Publication Date:
2019
Description:
〈p〉Publication date: August 2019〈/p〉
〈p〉〈b〉Source:〈/b〉 Nonlinear Analysis, Volume 185〈/p〉
〈p〉Author(s): Verena Bögelein, Pekka Lehtelä, Stefan Sturm〈/p〉
〈h5〉Abstract〈/h5〉
〈div〉〈p〉We investigate the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si1.gif"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mi〉m〈/mi〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msubsup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" altimg="si2.gif"〉〈msup〉〈mrow〉〈mi〉u〈/mi〉〈/mrow〉〈mrow〉〈mfrac〉〈mrow〉〈mi〉m〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/mfrac〉〈/mrow〉〈/msup〉〈mo〉∈〈/mo〉〈msubsup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo〉;〈/mo〉〈msubsup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mtext〉loc〈/mtext〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msubsup〉〈mrow〉〈mo〉(〈/mo〉〈mi〉Ω〈/mi〉〈mo〉)〈/mo〉〈/mrow〉〈mo〉)〈/mo〉〈/mrow〉〈/math〉 in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.〈/p〉〈/div〉
Print ISSN:
0362-546X
Electronic ISSN:
1873-5215
Topics:
Mathematics
