caa a22 4500 605496439 CHVBK 20210128105157.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s10231-014-0413-7 doi (NATIONALLICENCE)springer-10.1007/s10231-014-0413-7 A general class of free boundary problems for fully nonlinear parabolic equations [Elektronische Daten] [Alessio Figalli, Henrik Shahgholian] In this paper, we consider the fully nonlinear parabolic free boundary problem $$\begin{aligned} \left\{ \begin{array}{ll} F(D^2u) -\partial _{t} u=1 &{} \text {a.e. in }Q_1 \cap \Omega \\ |D^2 u| + |\partial _{t} u| \le K &{} \text {a.e. in }Q_1{\setminus }\Omega , \end{array} \right. \end{aligned}$$ F ( D 2 u ) - ∂ t u = 1 a.e. in Q 1 ∩ Ω | D 2 u | + | ∂ t u | ≤ K a.e. in Q 1 \ Ω , where $$K>0$$ K > 0 is a positive constant, and $$\Omega $$ Ω is an (unknown) open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that $$W_x^{2,n} \cap W_t^{1,n} $$ W x 2 , n ∩ W t 1 , n solutions are locally $$C_x^{1,1}\cap C_t^{0,1} $$ C x 1 , 1 ∩ C t 0 , 1 inside $$Q_1$$ Q 1 . A key starting point for this result is a new BMO-type estimate, which extends to the parabolic setting the main result inCaffarelli and Huang (Duke Math J 118(1):1-17, 2003). Once optimal regularity for $$u$$ u is obtained, we also show regularity for the free boundary $$\partial \Omega \cap Q_1$$ ∂ Ω ∩ Q 1 under the extra condition that $$\Omega \supset \{ u \ne 0 \}$$ Ω ⊃ { u ≠ 0 } , and a uniform thickness assumption on the coincidence set $$\{ u = 0 \}$$ { u = 0 } . Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Free boundaries nationallicence Regularity nationallicence Parabolic fully nonlinear nationallicence Figalli Alessio Mathematics Department, The University of Texas at Austin, 78712-1202, Austin, TX, USA aut Shahgholian Henrik Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/4(2015-08-01), 1123-1134 0373-3114 194:4<1123 2015 194 10231 https://doi.org/10.1007/s10231-014-0413-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0413-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Figalli Alessio Mathematics Department, The University of Texas at Austin, 78712-1202, Austin, TX, USA aut NATIONALLICENCE 700 1- Shahgholian Henrik Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/4(2015-08-01), 1123-1134 0373-3114 194:4<1123 2015 194 10231 SWISSBIB 560493258