\infty$-eigenvalue problem as a special case. Herein we prove existence of a minimiser $u_\infty$ with extra properties, derived as the limit of minimisers of approximating constrained $L^p$ problems as $p\to \infty$. A central contribution and novelty of this work is that $u_\infty$ is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence $\infty$-Laplacian. Our results are new even in the scalar case of the $\infty$-eigenvalue problem.
-
arXiv:2005.09637 (Published 2020-05-18)
On the inverse source identification problem in $L^\infty$ for fully nonlinear elliptic PDE
Comments: 14 pages. arXiv admin note: text overlap with arXiv:1811.02845Categories: math.APIn this paper we generalise the results proved in [N. Katzourakis, An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.
-
arXiv:1812.10988 (Published 2018-12-28)
On the numerical approximation of vectorial absolute minimisers in $L^\infty$
Comments: 20 pages, 54 figuresLet $\Omega$ be an open set. We consider the supremal functional \[ \tag{1} \label{1} \ \ \ \ \ \ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm D u \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq \Omega \text{ open}, \] applied to locally Lipschitz mappings $u : \mathbb R^n \supseteq \Omega \longrightarrow \mathbb R^N$, where $n,N\in \mathbb N$. This is the model functional of Calculus of Variations in $L^\infty$. The area is developing rapidly, but the vectorial case of $N\geq 2$ is still poorly understood. Due to the non-local nature of \eqref{1}, usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $n,N\geq 2$. Herein we present numerical experiments based on a new method recently proposed by the first author in the papers [33, 35].
-
arXiv:1812.10319 (Published 2018-12-26)
Inverse optical tomography through PDE constrained optimisation in $L^\infty$
Comments: 27 pagesCategories: math.APFluorescent Optical Tomography (FOT) is a new bio-medical imaging method with wider industrial applications. It is currently intensely researched since it is very precise and with no side effects for humans, as it uses non-ionising red and infrared light. Mathematically, FOT can be modelled as an inverse parameter identification problem, associated with a coupled elliptic system with Robin boundary conditions. Herein we utilise novel methods of Calculus of Variations in $L^\infty$ to lay the mathematical foundations of FOT which we pose as a PDE-constrained minimisation problem in $L^p$ and $L^\infty$.
-
arXiv:1812.10093 (Published 2018-12-25)
A minimisation problem in ${\mathrm{L}}^\infty$ with PDE and unilateral constraints
Comments: 26 pagesCategories: math.APWe study the minimisation of a cost functional which measures the misfit on the boundary of a domain between a component of the solution to a certain parametric elliptic PDE system and a prediction of the values of this solution. We pose this problem as a PDE-constrained minimisation problem for a supremal cost functional in ${\mathrm{L}}^\infty$, where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in ${\mathrm{L}}^p$ as $p\to\infty$ and the generalised Kuhn-Tucker theory to derive the relevant variational inequalities in ${\mathrm{L}}^p$ and ${\mathrm{L}}^\infty$. These results are motivated by the mathematical modelling of the novel bio-medical imaging method of Fluorescent Optical Tomography.
-
arXiv:1812.10069 (Published 2018-12-25)
On a vector-valued generalisation of viscosity solutions for general PDE systems
Comments: 34 pages, 6 figuresCategories: math.APWe propose a theory of non-differentiable solutions which applies to fully nonlinear PDE systems and extends the theory of viscosity solutions of Crandall-Ishii-Lions to the vectorial case. Our key ingredient is the discovery of a notion of extremum for maps which extends min-max and allows "nonlinear passage of derivatives" to test maps. This new PDE approach supports certain stability and convergence results, preserving some basic features of the scalar viscosity counterpart. In this first part of our two-part work we introduce and study the rudiments of this theory, leaving applications for the second part.
-
arXiv:1812.03378 (Published 2018-12-08)
Vectorial variational principles in $L^\infty$ and their characterisation through PDE systems
Comments: 12 pagesCategories: math.APWe 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 absolute minimisers and the concept of $\infty$-minimal maps introduced more recently by the second author. We prove that $C^1$ absolute minimisers characterise a divergence system with parameters probability measures and that $C^2$ $\infty$-minimal maps characterise Aronsson's PDE system. Since in the scalar case these different variational concepts coincide, it follows that the non-divergence Aronsson's equation has an equivalent divergence counterpart.
