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BEGIN:VEVENT
SUMMARY:Xavier Cabre (ICREA and Universitat Politecnica de Catalunya)
DTSTART:20210906T150000Z
DTEND:20210906T154500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/1
DESCRIPTION:Title: Stable solutions to semilinear elliptic equations are smooth up to di
mension 9\nby Xavier Cabre (ICREA and Universitat Politecnica de Catal
unya) as part of BIRS workshop: Nonlinear Potential Theoretic Methods in P
artial Differential Equations\n\n\nAbstract\nThe regularity of stable solu
tions to semilinear elliptic PDEs has been studied since the 1970's. In di
mensions $10$ and higher\, there exist singular stable energy solutions. I
n this talk I will describe a recent work in collaboration with Figalli\,
Ros-Oton\, and Serra\, where we prove that stable solutions are smooth up
to the optimal dimension $9$. This answers to an open problem posed by Bre
zis in the mid-nineties concerning the regularity of extremal solutions to
Gelfand-type problems.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Focardi (Università di Firenze)
DTSTART:20210906T154500Z
DTEND:20210906T163000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/2
DESCRIPTION:Title: On the regularity of singular sets of minimizers for the Mumford-Shah
energy\nby Matteo Focardi (Università di Firenze) as part of BIRS wo
rkshop: Nonlinear Potential Theoretic Methods in Partial Differential Equa
tions\n\n\nAbstract\nWe will survey the regularity theory of minimizers of
the Mumford-Shah functional\, focusing in particular on that of the corre
sponding singular sets. Starting with nowadays classical results\, we will
finally discuss more recent developments\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phuc Nguyen (Louisiana State University)
DTSTART:20210906T170000Z
DTEND:20210906T174500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/3
DESCRIPTION:Title: Choquet integrals\, capacitary inequalities\, and the Hardy-Littlewoo
d maximal function\nby Phuc Nguyen (Louisiana State University) as par
t of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Diffe
rential Equations\n\n\nAbstract\nWe obtain the boundedness of the Hardy-Li
ttlewood maximal function on $L^q$ type spaces defined via Choquet integra
ls associate to Sobolev capacities. The bounds are obtained in full range
of exponents including a weak type end-point bound. We also obtain a capac
itary inequality of Maz'ya type which resolves a problem proposed by D. Ad
ams. This talk is based on joint work with Keng Hao Ooi.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengfei Guan (McGill University)
DTSTART:20210906T174500Z
DTEND:20210906T183000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/4
DESCRIPTION:Title: Entropy quantities associated to Gauss curvature type flows\nby P
engfei Guan (McGill University) as part of BIRS workshop: Nonlinear Potent
ial Theoretic Methods in Partial Differential Equations\n\n\nAbstract\nWe
discuss the role of entropy functionals played in the study of Gauss curva
ture type flows: 1. the monotonicity of the associated entropies\, 2. diam
eter\, non-collapsing entropy points estimates\, 3. convergence. Similar
entropy functionals also exists for anisotropy type Gauss curvature flows.
\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Connor Mooney (University of California\, Irvine)
DTSTART:20210906T183000Z
DTEND:20210906T191500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/5
DESCRIPTION:Title: The Bernstein problem for equations of minimal surface type\nby C
onnor Mooney (University of California\, Irvine) as part of BIRS workshop:
Nonlinear Potential Theoretic Methods in Partial Differential Equations\n
\n\nAbstract\nThe Bernstein problem asks whether entire minimal graphs in
dimension N+1 are necessarily hyperplanes. This problem was solved in comb
ined works of Bernstein\, Fleming\, De Giorgi\, Almgren\, and Simons ("yes
" if N < 8)\, and Bombieri-De Giorgi-Giusti ("no" otherwise). We will disc
uss the analogue of this problem for graphical minimizers of anisotropic e
nergies. In particular\, we will discuss new examples of nonlinear entire
graphical minimizers in the case N = 6\, and recent joint work with Y. Yan
g towards constructing such examples in the lowest-possible-dimensional ca
