scholarly journals On the singular set of the parabolic obstacle problem

2006 ◽  
Vol 231 (2) ◽  
pp. 656-672 ◽  
Author(s):  
Adrien Blanchet
Keyword(s):  
Obstacle Problem ◽  
Singular Set

scholarly journals The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Agnid Banerjee ◽  
Donatella Danielli ◽  
Nicola Garofalo ◽  
Arshak Petrosyan
Keyword(s):  
Parabolic Equations ◽  
Obstacle Problem ◽  
Degenerate Parabolic Equations ◽  
Singular Set ◽  
Heat Operator ◽  
Complete Structure ◽  
Local Extension ◽  
Monotonicity Formulas ◽  
Degenerate Parabolic ◽  
Thin Obstacle

AbstractWe study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ | y | a for $$a \in (-1,1)$$ a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial _t - \Delta _x)^s$$ ( ∂ t - Δ x ) s for $$s \in (0,1)$$ s ∈ ( 0 , 1 ) . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ a = 0 ).

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

scholarly journals Singing to Emmanuel: The Wall Paintings of Sant Miquel in Terrassa and the 6th Century Artistic Reception of Byzantium in the Western Mediterranean

Arts ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 128
Author(s):  
Carles Sánchez Márquez
Keyword(s):  
19Th Century ◽  
Eastern Mediterranean ◽  
Western Mediterranean ◽  
Wall Paintings ◽  
Singular Set ◽  
Late 19Th Century ◽  
Early Date ◽  
Early Christian ◽  
Christian Art ◽  
Technical Study

Since the late 19th century the wall paintings of Sant Miquel in Terrassa have drawn attention due to their singularity. From the early studies of Josep Puig i Cadafalch (1867–1956) to the present, both the iconographic program and the chronology of the paintings have fueled controversy among scholars. In particular, chronological estimates range from the time of Early Christian Art to the Carolingian period. However, a recent technical study of the paintings seems to confirm an early date around the 6th century. This new data allows us to reassess the question in other terms and explore a new possible context for the paintings. First, it is very likely that the choice of iconographic topics was related to the debates on the Arian heresy that took place in Visigothic Spain during the 5th and 6th centuries. Secondly, the paintings of Sant Miquel should be reconsidered as a possible reception of a larger 6th-century pictorial tradition linked to the Eastern Mediterranean, which is used in a very particular way. However, thus far we ignore which were the means for this artistic transmission as well as the reasons which led the “doers” of Terrassa to select such a peculiar and unique repertoire of topics, motifs, and inscriptions. My paper addresses all these questions in order to propose a new Mediterranean framework for the making of this singular set of paintings.

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