-
arXiv:1811.02845 (Published 2018-11-07)
An $L^\infty$ regularisation strategy to the inverse source identification problem for elliptic equations
Comments: 20 pagesCategories: math.APIn this paper we utilise new methods of Calculus of Variations in $L^\infty$ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in $L^2$ is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain.
-
arXiv:1801.09756 (Published 2018-01-29)
Counterexamples in Calculus of Variations in $L^\infty$ through the vectorial Eikonal equation
Comments: 5 pagesCategories: math.APWe show that for any regular bounded domain $\Omega\subseteq \mathbb R^n$, there exist infinitely many global diffeomorphisms equal to the identity on $\partial \Omega$ which solve the Eikonal equation. We also provide explicit examples of such maps on annular domains. This implies that the $\infty$-Laplace system arising in vectorial Calculus of Variations in $L^\infty$ does not suffice to characterise either limits of $p$-Harmonic maps as $p\to \infty$, or absolute minimisers in the sense of Aronsson.
-
arXiv:1704.04492 (Published 2017-04-14)
Rigidity and flatness of the image of certain classes of mappings having tangential Laplacian
Comments: 15 pages, 2 figuresCategories: math.APIn this paper we consider the PDE system of vanishing normal projection of the Laplacian for $C^2$ maps $u : \mathbb{R}^n \supseteq \Omega \longrightarrow \mathbb{R}^N$: \[ [\![\mathrm{D} u]\!]^\bot \Delta u = 0 \ \, \text{ in }\Omega. \] This system has discontinuous coefficients and geometrically expresses the fact that the Laplacian is a vector field tangential to the image of the mapping. It arises as a constituent component of the $p$-Laplace system for all $p\in [2,\infty]$. For $p=\infty$, the $\infty$-Laplace system is the archetypal equation describing extrema of supremal functionals in vectorial Calculus of Variations in $L^\infty$. Herein we show that the image of a solution $u$ is piecewise affine if either the rank of $\mathrm{D} u$ is equal to one or $n=2$ and $u$ has additively separated form. As a consequence we obtain corresponding flatness results for $p$-Harmonic maps, $p\in [2,\infty]$.
-
arXiv:1703.03648 (Published 2017-03-10)
The Eigenvalue Problem for the $\infty$-Bilaplacian
Comments: 22 pagesCategories: math.APWe consider the problem of finding and describing minimisers of the Rayleigh quotient \[ \Lambda_\infty \, :=\, \inf_{u\in \mathcal{W}^{2,\infty}(\Omega)\setminus\{0\} }\frac{\|\Delta u\|_{L^\infty(\Omega)}}{\|u\|_{L^\infty(\Omega)}}, \] where $\Omega \subseteq \mathbb{R}^n$ is a bounded $C^{1,1}$ domain and $\mathcal{W}^{2,\infty}(\Omega)$ is a class of weakly twice differentiable functions satisfying either $u=0$ or $u=|\mathrm{D} u|=0$ on $\Omega$. Our first main result, obtained through approximation by $L^p$-problems as $p\to \infty$, is the existence of a minimiser $u_\infty \in \mathcal{W}^{2,\infty}(\Omega)$ satisfying \[ \left\{ \begin{array}{ll} \Delta u_\infty \, \in \, \Lambda_\infty \mathrm{Sgn}(f_\infty) & \text{ a.e. in }\Omega, \\ \Delta f_\infty \, =\, \mu_\infty & \text{ in }\mathcal{D}'(\Omega), \end{array} \right. \] for some $f_\infty\in L^1(\Omega)\cap BV_{\text{loc}}(\Omega)$ and a measure $\mu_\infty \in \mathcal{M}(\Omega)$, for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue $\Lambda_\infty$ on the domain, establishing the validity of a Faber-Krahn type inequality: among all $C^{1,1}$ domains with fixed measure, the ball is a strict minimiser of $\Omega \mapsto \Lambda_\infty(\Omega)$. This result is shown to hold true for either choice of boundary conditions and in every dimension.