se N = 4.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iwona Chlebicka (University of Warsaw)
DTSTART:20210907T130000Z
DTEND:20210907T134500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/6
DESCRIPTION:Title: Potential estimates for solutions to quasilinear elliptic problems wi
th general growth. Scalar and vectorial case\nby Iwona Chlebicka (Univ
ersity of Warsaw) as part of BIRS workshop: Nonlinear Potential Theoretic
Methods in Partial Differential Equations\n\n\nAbstract\nWe consider measu
re data elliptic problems involving a second order operator exhibiting Orl
icz growth and having measurable coefficients. As known in the $p$-Laplace
case\, pointwise estimates for solutions expressed with the use of nonlin
ear potential are powerful tools in the study of the local behaviour of th
e solutions. Not only we provide such estimates expressed in terms of a po
tential of generalized Wolff type\, but also we investigate their regulari
ty consequences. For scalar equations we do not need to impose any structu
ral conditions on the the operator and we study generalized $A$-harmonic f
unctions being distributional solutions to problems with nonnegative measu
re. Lower and upper estimates we provide are sharp in the sense that the p
otential cannot be substituted with a better one. As a consequence we get
a bunch of sharp criteria for continuity or H\\"older continuity of the so
lutions. For systems we impose typical assumptions of the Uhlenbeck-type s
tructure of the operator and separated variables\, whereas the measure can
be signed as another notion of very weak solutions is employed. In this c
ase the upper bound is shown with the same potential as in the scalar case
and presented together with its precise consequences for the local behavi
our of solutions. The talk is based on joint works:(scalar) with F.~Gianne
tti and A.~Zatorska-Goldstein [arXiv:2006.02172] and (vectorial) with Y.~Y
oun and A.~Zatorska-Goldstein\, [arXiv:2102.09313]\, [arXiv:2106.11639].\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Schwarzacher (Charles University)
DTSTART:20210907T134500Z
DTEND:20210907T143000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/7
DESCRIPTION:Title: Construction of a right inverse for the divergence in non-cylindrical
time dependent domains\nby Sebastian Schwarzacher (Charles University
) as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Parti
al Differential Equations\n\n\nAbstract\nWe discuss the construction of a
stable right inverse for the divergence operator in non-cylindrical domain
s in space-time. The domains are assumed to be Hölder regular in space an
d evolve continuously in time. The inverse operator is of Bogovskij type\,
meaning that it attains zero boundary values. We provide estimates in Sob
olev spaces of positive and negative order with respect to both time and s
pace variables. The regularity estimates on the operator depend on the ass
umed Hölder regularity of the domain. The results can naturally be connec
ted to the known theory for Lipschitz domains. As an application\, we prov
e refined pressure estimates for weak and very weak solutions to Navier--S
tokes equations in time dependent domains. This is a joint work with Olli
Saari.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuomo Kuusi (University of Helsinki)
DTSTART:20210907T143000Z
DTEND:20210907T151500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/8
DESCRIPTION:Title: Homogenization\, linearization and large-scale regularity for nonline
ar elliptic equations\nby Tuomo Kuusi (University of Helsinki) as part
of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Differ
ential Equations\n\n\nAbstract\nWe will consider nonlinear\, uniformly ell
iptic equations with variational structure and random\, highly oscillating
coefficients and discuss the corresponding stochastic homogenization theo
ry. After recalling basic ideas on how to get quantitative rates of homoge
nization for nonlinear uniformly convex problems\, we will discuss our rec
ent work\, jointly with S. Armstrong and S. Ferguson\, showing that homoge
nization and linearization commute. This is in the sense that the lineariz
ed equation homogenizes to the linearization of the homogenized equation (
linearized around the corresponding solution of the homogenized equation).
