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SUMMARY:Moshe Marcus (Technion\, Israel Institute of Technology\, Israel.)
 )
DTSTART:20201112T093000Z
DTEND:20201112T103000Z
DTSTAMP:20260423T005731Z
UID:WOPARA/16
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WOPARA/16/">
 Boundary value problems for Schrodinger equations with Hardy type potentia
 ls</a>\nby Moshe Marcus (Technion\, Israel Institute of Technology\, Israe
 l.)) as part of Webinar on PDE and related areas\n\n\nAbstract\nWe conside
 r equations of the form− $L_V u=\\tau$ where $L_V= \\Delta +V$ and $\\ta
 u$ is a Radon measure in a Lipschitz bounded domain $D\\subset\\mathbb{R}^
 N$.  The assumptions on $V$ include the condition $|V(x)|\\leq a \\delta (
 x)^{-2}$ - where $a >0$ and $\\delta (x) = \\mbox{dist} (x\,\\partial D)$.
  An additional condition guarantees the existence of a ground state $\\Phi
 _V$. The model example is $V=\\gamma \\delta (x)^{-2}$ where $\\gamma< cH$
  ($cH$=Hardy constant). \n\nFor positive solutions of the equation we defi
 ne the $L_V$ boundary trace. If $\\int_D \\Phi_V d\\tau<\\infty$\, the bou
 ndary trace is well defined as a positive\, bounded measure $\\nu$ on $\\p
 artial D$.  We consider the corresponding b.v.p.\, namely\, $-L+V u=\\tau$
  in $D$\,$u=\\nu$ on $\\partial D$. We show that\, for $\\tau$ and $\\nu$ 
 as above\, the b.v.p.  has a unique  solution.  Further\,  under  some  co
 nditions  on  the  ground  state  -satisfied for a large family of potenti
 als - we obtain sharp two sided estimates of positive solutions of the b.v
 .p.  Finally we discuss some applications to semilinear problems involving
  the operator $L_V$.\n
LOCATION:https://researchseminars.org/talk/WOPARA/16/
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