-
arXiv:1701.07415 (Published 2017-01-25)
On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functions
Comments: 21 pages, 5 figuresIn [KP16] (https://arxiv.org/pdf/1605.07880) the authors introduced a second order variational problem in $L^{\infty}$. The associated equation, coined the $\infty$-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by $\Delta^2_\infty u\, := (\Delta u)^3 | D (\Delta u) |^2 = 0.$ In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call $\infty$-Biharmonic functions. For fixed $p$ we design a mixed finite element scheme for the pre-limiting equation, the $p$-Bilaplacian $\Delta^2_p u\, := \Delta(| \Delta u |^{p-2} \Delta u) = 0.$ We prove convergence of the numerical solution to the weak solution of $\Delta^2_p u = 0$ and show that we are able to pass to the limit $p\to\infty$. We perform various tests aimed at understanding the nature of solutions of $\Delta^2_\infty u$ and in 1-$d$ we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of $\mathcal D$-solutions.
-
arXiv:1701.03348 (Published 2017-01-12)
Existence, Uniqueness and Structure of Second Order absolute minimisers
Comments: 16 pagesCategories: math.APLet $\Omega \subseteq \mathbb{R}^n$ be a bounded open $C^{1,1}$ set. In this paper we prove the existence of a unique second order absolute minimiser $u_\infty$ of the functional \[ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm{F}(\cdot, \Delta u) \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq \Omega \text{ measurable}, \] with prescribed boundary conditions for $u$ and $\mathrm{D} u$ on $\partial \Omega$ and under natural assumptions on $\mathrm{F}$. We also show that $u_\infty$ is partially smooth and there exists a harmonic function $f_\infty \in L^1(\Omega)$ such that \[ \mathrm{F}(x, \Delta u_\infty(x)) \, =\, e_\infty\, \mathrm{sgn}\big(f_\infty(x)\big) \] for all $x \in \{f_\infty \neq 0\}$, where $e_\infty$ is the infimum of the global energy.
-
arXiv:1611.05936 (Published 2016-11-18)
A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in $L^\infty$
Comments: 13 pagesCategories: math.APLet $n,N\in \mathbb{N}$ with $\Omega \subseteq \mathbb{R}^n$ open. Given $H \in C^2(\Omega \times \mathbb{R}^N\times \mathbb{R}^{Nn}),$ we consider the functional \[ \tag{1} \label{1} E_\infty (u,\mathcal{O})\, :=\, \underset{\mathcal{O}}{\mathrm{ess}\,\sup}\, H (\cdot,u,\mathrm{D} u) ,\ \ \ u\in W^{1,\infty}_\text{loc}(\Omega,\mathbb{R}^N),\ \ \ \mathcal{O} \Subset \Omega. \] The associated PDE system which plays the role of Euler-Lagrange equations in $L^\infty$ is \[ \label{2} \tag{2} \left\{ \begin{array}{r} H_{P}(\cdot, u, \mathrm{D}u)\, \mathrm{D} \big(H(\cdot, u, \mathrm{D} u)\big) \, = \, 0, \ \ \ H(\cdot, u, \mathrm{D} u) \, [\![H_{P}(\cdot, u, \mathrm{D} u)]\!]^\bot \Big(\mathrm{Div}\big(H_{P}(\cdot, u, \mathrm{D} u)\big)- H_{\eta}(\cdot, u, \mathrm{D} u)\Big)\, =\, 0, \end{array} \right. \] where $[\![A]\!]^\bot := \mathrm{Proj}_{R(A)^\bot}$. Herein we establish that generalised solutions to \eqref{2} can be characterised as local minimisers of \eqref{1} for appropriate classes of affine variations of the energy. Generalised solutions to \eqref{2} are understood as $\mathcal{D}$-solutions, a general framework recently introduced by one of the authors.