This procedure can be iterated to show higher regularity of the homogeniz
ed Lagrangian as well as large-scale regularity for minimizers.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Verena Bogelein (Paris-Lodron-University Salzburg)
DTSTART:20210907T154000Z
DTEND:20210907T162500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/9
DESCRIPTION:Title: Higher regularity in congested traffic dynamics\nby Verena Bogele
in (Paris-Lodron-University Salzburg) as part of BIRS workshop: Nonlinear
Potential Theoretic Methods in Partial Differential Equations\n\n\nAbstrac
t\nWe consider an elliptic system that is motivated by a congested traffic
dynamics problem. It has the form\n$$ \\mathrm{div}\\bigg((|Du|-1)_+^{p-1
}\\frac{Du}{|Du|}\\bigg)=f\,$$\nand falls into the context of very degener
ate problems. Continuity properties of the gradient have been investigated
in the scalar case by Santambrogio & Vespri and Colombo & Figalli. \nIn t
his talk we establish the optimal regularity of weak solutions in the vect
orial case for any $p>1$. This is joint work with F. Duzaar\, R. Giova and
A. Passarelli di Napoli.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristiana De Filippis (Università di Torino)
DTSTART:20210907T162500Z
DTEND:20210907T171000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/10
DESCRIPTION:Title: Perturbations beyond Schauder\nby Cristiana De Filippis (Univers
ità di Torino) as part of BIRS workshop: Nonlinear Potential Theoretic Me
thods in Partial Differential Equations\n\n\nAbstract\nSchauder estimates
hold in the nonuniformly elliptic setting. Specifically\, first derivative
s of solutions to nonuniformly elliptic variational problems and elliptic
equations are locally H\\"older continuous\, provided coefficients are loc
ally H\\"older continuous. In this talk I will present new regularity resu
lts for minima of nonuniformly elliptic functionals with emphasis on delic
ate borderline regulairty criteria. My talk is based on papers:\n-C. De Fi
lippis\, Quasiconvexity and partial regularity via nonlinear potentials. P
reprint (2021)\;\n-C. De Filippis\, G. Mingione\, Lipschitz bounds and non
autonomous integrals. Arch. Ration. Mech. Anal.\, to appear\; C. De Filipp
is\, G. Mingione\, Nonuniformly elliptic Schauder estimates. Preprint (202
1).\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ovidiu Savin (Columbia University)
DTSTART:20210907T173500Z
DTEND:20210907T182000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/11
DESCRIPTION:Title: The multiple membrane problem\nby Ovidiu Savin (Columbia Univers
ity) as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Pa
rtial Differential Equations\n\n\nAbstract\nFor a positive integer $N$\, t
he $N$-membranes problem describes the equilibrium position of $N$ ordered
elastic membranes subject to forcing and boundary conditions. If the heig
hts of the membranes are described by real functions $u_1\, u_2\,...\,u_N$
\, then the problem can be understood as a system of $N-1$ coupled obstacl
e problems with interacting free boundaries which can cross each other. Wh
en $N=2$ there is only one free boundary and the problem is equivalent to
the classical obstacle problem. I will discuss a work in collaboration wit
h Hui Yu about the regularity of the free boundaries in the two dimensiona
l case.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniela De Silva (Barnard College - Columbia University)
DTSTART:20210907T182000Z
DTEND:20210907T190500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/12
DESCRIPTION:Title: Inhomogeneous global minimizers to the one-phase free boundary probl
em\nby Daniela De Silva (Barnard College - Columbia University) as par
t of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Diffe
rential Equations\n\n\nAbstract\nGiven a global 1-homogeneous minimizer $U
_0$ to the Alt-Caffarelli energy functional\, with $sing(F(U_0)) = \\{0\\}
$\, we provide a foliation of the half-space $\\mathbb R^{n} \\times [0\,+
\\infty)$ with dilations of graphs of global minimizers $\\underline U \\l
eq U_0 \\leq \\bar U$ with analytic free boundaries at distance 1 from the
origin. This is a joint work with D. Jerison and H. Shahgholian.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (Caltech / University of Munich)
DTSTART:20210908T133000Z
DTEND:20210908T141500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/13
DESCRIPTION:Title: Which magnetic fields support a zero mode?\nby Rupert Frank (Cal
tech / University of Munich) as part of BIRS workshop: Nonlinear Potential
Theoretic Methods in Partial Differential Equations\n\n\nAbstract\nMotiva
ted by the question from mathematical physics about the size of magnetic f
ields that support zero modes for the three dimensional Dirac equation\, w
e study a certain conformally invariant spinor equation. We state some con
jectures and present results in their support. Those concern\, in particul
