Parabolic comparison principle and quasiminimizers in metric measure spaces (original) (raw)
Regularity of quasi-minimizers on metric spaces
manuscripta mathematica, 2001
Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasiminimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally Hölder continuous, if the space is doubling and supports a Poincaré inequality. J. Kinnunen:
PARABOLIC VARIATIONAL PROBLEMS AND REGULARITY IN METRIC SPACES
In this paper we study variational problems related to the heat equation in metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is the proof of a scale-invariant Harnack inequality for functions in parabolic De Giorgi classes.
A comparison principle for minimizers
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000
We give some conditions that ensure the validity of a Comparison principle for the minimizers of integral functionals, without assuming the validity of the Euler-Lagrange equation. We deduce a weak maximum principle for (possibly) degenerate elliptic equations and, together with a generalization of the bounded slope condition, the Lipschitz continuity of minimizers. To prove the main theorem we give a result on the existence of a representative of a given Sobolev function that is absolutely continuous along the trajectories of a suitable autonomous system. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Un principe de comparaison pour les minima Résumé. Nous donnons des conditions qui assurent la validité d'un principe de comparaison pour les minimums d'une fonctionnelle intégrale qui ne satisfont pas nécessairement à l'équation d'Euler-Lagrange. Nous en déduisons un principe de maximum faible pour les équations elliptiques (éventuellement) dégénerées et, en généralisant la condition de la pente bornée, la Lipschitz continuité des minimums. La preuve du théorème principal se base sur l'éxistence d'un représentant d'une fonction de Sobolev donnée qui est absolument continu sur les trajectoires d'un système autonome convenable. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée Nous fixons un ouvert borné Ω de R n. La fonction L(x, z, p) est définie dans Ω × R × R n et est une fonction dans W 1,q (Ω), q 1. La fonctionū est dans W 1,q (Ω) et on pose W 1,q u (Ω) =ū + W 1,q 0 (Ω). Dans cette partie nous nous référons aux hypothèses A, A , B et D du texte anglais qui suit. THÉORÈME PRINCIPAL 1 ([4]).-On suppose que (L,) satisfait à l'hypothèse B. Soit w un minimum de I(u) = Ω L x, u(x), ∇u(x) dx dans W 1,q u (Ω). Si w dans ∂Ω, alors w presque partout dans Ω. Note présentée par Haïm BRÉZIS.
Moser iteration for (quasi)minimizers on metric spaces
manuscripta mathematica, 2006
We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub-and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete.
A Comparison Principle and the Lipschitz Continuity for Minimizers
Nonconvex Optimization and Its Applications, 2001
We give some conditions that ensure the validity of a Comparison Principle for the minimizers of integral functionals, without assuming the validity of the Euler-Lagrange equation. We deduce a weak Maximum Principle for (possibly) degenerate elliptic equations and, together with a generalization of the Bounded Slope Condition, a result on the Lipschitz continuity of minimizers.
Potential theory of quasiminimizers
Annales- Academiae Scientiarum Fennicae Mathematica
We study nonlinear potential theory related to quasiminimizers on a metric measure space equipped with a doubling measure and supporting a Poincare inequality. Our objective is to show that quasiminimizers create an interesting potential theory with new features although from the potential theoretic point of view they have several drawbacks: They do not provide a unique solution to the Dirichlet problem, they do not obey the comparison principle and they do not form a sheaf. However, many potential theoretic concepts such as harmonic functions, superharmonic functions and the Poisson modiflcation have their counterparts in the theory of quasiminimizers and, in particular, we are interested in questions related to regularity, convergence and polar sets.
Some remarks on weak generalizations of minima and quasi efficiency
arXiv: Optimization and Control, 2019
In this note, we remark, with sufficient mathematical rigor, that many weak generalizations of the usual minimum available in the literature are not true generalizations. Motivated by the Ekeland Variational Principle, we provide, first time, the criteria for weaker generalizations of the usual minimum. Further, we show that the quasi efficiency, recently used in Bhatia et al. (Optim. Lett. 7, 127-135 (2013)) and introduced in Gupta et al. ( Bull. Aust. Math. Soc. 74, 207-218 (2006)) is not a true generalization of the usual efficiency. Since the former paper relies heavily on the results of later one, so we discuss the later paper. We show that the necessary optimality condition is a consequence of the local Lipschitzness and sufficiency result is trivial in the later paper. Consequently, the duality results of the same paper are also inconsistent.
The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces
Journal de Mathématiques Pures et Appliquées, 2010
We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincaré inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.
Advances in Calculus of Variations
In this article, we consider the (double) minimization problem min { P ( E ; Ω ) + λ W p ( E , F ) : E ⊆ Ω , F ⊆ R d , | E ∩ F | = 0 , | E | = | F | = 1 } , \min\{P(E;\Omega)+\lambda W_{p}(E,F):E\subseteq\Omega,\,F\subseteq\mathbb{R}^{d},\,\lvert E\cap F\rvert=0,\,\lvert E\rvert=\lvert F\rvert=1\}, where λ ⩾ 0 \lambda\geqslant 0 , p ⩾ 1 p\geqslant 1 , Ω is a (possibly unbounded) domain in R d \mathbb{R}^{d} , P ( E ; Ω ) P(E;\Omega) denotes the relative perimeter of 𝐸 in Ω and W p W_{p} denotes the 𝑝-Wasserstein distance. When Ω is unbounded and d ⩾ 3 d\geqslant 3 , it is an open problem proposed by Buttazzo, Carlier and Laborde in the paper On the Wasserstein distance between mutually singular measures. We prove the existence of minimizers to this problem when the dimension d ⩾ 1 d\geqslant 1 , 1 p + 2 d > 1 \frac{1}{p}+\frac{2}{d}>1 , Ω = R d \Omega=\mathbb{R}^{d} and 𝜆 is sufficiently small.