Searcharxiv⌕ Search

arXiv subjects

Nikos Katzourakis

Publications and source records attributed to Nikos Katzourakis.

51 records · Page 3Linked to original sources
arXiv2016

Existence of 1D vectorial Absolute Minimisers in $L^\infty$ under minimal assumptions

We prove the existence of vectorial Absolute Minimisers in the sense of Aronsson to the supremal functional $E_\infty(u,Ω') = \|\mathscr{L}(\cdot,u,D u)\|_{L^\infty(Ω')}$, $Ω'\Subset Ω$, applied to $W^{1,\infty}$ maps $u:Ω\subseteq \mathbb{R}\longrightarrow \mathbb{R}^N$ with given boundary values. The assumptions on $\mathscr{L}($ are minimal, improving earlier existence results previously established by Barron-Jensen-Wang and by the second author.

math.AP↗
arXiv2016

On the Well-Posedness of Global Fully Nonlinear First Order Elliptic Systems

In the very recent paper [K1], the second author proved that for any $ f\in L^2(\mathbb{R}^n,\mathbb{R}^N)$, the fully nonlinear first order system $F(\cdot,\mathrm{D} u) =f$ is well posed in the so-called J.L. Lions space and moreover the unique strong solution $u:\mathbb{R}^n\longrightarrow \mathbb{R}^N$ to the problem satisfies a quantitative estimate. A central ingredient in the proof was the introduction of an appropriate notion of ellipticity for $F$ inspired by Campanato's classical work in the 2nd order case. Herein we extend the results of [K1] by introducing a new strictly weaker ellipticity condition and by proving well posedness in the same "energy" space.

math.AP↗
arXiv2016

Existence and Uniqueness of Global Strong Solutions to Fully Nonlinear Second Order Elliptic Systems

We consider the problem of existence and uniqueness of strong a.e. solutions $u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE system \[\label{1} \tag{1} F(\cdot,D^2u ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \] when $ f\in L^2(\mathbb{R}^n)^N$ and $F$ is a Carathéodory map. \eqref{1} has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato's ellipticity condition on $F$. By introducing a new much weaker notion of ellipticity, we prove solvability of \eqref{1} in a tailored Sobolev "energy" space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a "perturbation device" which allows to use Campanato's near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form \eqref{1}.

math.AP↗
arXiv2015

On the Numerical Approximation of $\infty$-Harmonic Mappings

Given a map $u : Ω\subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, the $\infty$-Laplacian is the system \[ \label{1} Δ_\infty u \, :=\, \Big(\text{D}u \otimes \text{D}u + |\text{D}u|^2 [\text{D}u]^\bot \! \otimes I \Big) : \text{D}^2 u\, = \, 0. \tag{1} \] \eqref{1} is the model system of vectorial Calculus of Variations in $L^\infty$ and arises as the "Euler-Lagrange" equation in relation to the supremal functional \[ \label{2} E_\infty(u,Ω)\, :=\, \| \text{D}u \|_{L^\infty(Ω)}. \tag{2} \] The scalar case of \eqref{1} has been introduced by Aronsson in the 1960s and by now is relatively classical and well understood. The general system \eqref{1} has been discovered and studied by the first author in a series of recent papers. Supremal functionals are fundamental for applications because they provide more realistic models as opposed to conventional integral models. Herein we provide numerical approximations of solutions to the Dirichlet problem when $n=2$ and $N=2,3$ for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in $L^\infty$ and provide insights on the structure of general solutions and the natural separation to phases they present.

math.AP↗
arXiv2015

Mollification of $\mathcal{D}$-solutions to Fully Nonlinear PDE Systems

In a recent paper (arXiv:1501.06164) the author has introduced a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows the interpretation of merely measurable maps as solutions. This approach is duality-free and builds on the probabilistic representation of limits of difference quotients via Young measures over certain compactifications of the "state space". Herein we establish a systematic regularisation scheme of this notion of solution which, by analogy, is the counterpart of the usual mollification by convolution of weak solutions and of the mollification by sup/inf convolutions of viscosity solutions.