ar\, two novel Sobolev inequalities for spinors and vector fields. The tal
k is based on joint work with Michael Loss.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Kristensen (University of Oxford)
DTSTART:20210908T141500Z
DTEND:20210908T150000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/14
DESCRIPTION:Title: Decompositions of sequences of PDE constrained maps\nby Jan Kris
tensen (University of Oxford) as part of BIRS workshop: Nonlinear Potentia
l Theoretic Methods in Partial Differential Equations\n\n\nAbstract\nIt is
a convenient and well-known fact that for exponents p>1\, any Lp-weakly c
onverging sequence of PDE constrained\nmaps admits a decomposition into se
quences of PDE constrained maps where one converges in measure (no oscilla
tion) and\nthe other is p-equi-integrable (no concentration). For p=1 the
relevant corresponding result concerns weakly* convergent sequences\nof PD
E constrained measures and is false: the oscillation and concentration can
not be separated while simultaneously satisfying\nthe PDE constraint. In t
his talk we explain how the concentration regardless of the failure of a d
ecomposition result retains its PDE\ncharacter. The presented results are
parts of joint works with Andre Guerra and Bogdan Raita.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Salani (Università di Firenze)
DTSTART:20210908T152500Z
DTEND:20210908T161000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/15
DESCRIPTION:Title: The intimate relationship between log-concavity and heat flow\nb
y Paolo Salani (Università di Firenze) as part of BIRS workshop: Nonlinea
r Potential Theoretic Methods in Partial Differential Equations\n\n\nAbstr
act\nThe talk will be based on some papers in collaboration with Kazuhiro
Ishige (The University of Tokyo) and Asuka Takatsu (Tokyo Metropolitan Uni
versity) where we investigate the preservation of concavity properties by
heat flow. Surprisingly\, we have recently proved that there exist concavi
ties stronger than log-concavity that are preserved by the Dirichlet heat
flow\, however\, when we consider a suitable class of concavities\, log-co
ncavity remains the strongest possible. Moreover\, in our latest paper\, w
e prove that\, when starting with an initial datum which shares any concav
ity weaker than log-concavity\, then the solution may lose immediately any
reminiscence of concavity. In this way we almost complete the study of pr
eservation of concavity by the Dirichlet heat flow\, started by Brascamp a
nd Lieb in 1976.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jana Bjorn (Linkoeping University)
DTSTART:20210908T161000Z
DTEND:20210908T165500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/16
DESCRIPTION:Title: Fine potential theory via analysis on metric spaces\nby Jana Bjo
rn (Linkoeping University) as part of BIRS workshop: Nonlinear Potential T
heoretic Methods in Partial Differential Equations\n\n\nAbstract\nWeshow
how p-harmonic functions and Sobolev spaces on metric spaces\, based on up
per gradients\, naturally lead to fine potential theory\, even in the sett
ing of Euclidean spaces.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Weth (Goethe-University Frankfurt am Main)
DTSTART:20210908T172000Z
DTEND:20210908T180500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/17
DESCRIPTION:Title: Morse index versus radial symmetry for fractional Dirichlet problems
\nby Tobias Weth (Goethe-University Frankfurt am Main) as part of BIRS
workshop: Nonlinear Potential Theoretic Methods in Partial Differential E
quations\n\n\nAbstract\nI will discuss a new estimate\, obtained in joint
work with M.\nM. Fall\, P.A. Feulefack and R.Y. Temgoua\,\non the Morse in
dex of radially symmetric sign changing solutions to\nsemilinear fractiona
l Dirichlet\nproblems in the unit ball. In particular\, the result applies
to the\nDirichlet eigenvalue problem for the\nfractional Laplacian and im
plies that eigenfunctions corresponding to\nthe second Dirichlet eigenvalu
e\nare antisymmetric. This resolves a conjecture of Banuelos and Kulczycki
.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Scott Armstrong (Scott Armstrong)
DTSTART:20210908T180500Z
DTEND:20210908T185000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/18
DESCRIPTION:Title: Optimal doubling inequalities for periodic elliptic equations\nb
y Scott Armstrong (Scott Armstrong) as part of BIRS workshop: Nonlinear Po
tential Theoretic Methods in Partial Differential Equations\n\n\nAbstract\
nI will discuss recent work with T. Kuusi and C. Smart on quantitative uni
que continuation for solutions of periodic elliptic equations on large sca
les.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dominic Breit (Heriot-Watt University)
DTSTART:20210909T133000Z
DTEND:20210909T141500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/19
