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SUMMARY:Hans-Christoph Grunau (Otto-von-Guericke-Universität Magdeburg)
DTSTART:20210713T170000Z
DTEND:20210713T180000Z
DTSTAMP:20260423T035031Z
UID:OSGA/70
DESCRIPTION:Title: Bo
undary value problems for the Willmore and the Helfrich functional for sur
faces of revolution\nby Hans-Christoph Grunau (Otto-von-Guericke-Unive
rsität Magdeburg) as part of Online Seminar "Geometric Analysis"\n\n\nAbs
tract\nThis talk concerns joint works with A. Dall'Acqua\, K. Deckelnick\,
M. Doemeland\, S. Eichmann\, and S. Okabe.\n\nA special form of the Helf
rich energy for a sufficiently smooth (two dimensional) surface $ S \\sub
set \\mathbb{R} ^3 $ (with or without boundary) is defined by\n $$\n
{\\mathscr H}_\\varepsilon(S) := \\int_{S} H^2 \\\, d S + \\varepsilon
\\int_{S} \\\, d S \,\n $$\n where $H$ denotes the mean curvature of
$S$.\n The first integral may be considered as a bending energy and th
e second as\n surface (stretching) energy. ${\\mathscr W} (S):={\\maths
cr H}_0 (S)$ is\n called the Willmore functional.\n We consider surf
aces of revolution $ S $\n $$\n (x\,\\varphi)\\mapsto \\big(x\,
u(x)\\cos \\varphi\, u(x)\\sin \\varphi \\big) \\\, \,\n \\quad x\
\in[-1\,1]\,~\\varphi\\in[0\,2\\pi]\,\n $$\n with smooth strictly po
sitive profile curve $u$ subject to Dirichlet\n boundary conditions\n
$$\n u(-1)=\\alpha\,\\quad u(1)=\\beta\,\\quad u'(\\pm1)=0\n $$\n
and aim at minimising ${\\mathscr H}_\\varepsilon$. Thanks to these bou
ndary conditions the Gauss curvature integral $\\int_{S} K\\\, d S $ beco
mes a constant and needs not to be considered.\n\nIn the first part of th
e lecture I shall consider the Willmore case\, i.e.\n $\\varepsilon=0$.
After briefly recalling minimisation in the symmetric case\n $\\alpha=
\\beta$ (see [1\,4]) I shall show how much more complicated the problem\n
becomes for $\\alpha\\not=\\beta$. Only when $\\alpha$ and $\\beta$ do
not differ\n too much\, the profile curve will remain a graph while in
general it will\n become a nonprojectable curve\, see [3].\n\nIn the se
cond part\, ${\\mathscr H}_\\varepsilon$ is considered for\n $\\varepsi
lon\\in[0\,\\infty)$\, but again in the symmetric setting $\\alpha=\\beta
$. For $\\alpha \\ge \\alpha_m = c_m \\cosh(\\frac{1}{c_m})\\approx 1.895$
with $c_m\\approx 1.564$ the unique solution of the equation\n$\n\\frac{
2}{c} = 1 + e ^ {-2/ c}\n$\, when one has a catenoid $v_\\alpha$ which
globally minimises the surface\nenergy\, we find minimisers $u_\\varepsil
on$ for any $\\varepsilon\\ge 0$\nand show uniform and locally smooth conv
ergence $u_\\varepsilon \\to v_\\alpha$ under the singular limit\n$\\varep
silon \\to \\infty$. These results are collected in [2].\n\nAt the end I s
hall briefly mention recent work on obstacle problems [5].\n \n\n
\n \n[1] A. Dall'Acqua\, K. Deckelnick\, and H.-Ch. Grunau\,\n Cl
assical solutions to the Dirichlet problem for Willmore\n surfaces of r
evolution\, Adv. Calc. Var. 1 (2008)\, 379-397.\n\n[2] K.
Deckelnick\, H.-Ch. Grunau\, and M. Doemeland\, Boundary value problems fo
r the Helfrich functional for surfaces of revolution\n - Existence
and asymptotic behaviour\, Calc. Var. Partial Differ. Equ. 60<
/b> (2021)\, Article number 32.\n\n[3] S. Eichmann and H.-Ch. Grunau\,\n
Existence for Willmore surfaces of revolution satisfying non-symmetric
Dirichlet boundary conditions\,\n Adv. Calc. Var. 12 (20
19)\, 333–361.\n\n[4] H.-Ch. Grunau\, The asymptotic shape of a boundary
layer of symmetric\n Willmore surfaces of revolution.\n In: C. Band
le et al. (eds.)\, Inequalities and Applications 2010.\n Internatio
nal Series of Numerical Mathematics 161 (2012)\, 19-29.\n\n[5]
H.-Ch. Grunau and S. Okabe\,\n Willmore obstacle problems under Dirich
let boundary conditions\, submitted.\n
LOCATION:https://researchseminars.org/talk/OSGA/70/
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