math.AP↗
arXiv2015

Remarks on the validity on the maximum principle for the $\infty$-Laplacian

In this note we give three counter-examples which show that the Maximum Principle generally fails for classical solutions of a system and a single equation related to the $\infty$-Laplacian. The first is the tangential part of the $\infty$-Laplace system and the second is the scalar $\infty$-Laplace equation perturbed by a linear gradient term. The interpretations of the Maximum Principle for the system are that of the Convex Hull Property and also of the Maximum Principle of the modulus of the solution.

math.AP↗
arXiv2015

On the Dirichlet Problem for Fully Nonlinear Elliptic Hessian Systems

We consider the problem of existence and uniqueness of strong solutions $u: Ω\subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ in $(H^{2}\cap H^{1}_0)(Ω)^N$ to the problem \[\label{1} \tag{1} \left\{ \begin{array}{l} F(\cdot,D^2u ) \,=\, f, \ \ \text{ in }Ω,\\ \hspace{31pt} u\,=\, 0, \ \ \text{ on }\partial Ω, \end{array} \right. \] when $ f\in L^2(Ω)^N$, $F$ is a Carathéodory map and $Ω$ is convex. \eqref{1} has been considered by several authors, firstly by Campanato and under Campanato's ellipticity condition. By employing a new weaker notion of ellipticity introduced in recent work of the author [K2] for the respective global problem on $\mathbb{R}^n$, we prove well-posedness of \eqref{1}. Our result extends existing ones under hypotheses weaker than those known previously. An essential part of our analysis in an extension of the classical Miranda-Talenti inequality to the vector case of 2nd order linear hessian systems with rank-one convex coefficients.

math.AP↗
arXiv2015

On the Loss of Compactness in the Vectorial Heteroclinic Connection Problem

We give an alternative proof of the theorem of Alikakos-Fusco [AF] concerning existence of heteroclinic solutions to a Hamiltonian ODE system on the whole real line which arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of [AF] and establishes existence by analysing the loss of compactness in minimising sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.

math.CA↗
arXiv2015

On Linear Degenerate Elliptic PDE Systems with Constant Coefficients

Let $\textbf{A}$ be a symmetric convex quadratic form on $\mathbb{R}^{Nn}$ and $Ω\Subset \mathbb{R}^n$ a bounded convex domain. We consider the problem of existence of solutions $u: Ω\subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the problem \[ \tag{1} \label{1} \begin{split} \sum_{β=1}^N\sum_{i,j=1}^n \textbf{A}_{αi βj}\, D^2_{ij}u_β\,=\, f_α, \text{ in }Ω,\quad \ u\,=\, 0, \text{ on }\partial Ω, \end{split} \] when $ f\in L^2(Ω,\mathbb{R}^N)$. \eqref{1} is degenerate elliptic and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not have even distributional solutions. By introducing an extension of distributions adapted to \eqref{1}, we prove existence, partial regularity and by imposing an extra condition uniqueness as well. The satisfaction of the boundary condition is also an issue due to the low regularity of the solution. The motivation to study \eqref{1} and the method of the proof arose from recent work of the author [K4, arXiv:1501.06164] on generalised solutions for fully nonlinear systems.

math.AP↗
arXiv2014

Existence and Uniqueness of Global Solutions to Fully Nonlinear First Order Elliptic Systems

Let $F : \mathbb{R}^n \times \mathbb{R}^{N\times n} \rightarrow \mathbb{R}^N$ be a Caratheodory map. In this paper we consider the problem of existence and uniqueness of weakly differentiable global strong a.e. solutions $u: \mathbb{R}^n \longrightarrow \mathbb{R}^N$ to the fully nonlinear PDE system \[\label{1} \tag{1} F(\cdot,Du ) \,=\, f, \ \ \text{ a.e. on }\mathbb{R}^n, \] when $ f\in L^2(\mathbb{R}^n)^N$. This problem has not been considered before. By introducing an appropriate notion of ellipticity, we prove existence of solution to \eqref{1} in a tailored Sobolev "energy" space (known also as the J.L. Lions space) and a uniqueness a priori estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods and a "perturbation device" which allows to use of Campanato's notion of near operators, an idea developed for the 2nd order case.