DESCRIPTION:Title: Global Besov regularity for nonlinear elliptic problems\nby Domi
nic Breit (Heriot-Watt University) as part of BIRS workshop: Nonlinear Pot
ential Theoretic Methods in Partial Differential Equations\n\n\nAbstract\n
We prove global Besov estimates for the p-Laplacian with right-hand side i
n divergence form under optimal assumptions on the regularity of the bound
ary of the domain $\\Omega$. In particular\, we show that $B^s_{\\varrho\,
q}(\\Omega)$-regularity transfers from the forcing $F$ to the non-linear f
lux $|\\nabla u|^{p-2}\\nabla u$ provided the boundary belongs to the clas
s $B^{s+1-1/q}_{\\varrho\,q}$ and has a small Lipschitz constant. In the l
inear case $p=2$ this recovers a sharp result from Maz'ya-Shaposhnikova.\n
This is a joint work with A. Balci and L. Diening.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frédéric Robert (Universite de Lorraine)
DTSTART:20210909T141500Z
DTEND:20210909T150000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/20
DESCRIPTION:by Frédéric Robert (Universite de Lorraine) as part of BIRS
workshop: Nonlinear Potential Theoretic Methods in Partial Differential Eq
uations\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filippo Gazzola (Politecnico di Milano)
DTSTART:20210909T152500Z
DTEND:20210909T161000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/21
DESCRIPTION:Title: Long-time behavior of partially damped systems modeling degenerate p
lates with piers\nby Filippo Gazzola (Politecnico di Milano) as part o
f BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Differen
tial Equations\n\n\nAbstract\nWe consider a partially damped nonlinear bea
m-wave system of evolution PDE's modeling the dynamics of a degenerate pla
te. The plate can move both vertically and torsionally and\, consequently\
, the solution has two components. We show that the component from the dam
ped beam equation always vanishes asymptotically while the component from
the (undamped) wave equation does not. In case of small energies we show t
hat the first component vanishes at exponential rate. Our results highligh
t that partial damping is not enough to steer\nthe whole solution to rest
and that the partially (controlled) damped system can be less stable than
the undamped system. Hence\, the model and the behavior of the solution en
ter in the framework of the so-called indirect damping and destabilization
paradox. These phenomena are valorized by a physical interpretation leadi
ng to possible new explanations of the Tacoma Narrows Bridge collapse and
to possible damages due to the damping control parameter. This is joint wo
rk with A. Soufyane (Sharjah\, UAE)\, based on a previous model developed
with M. Garrione (Milano\, Italy).\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tadele Mengesha (The University of Tennessee\, Knoxville)
DTSTART:20210909T161000Z
DTEND:20210909T165500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/22
DESCRIPTION:Title: Calderon-Zygmund type estimates for nonlocal PDEs with Holder contin
uous kernel\nby Tadele Mengesha (The University of Tennessee\, Knoxvil
le) as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Par
tial Differential Equations\n\n\nAbstract\nIn this talk I will present a r
esult on Sobolev regularity of weak solutions to linear nonlocal equations
. The theory we develop is concerned with obtaining higher integrability a
nd differentiability of solutions of linear nonlocal equations. In additio
n to the standard conditions on the coefficient symmetricity and elliptici
ty\, if we assume uniformly Holder continuity of the coefficient\, then we
ak solutions from the energy space that correspond to highly integrable ri
ght hand side will have an improved Sobolev regularity\nalong the differen
tiability scale in addition to the expected integrability gain. This resu
lt is consistent with self-improving properties of nonlocal equations that
has been observed by other earlier works. To prove our result\, we use a
perturbation argument where optimal regularity of solutions of a simpler e
quation is systematically used to derive an improved regularity for the so
lution of the nonlocal equation. This is a joint work with Armin Schikorra
and Sasikarn Yeepo.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Diening (Bielefeld University)
DTSTART:20210909T172000Z
DTEND:20210909T180500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/23
DESCRIPTION:Title: Elliptic Equations with Degenerate Weights\nby Lars Diening (Bie
lefeld University) as part of BIRS workshop: Nonlinear Potential Theoretic
Methods in Partial Differential Equations\n\n\nAbstract\nWe obtain new lo
cal Calderon-Zygmund estimates for elliptic equations with matrix-valued w
eights for linear as well as non-linear equations. We introduce a novel $\
\log-BMO$ condition on the weight. In particular\, we assume smallness of
the logarithm of the matrix-valued weight in $BMO$. This allows to include
