Boundary Regularity for the Porous Medium Equation
Björn, A., Björn, J., Gianazza, U., & Siljander, J. (2018). Boundary Regularity for the Porous Medium Equation. Archive for Rational Mechanics and Analysis, 230(2), 493-538. https://doi.org/10.1007/s00205-018-1251-3
Published in
Archive for Rational Mechanics and AnalysisDate
2018Copyright
© The Author(s) 2018
We study the boundary regularity of solutions to the porous medium equation ut=Δum in the degenerate range m>1 . In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general—not necessarily cylindrical—domains in Rn+1 . One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/superparabolic functions and weak sub/supersolutions.
...


Publisher
SpringerISSN Search the Publication Forum
0003-9527Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/28060705
Metadata
Show full item recordCollections
License
Related items
Showing items with similar title or keywords.
-
Lower semicontinuity of weak supersolutions to the porous medium equation
Avelin, Benny; Lukkari, Teemu (American Mathematical Society, 2015)Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that nonnegative weak supersolutions become lower semicontinuous after redefinition on a set ... -
Regularization and finite element approximation of the wave equation with Dirichlet boundary data
Lasiecka, I.; Sokołowski, J.; Neittaanmäki, Pekka (Polish Academy of Sciences, Institute of Mathematics, 1990)A numerical method for solving the wave equation with nonhomogenuous, nonsmooth Dirichlet boundary condition is proposed. Convergence of the method is proved and some erràr estimates are derived [L-S-2]. The method ... -
On the local and global regularity of tug-of-war games
Heino, Joonas (University of Jyväskylä, 2018)This thesis studies local and global regularity properties of a stochastic two-player zero-sum game called tug-of-war. In particular, we study value functions of the game locally as well as globally, that is, close to ... -
A two-phase problem with Robin conditions on the free boundary
Guarino Lo Bianco, Serena; La Manna, Domenico Angelo; Velichkov, Bozhidar (Les Éditions de l'École polytechnique, 2021)We study for the first time a two-phase free boundary problem in which the solution satisfies a Robin boundary condition. We consider the case in which the solution is continuous across the free boundary and we prove an ... -
On some partial data Calderón type problems with mixed boundary conditions
Covi, Giovanni; Rüland, Angkana (Elsevier, 2021)In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal ...