Relevant bibliographies by topics / Epiperimetric inequality

Academic literature on the topic 'Epiperimetric inequality'

Author: Grafiati
Published: 10 March 2023

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Journal articles on the topic "Epiperimetric inequality"

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Colombo, Maria, Luca Spolaor, and Bozhidar Velichkov. "A logarithmic epiperimetric inequality for the obstacle problem." Geometric and Functional Analysis 28, no. 4 (May 11, 2018): 1029–61. http://dx.doi.org/10.1007/s00039-018-0451-1.

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Rivière, Tristan. "A lower-epiperimetric inequality for area-minimizing surfaces." Communications on Pure and Applied Mathematics 57, no. 12 (September 24, 2004): 1673–85. http://dx.doi.org/10.1002/cpa.20047.

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Geraci, Francesco. "An epiperimetric inequality for the lower dimensional obstacle problem." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 39. http://dx.doi.org/10.1051/cocv/2018024.

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In this paper we give a proof of an epiperimetric inequality in the setting of the lower dimensional obstacle problem. The inequality was introduced by Weiss [Invent. Math.138(1999) 23–50) for the classical obstacle problem and has striking consequences concerning the regularity of the free-boundary. Our proof follows the approach of Focardi and Spadaro [Adv. Differ. Equ.21(2015) 153–200] which uses an homogeneity approach and aΓ-convergence analysis.
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Shi, Wenhui. "An epiperimetric inequality approach to the parabolic Signorini problem." Discrete & Continuous Dynamical Systems - A 40, no. 3 (2020): 1813–46. http://dx.doi.org/10.3934/dcds.2020095.

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Spolaor, Luca, and Bozhidar Velichkov. "On the logarithmic epiperimetric inequality for the obstacle problem." Mathematics in Engineering 3, no. 1 (2021): 1–42. http://dx.doi.org/10.3934/mine.2021004.

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Colombo, Maria, Luca Spolaor, and Bozhidar Velichkov. "On the asymptotic behavior of the solutions to parabolic variational inequalities." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 768 (November 1, 2020): 149–82. http://dx.doi.org/10.1515/crelle-2019-0041.

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AbstractWe consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Łojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon ([22]) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.
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Engelstein, Max, Luca Spolaor, and Bozhidar Velichkov. "(Log-)epiperimetric inequality and regularity over smooth cones for almost area-minimizing currents." Geometry & Topology 23, no. 1 (March 5, 2019): 513–40. http://dx.doi.org/10.2140/gt.2019.23.513.

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Spolaor, Luca, and Bozhidar Velichkov. "An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2‐Dimensional Case." Communications on Pure and Applied Mathematics 72, no. 2 (August 3, 2018): 375–421. http://dx.doi.org/10.1002/cpa.21785.

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Banerjee, A., D. Danielli, N. Garofalo, and A. Petrosyan. "The regular free boundary in the thin obstacle problem for degenerate parabolic equations." St. Petersburg Mathematical Journal 32, no. 3 (May 11, 2021): 449–80. http://dx.doi.org/10.1090/spmj/1656.

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This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t − Δ x ) s (\partial _t - \Delta _x)^s for s ∈ ( 0 , 1 ) s \in (0,1) . The regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. The approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. By using several results from the elliptic theory, including the epiperimetric inequality, the optimal regularity is established for solutions as well as the H 1 + γ , 1 + γ 2 H^{1+\gamma ,\frac {1+\gamma }{2}} regularity of the free boundary near such regular points.
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Garofalo, Nicola, Arshak Petrosyan, and Mariana Smit Vega Garcia. "An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients." Journal de Mathématiques Pures et Appliquées 105, no. 6 (June 2016): 745–87. http://dx.doi.org/10.1016/j.matpur.2015.11.013.

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Dissertations / Theses on the topic "Epiperimetric inequality"

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GERACI, FRANCESCO. "The Classical Obstacle Problem for nonlinear variational energies and related problems." Doctoral thesis, 2017. http://hdl.handle.net/2158/1079281.

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In this thesis we investigate the classical obstacal problem for nonlinear variational energies and related problems. We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a type-Dini continuity property. These formulae are used to obtain the regularity of free boundary points following the approaches by Caffarelli, Monneau and Weiss. We develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies. The key tools are optimal C 1,1 regularity, which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields, and the results in Focardi et al. (2015) concerning the obstacle problem for quadratic energies with Lipschitz coefficients. Furthermore, we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional, or more generally to area-type functionals, as well. We prove also an epiperimetric inequality for the fractional obstacle problem thus extending the pioneering results by Weiss (1999) on the classical obstacle problem and the results of Focardi and Spadaro (2016) in the thin obstacle problem. We deduce the regularity of a suitable subset of the free boundary as a consequence of a decay estimate of a boundary adjusted energy “à la Weiss”, the non degeneracy of the solution and the uniqueness of the limits of suitable rescaled funciotns.
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Book chapters on the topic "Epiperimetric inequality"

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Velichkov, Bozhidar. "An Epiperimetric Inequality Approach to the Regularity of the One-Phase Free Boundaries." In Lecture Notes of the Unione Matematica Italiana, 189–221. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-13238-4_12.

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AbstractThroughout this section, we will use the notation $$\displaystyle W_0(u)=\int _{B_1}|\nabla u|{ }^2\,dx-\int _{\partial B_1}u^2\,d\mathcal {H}^{d-1}\qquad \text{and}\qquad W(u)=W_0(u)+|\{u>0\}\cap B_1|, $$ W 0 ( u ) = ∫ B 1 | ∇ u | 2 d x − ∫ ∂ B 1 u 2 d ℋ d − 1 and W ( u ) = W 0 ( u ) + | { u > 0 } ∩ B 1 | , where B1 is the unit ball in $$\mathbb {R}^d$$ ℝ d , d ≥ 2 and u ∈ H1(B1).
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