degenerate\, discontinuous weights. The assumption on the smallness param
eter is sharp and linear in terms of the integrability exponent of the gra
dient. This is a novelty even in the linear setting with non-degenerate we
ights compared to previously known results\, where the dependency was expo
nential. We also consider regularity up to the boundary. The exponent of i
ntegrability depends again linearly on the smallness condition on the boun
dary.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert McOwen (Northeastern Univeristy)
DTSTART:20210909T180500Z
DTEND:20210909T185000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/24
DESCRIPTION:Title: Gilbarg-Serrin Equation and Lipschitz Regularity\nby Robert McOw
en (Northeastern Univeristy) as part of BIRS workshop: Nonlinear Potential
Theoretic Methods in Partial Differential Equations\n\n\nAbstract\nWe dis
cuss conditions for Lipschitz and $C^1$ regularity for a uniformly ellipti
c equation in divergence form with coefficients that were introduced by Gi
lbarg & Serrin. In particular\, we find cases where Lipschitz regularity h
olds but the coefficients are not Dini continuous\, or do not even have Di
ni mean oscillation. The form of the coefficients also enables us to obtai
n specific conditions and examples for which there exists a weak solution
that is not Lipschitz continuous. (This is joint work with V.G.Maz’ya.)\
n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florica Cîrstea (University of Sydney)
DTSTART:20210910T133000Z
DTEND:20210910T141500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/25
DESCRIPTION:Title: Anisotropic elliptic equations with gradient-dependent lower order t
erms and L1 data\nby Florica Cîrstea (University of Sydney) as part o
f BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Differen
tial Equations\n\n\nAbstract\nFor every summable function $f$\, we prove t
he existence of a weak solution for a general class of Dirichlet anisotrop
ic elliptic problems in a bounded open subset $\\Omega$ of $\\mathbb R^N$.
The principal part is a divergence-form nonlinear anisotropic operator $\
\mathcal A$\, the prototype of which is $$\\mathcal A u=-\\sum_{j=1}^N \\p
artial_j(|\\partial_j u|^{p_j-2}\\partial_j u)$$ with $p_j>1$ for all $1\\
leq j\\leq N$ and $\\sum_{j=1}^N (1/p_j)>1$. As a novelty\, our lower orde
r terms involve a new class of operators $\\mathfrak B$ such that $\\mathc
al{A}-\\mathfrak{B}$ is bounded\, coercive and pseudo-monotone from $W_0^{
1\,\\overrightarrow{p}}(\\Omega)$ into its dual\, as well as a gradient-de
pendent nonlinearity with an ``anisotropic natural growth" in the gradient
and a good sign condition. This is joint work with Barbara Brandolini (Un
iversita degli Studi di Palermo\, Italy).\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Premoselli (Université Libre de Bruxelles)
DTSTART:20210910T141500Z
DTEND:20210910T150000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/26
DESCRIPTION:Title: Towers of bubbles for critical stationary Schrodinger equations in l
arge dimensions\nby Bruno Premoselli (Université Libre de Bruxelles)
as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial
Differential Equations\n\n\nAbstract\nIn this talk we consider perturbati
ons of critical stationary Schrodinger equations\, such as Yamabe-type equ
ations on manifolds or Brézis-Nirenberg-type equations on bounded open se
ts. We are interested in the blow-up behavior of such equations\; in parti
cular in how blowing-up solutions may develop « multi-bubble blow-up »\,
that is how several interacting concentrating peaks may appear.\n In dime
nsions larger than 7\, on a locally conformally flat manifold\, we constru
ct positive blowing-up solutions of such equations that behave like towers
of bubbles concentrating at a critical point of the mass function. The re
sult does not assume any symmetry on the underlying manifold. The construc
tion is performed by combining finite-dimensional reduction methods with a
linear bubble-tree analysis. Our approach works both in the positive and
sign-changing case: as a byproduct of our analysis we prove the existence\
, on a generic bounded open set of $\\mathbb{R}^n$\, of blowing-up solutio
ns of the Brézis-Nirenberg equation that behave like towers of bubbles of
alternating signs.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Van Schaftingen (Université catholique de Louvain)
DTSTART:20210910T152500Z
DTEND:20210910T161000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/27
DESCRIPTION:Title: Ginzburg–Landau functionals on planar domains for a general compac
t vacuum manifold\nby Jean Van Schaftingen (Université catholique de
Louvain) as part of BIRS workshop: Nonlinear Potential Theoretic Methods i
n Partial Differential Equations\n\n\nAbstract\nGinzburg–Landau type fun
ctionals provide a relaxation scheme to construct harmonic maps in the pre
