We study the two dimensional least gradient problem in convex polygonal sets in the plane, $Ω$. We show the existence of solutions when the boundary data $f$ are attained in the trace sense. The main difficulty here is a possible discontinuity of $f$. Moreover, due to the lack of strict convexity of $Ω$, the classical results are not applicable. We state the admissibility conditions on the boundary datum $f$, that are sufficient for establishing an existence result. One of them is that $f\in BV(\partialΩ)$. The solutions are constructed by a limiting process, which uses solutions to known problems
In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.
This article is devoted to obtain the $Γ$-limit, as $ε$ tends to zero, of the family of functionals $$F_ε(u)=\int_Ωf\Bigl(x,\frac{x}ε,..., \frac{x}{ε^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.
In this paper, we consider the following free boundary problem $$ (P)\left\{\begin{array}{ll} Δu = λϕ(x)\Sum_{i=1}^n H(u-μ_i )& \quad \mbox{ in }\ Ω=Ω_2\setminus \overlineΩ_1, \\[0.3cm]u =0 &\quad \mbox{ on } \partial Ω_2, \\[0.3cm]u =M &\quad \mbox{ on } \partial Ω_1. \end{array} \right. $$ The domain $Ω$ is a convex ring where $Ω_1$ and $Ω_2$ are bounded convex domain in $\mathbb{R}^N,$ $N\geq 2,$ $H$ is the Heaviside step function, $λ, M,μ_i$ $(i=1,..,n)$ are a given positive real parameters and $ϕ$ is a given function . We show that under suitable conditions, there exists a solution to pro…
For $0<p<+\infty$, we prove a global $W^{2,p}$-estimate for potentials of optimal transport maps between convex domains in the plane. Among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials.
Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.
We study the two dimensional least gradient problem in a convex, but not necessary strictly convex region. We look for solutions in the space of $BV$ functions satisfying the boundary data $f$ in trace sense. We assume that $f$ is in $BV$ too. We state admissibility conditions on the trace and on the domain that are sufficient for existence of solutions.
We consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\fracμ{2}\f…
In convex planar domains, given an initial vorticity with one sign, we study the regularity and geometric properties of the dynamically stable solutions to the Euler equations in the coadjoint orbit of the initial vorticity. These flows have elliptic stagnation points. Under some nondegeneracy conditions on the data, we show they are Holder continuous and have convex level curves. We also give a detailed description for the set of stagnation points. If the initial vorticity has nice level set topology, these stable solutions are in the L^\infty-strong closure of the coadjoint orbit. We also de…
In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of the boundary and the normal derivative of f on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex domains.
In this paper, we study the nonlinear stability of the composite wave consisting of planar rarefaction and planar contact waves for viscous conservation laws with degenerate flux under multi-dimensional periodic perturbations. To the level of our knowledge, it is the first stability result of the composite wave for conservation laws in several dimensions. Moreover, the perturbations studied in the present paper are periodic, which keep constantly oscillating at infinity. Suitable ansatz is constructed to overcome the difficulty caused by this kind of perturbation and delicate estimates are don…
We study a rate-independent system with non-convex energy and in the case of a time-discontinuous loading. We prove existence of the rate-dependent viscous regularization by time-incremental problems, while the existence of the so called parameterized $BV$-solutions is obtained via vanishing viscosity in a suitable parameterized setting. In addition, we prove that the solution set is compact.
We study two special cases of the planar least gradient problem. In the first one, the boundary conditions are imposed on a part of the strictly convex domain. In the second case, we impose the Dirichlet data on the boundary of a rectangle, an example of convex but not strictly convex domain. We show the existence of solutions and study their properties for particular cases of data.
In this paper we consider the second eigenfunction of the Laplacian with Dirichlet boundary conditions in convex domains. If the domain has \emph{large eccentricity} then the eigenfunction has \emph{exactly} two nondegenerate critical points (of course they are one maximum and one minimum). The proof uses some estimates proved by Jerison ([Jer95a]) and Grieser-Jerison ([GJ96]) jointly with a topological degree argument. Analogous results for higher order eigenfunctions are proved in rectangular-like domains considered in [GJ09].
We study the two dimensional least gradient problem in a convex polygonal set in the plane. We show existence of solutions when the boundary data are attained in the trace sense. Due to the lack of strict convexity, the classical results are not applicable. We state the admissibility conditions on the continuous boundary datum $f$ that are sufficient for establishing an existence and uniqueness result. The solutions are constructed by a limiting process, which uses the well-known geometry of superlevel sets of least gradient functions.