math.AP↗
arXiv2014

Optimal Infinity-Quasiconformal Immersions

For a Hamiltonian $K \in C^2(\mathbb{R}^{N \times n})$ and a map $u:Ω\subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,Ω) \ :=\ \big\|K(Du)\big\|_{L^\infty(Ω)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system \[ \label{2} A_\infty u \, :=\, \Big(K_P \otimes K_P + K[K_P]^\bot K_{PP}\Big)(Du):D^2 u \, = \, 0. \tag{2} \] Here $K_P$ is the derivative and $[K_P]^\bot$ is the projection on its nullspace. \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author \cite{K1}-\cite{K6}. Herein we apply our results to Geometric Analysis by choosing as $K$ the dilation function \[ K(P)={|P|^2}{\det(P^\top P)^{-1/n}} \] which measures the deviation of $u$ from being conformal. Our main result is that appropriately defined minimisers of \eqref{1} solve \eqref{2}. Hence, PDE methods can be used to study optimised quasiconformal maps. Nonconvexity of $K$ and appearance of interfaces where $[K_P]^\bot$ is discontinuous cause extra difficulties. When $n=N$, this approach has previously been followed by Capogna-Raich \cite{CR} and relates to Teichmüller's theory. In particular, we disprove a conjecture appearing therein.

math.AP↗
arXiv2014

Nonuniqueness in Vector-Valued Calculus of Variations in $L^\infty$ and some Linear Elliptic Systems

For a Hamiltonian $H \in C^2(\mathbb{R}^{N \times n})$ and a map $u:Ω\subseteq \mathbb{R}^n /!\longrightarrow \mathbb{R}^N$, we consider the supremal functional \[ \label{1} \tag{1} E_\infty (u,Ω) \ :=\ \big\|H(Du)\big\|_{L^\infty(Ω)} . \] The "Euler-Lagrange" PDE associated to \eqref{1} is the quasilinear system \[ \label{2} \tag{2} A_\infty u := \Big(H_P \otimes H_P + H[H_P]^\bot /! H_{PP}\Big)(Du):D^2 u = 0. \] \eqref{1} and \eqref{2} are the fundamental objects of vector-valued Calculus of Variations in $L^\infty$ and first arose in recent work of the author [K1]. Herein we show that the Dirichlet problem for \eqref{2} admits for all $n=N\geq 2$ infinitely-many smooth solutions on the punctured ball, in the case of $H(P)=|P|^2$ for the $\infty$-Laplacian and of $H(P)= {|P|^2}{\det(P^\top /! P)^{-1/n}}$ for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system $A(x):D^2u =0$ follows as a corollary. Hence, the celebrated $L^\infty$ scalar uniqueness theory of Jensen [J] has no counterpart when $N\geq 2$. The key idea in the proofs is to recast \eqref{2} as a first order differential inclusion $Du(x) \in \mathcal{K} \subseteq \mathbb{R}^{n\times n}$, $x\in Ω$.

math.AP↗
arXiv2014

Nonsmooth Convex Functionals and Feeble Viscosity Solutions of singular Euler-Lagrange Equations

Let F be nonnegative, convex and smooth off a compact set K. We prove that continuous local minimisers of convex functionals are "very weak" viscosity solutions in the sense of Juutinen-Lindqvist of the highly singular Euler-Lagrange PDE expanded. The hypotheses on F do not guarrantee existence of minimising weak solutions and include the singular p-Laplacian for 1<p<2. A much deeper converse is also true, if K={0} and extra natural assumptions are satisfied. Our main advance is that we introduce systematic "flat" sup-convolution regularisations which apply to general singular nonlinear PDEs in order to cancel the strong singularity of F. As an application we extend a classical theorem of Calculus of Variations regarding existence for the Dirichlet problem. These results extends previous work of Julin-Juutinen and Juutinen-Lindqvist-Manfredi.

math.AP↗