sence of topological obstructions. They arise in superconductivity models\
, in liquid crystal models (Landau–de Gennes functional) and in the gene
ration of cross-fields in meshing. For a general compact manifold target s
pace we describe the asymptotic number\, type and location of singularitie
s that arise in minimizers. We cover in particular the case where the fund
amental group of the vacuum manifold in nonabelian and hence the singulari
ties cannot be characterized univocally as elements of the fundamental gro
up. We obtain similar results for $p$–harmonic maps with $p<2$ going to
$2$. The results unify the existing theory and cover new situations and pr
oblems.\nThis is a joint work with Antonin Monteil (Paris-Est Créteil\, F
rance)\, Rémy Rodiac (Paris–Saclay\, France) and Benoît Van Vaerenberg
h (UCLouvain).\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Nitsch (Università di Napoli "Federico II")
DTSTART:20210910T161000Z
DTEND:20210910T165500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/28
DESCRIPTION:Title: Some optimization problems in thermal insulation\nby Carlo Nitsc
h (Università di Napoli "Federico II") as part of BIRS workshop: Nonlinea
r Potential Theoretic Methods in Partial Differential Equations\n\n\nAbstr
act\nOptimal insulation consists in finding the ``best" displacement of a
prescribed volume of insulating material around a given conductor. Accordi
ng to circumstances\, the ``best" configuration can be the one which minim
izes the heat dispersion\, maximizes the heat content\, minimizes the heat
rate loss etc. \nWe provide a flavor of the state of the art\, and then w
e focus on the case of prescribed heat source (inside the conductor)\, wit
h convective heat transfer across the solid and the environment. This corr
esponds to consider the stationary heat equation inside both conductor & i
nsulator together with Robin boundary conditions at the external boundary.
We aim at maximizing the heat content (the $L^1$ norm of the solution) am
ong all the possible distributions of insulating material with fixed mass\
, and we prove an optimal upper bound in terms of geometric quantities alo
ne. Eventually we prove a conjecture according to which the ball surrounde
d by a uniform distribution of insulating material maximizes the heat cont
ent.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Beck (Augsburg University)
DTSTART:20210910T172000Z
DTEND:20210910T180500Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/29
DESCRIPTION:Title: Lipschitz bounds and non-uniform ellipticity\nby Lisa Beck (Augs
burg University) as part of BIRS workshop: Nonlinear Potential Theoretic M
ethods in Partial Differential Equations\n\n\nAbstract\nn this talk we con
sider a large class of non-uniformly elliptic variational problems and dis
cuss optimal conditions guaranteeing the local Lipschitz regularity of sol
utions in terms of the regularity of the data. The analysis covers the mai
n model cases of variational integrals of anisotropic growth\, but also of
fast growth of exponential type investigated in recent years. The regular
ity criteria are established by potential theoretic arguments\, involve na
tural limiting function spaces on the data\, and reproduce\, in this very
general context\, the classical and optimal ones known in the linear case
for the Poisson equation. The results presented in this talk are part of a
joined project with Giuseppe Mingione (Parma)\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Camillo De Lellis (Institute for Advanced Study)
DTSTART:20210910T180500Z
DTEND:20210910T185000Z
DTSTAMP:20260422T185229Z
UID:BIRS_21w5100/30
DESCRIPTION:Title: Locally dissipative solutions of the Euler equations\nby Camillo
De Lellis (Institute for Advanced Study) as part of BIRS workshop: Nonlin
ear Potential Theoretic Methods in Partial Differential Equations\n\n\nAbs
tract\nThe Onsager conjecture\, recently solved by Phil Isett\, states tha
t\, below a certain threshold regularity\, Hoelder continuous solutions of
the Euler equations might dissipate the kinetic energy. The original work
of Onsager was motivated by the phenomenon of anomalous dissipation and a
rigorous mathematical justification of the latter should show that the en
ergy dissipation in the Navier-Stokes equations is\, in a suitable statist
ical sense\, independent of the viscosity. In particular it makes much mor
e sense to look for solutions of the Euler equations which\, besides dissi
pating the total kinetic energy\, satisfy as well a suitable form of loc
al energy inequality. Such solutions were first shown to exist by Laszlo S
zekelyhidi Jr. and myself. In this talk I will review the methods used so
far to approach their existence and the most recent results by Isett and b
y Hyunju Kwon and myself.\n
LOCATION:https://researchseminars.org/talk/BIRS_21w5100/30/
END:VEVENT
END:VCALENDAR