The Hardy constant of a simply connected domain $Ω\subset\R^2$ is the best constant for the inequality \[ \int_Ω|\nabla u|^2dx \geq c\int_Ω \frac{u^2}{{\rm dist}(x,\partialΩ)^2}\, dx \;, u\in C^{\infty}_c(Ω). \] After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In this work we determine the Hardy constant of an arbitrary quadrilateral in the plane. In particular we show that the Hardy constant is the same as that of a certain infinite sectorial region which has been …
We consider Morrey's open question whether rank-one convexity already implies quasiconvexity in the planar case. For some specific families of energies, there are precise conditions known under which rank-one convexity even implies polyconvexity. We will extend some of these findings to the more general family of energies $W:\operatorname{GL}^+(n)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split, i.e. \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F)\,, \] which is the natural finite extension of isotropic …
Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation
University of Science and Technology of China（中国科学技术大学）；Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish back-projection type inversion formulas that recover any initial data with support in $\Om$ modulo an explicitly computed smoothing integral operator $\K_\Om$. For circular and elliptical domains the operator $\K_\Om$ is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtaine…
Strictly stable solutions in uniformly convex planar domains may have nonconvex superlevel sets
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Institute of mathematics, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学研究所）
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally …
We study the Dirichlet problem for a graph $Σ$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $Γ=\partial Ω$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $Γ$ satisfies $h\geq H$, also suffices when $Ω$ is strictly convex, to prove the strict convexity of $Σ$.
We prove the sharp inequality \[ J(Ω) := \frac{λ_1(Ω)}{h_1(Ω)^2} < \frac{π^2}{4},\] where $Ω$ is any planar, convex set, $λ_1(Ω)$ is the first eigenvalue of the Laplacian under Dirichlet boundary conditions, and $h_1(Ω)$ is the Cheeger constant of $Ω$. The value on the right-hand side is optimal, and any sequence of convex sets with fixed volume and diameter tending to infinity is a maximizing sequence. Morever, we discuss the minimization of $J$ in the same class of subsets: we provide a lower bound which improves the generic bound given by Cheeger's inequality, we show the existence of a min…
For $p\in (1,2]$ and a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ let $μ_p(\barΩ,\cdot)$ be the $p$-capacitary curvature measure (generated by the closure $\barΩ$ of $Ω$) on the unit circle $\mathbb S^1$. This paper shows that such a problem of prescribing $μ_p$ on a planar convex domain: "Given a finite, nonnegative, Borel measure $μ$ on $\mathbb S^1$, find a bounded, convex, nonempty, open set $Ω\subset\mathbb R^2$ such that $dμ_p(\barΩ,\cdot)=dμ(\cdot)$" is solvable if and only if $μ$ has centroid at the origin and its support $\mathrm{supp}(μ)$ does not comprise any pair of …
We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In the physically interesting case of a rectangular domain, a similar result holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions with bounded initial vorticity. The main tool is a new BMO regularity estimate for the Dirichlet …
The Hardy constant of a simply connected domain $Ω\subset\mathbf{R}^2$ is the best constant for the inequality \[ \int_Ω|\nabla u|^2dx \geq c\int_Ω \frac{u^2}{{\rm dist}(x,\partialΩ)^2}\, dx \; , \;\;\quad u\in C^{\infty}_c(Ω). \] After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In \cite{BT} we have determined the Hardy constant of an arbitrary quadrilateral in the plane. In this work we continue our investigation and we compute the Hardy constant for other non-conv…
Convergence of Discontinuous Galerkin Methods for Quasiconvex and Relaxed Variational Problems
Computer, Electrical, Mathematical Sciences and Engineering Division, KAUST（计算机、电气、数学科学与工程系，阿卜杜拉国王科技大学）；Department of Mathematics, University of Sussex（萨塞克斯大学数学系）；Institute of Applied & Computational Mathematics, Foundation for Research & Technology-Hellas（应用与计算数学研究所，希腊研究与技术基金会）；Department of Mathematics and Applied Mathematics, University of Crete（克里特大学数学与应用数学系）；Faculty of Civil Engineering-Czech Technical University of Prague（捷克布拉格工业大学土木工程学院）
For a sufficiently regular open bounded set $D \subset R^2$ let us consider the equation $(-Δ)^{1/2} φ(x) = 1$, $x \in D$ with the Dirichlet exterior condition $φ(x) = 0$, $x \in D^c$. $φ$ is the expected value of the first exit time from $D$ of the Cauchy process in $R^2$. We prove that if $D \subset R^2$ is a convex bounded domain then $φ$ is concave on $D$. To show it we study the Hessian matrix of the harmonic extension of $φ$. The key idea of the proof is based on a deep result of Hans Lewy concerning determinants of Hessian matrices of harmonic functions.
We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Radó-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte's embedding theorem for planar graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend thes…
We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ close to the unit disk where this estimate fails to be extended to the interv…
