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BEGIN:VEVENT
SUMMARY:Xavier Cabre (ICREA and Universitat Politecnica de Catalunya)
DTSTART:20210906T150000Z
DTEND:20210906T154500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/1
DESCRIPTION:Title: Stable solutions to semilinear elliptic equations are smooth u
p to dimension 9\nby Xavier Cabre (ICREA and Universitat Politecnica d
e Catalunya) as part of BIRS workshop: Nonlinear Potential Theoretic Metho
ds in Partial Differential Equations\n\n\nAbstract\nThe regularity of stab
le solutions to semilinear elliptic PDEs has been studied since the 1970's
. In dimensions $10$ and higher\, there exist singular stable energy solut
ions. In this talk I will describe a recent work in collaboration with Fig
alli\, Ros-Oton\, and Serra\, where we prove that stable solutions are smo
oth up to the optimal dimension $9$. This answers to an open problem posed
by Brezis in the mid-nineties concerning the regularity of extremal solut
ions to Gelfand-type problems.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Focardi (Università di Firenze)
DTSTART:20210906T154500Z
DTEND:20210906T163000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/2
DESCRIPTION:Title: On the regularity of singular sets of minimizers for the Mumfo
rd-Shah energy\nby Matteo Focardi (Università di Firenze) as part of
BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Differenti
al Equations\n\n\nAbstract\nWe will survey the regularity theory of minimi
zers of the Mumford-Shah functional\, focusing in particular on that of th
e corresponding singular sets. Starting with nowadays classical results\,
we will finally discuss more recent developments\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phuc Nguyen (Louisiana State University)
DTSTART:20210906T170000Z
DTEND:20210906T174500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/3
DESCRIPTION:Title: Choquet integrals\, capacitary inequalities\, and the Hardy-Li
ttlewood maximal function\nby Phuc Nguyen (Louisiana State University)
as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Partia
l Differential Equations\n\n\nAbstract\nWe obtain the boundedness of the H
ardy-Littlewood maximal function on $L^q$ type spaces defined via Choquet
integrals associate to Sobolev capacities. The bounds are obtained in full
range of exponents including a weak type end-point bound. We also obtain
a capacitary inequality of Maz'ya type which resolves a problem proposed b
y D. Adams. This talk is based on joint work with Keng Hao Ooi.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pengfei Guan (McGill University)
DTSTART:20210906T174500Z
DTEND:20210906T183000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/4
DESCRIPTION:Title: Entropy quantities associated to Gauss curvature type flows\nby Pengfei Guan (McGill University) as part of BIRS workshop: Nonlinear
Potential Theoretic Methods in Partial Differential Equations\n\n\nAbstra
ct\nWe discuss the role of entropy functionals played in the study of Gaus
s curvature type flows: 1. the monotonicity of the associated entropies\,
2. diameter\, non-collapsing entropy points estimates\, 3. convergence. S
imilar entropy functionals also exists for anisotropy type Gauss curvature
flows.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Connor Mooney (University of California\, Irvine)
DTSTART:20210906T183000Z
DTEND:20210906T191500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/5
DESCRIPTION:Title: The Bernstein problem for equations of minimal surface type\nby Connor Mooney (University of California\, Irvine) as part of BIRS wo
rkshop: Nonlinear Potential Theoretic Methods in Partial Differential Equa
tions\n\n\nAbstract\nThe Bernstein problem asks whether entire minimal gra
phs in dimension N+1 are necessarily hyperplanes. This problem was solved
in combined works of Bernstein\, Fleming\, De Giorgi\, Almgren\, and Simon
s ("yes" if N < 8)\, and Bombieri-De Giorgi-Giusti ("no" otherwise). We wi
ll discuss the analogue of this problem for graphical minimizers of anisot
ropic energies. In particular\, we will discuss new examples of nonlinear
entire graphical minimizers in the case N = 6\, and recent joint work with
Y. Yang towards constructing such examples in the lowest-possible-dimensi
onal case N = 4.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Iwona Chlebicka (University of Warsaw)
DTSTART:20210907T130000Z
DTEND:20210907T134500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/6
DESCRIPTION:Title: Potential estimates for solutions to quasilinear elliptic prob
lems with general growth. Scalar and vectorial case\nby Iwona Chlebick
a (University of Warsaw) as part of BIRS workshop: Nonlinear Potential The
oretic Methods in Partial Differential Equations\n\n\nAbstract\nWe conside
r measure data elliptic problems involving a second order operator exhibit
ing Orlicz growth and having measurable coefficients. As known in the $p$-
Laplace case\, pointwise estimates for solutions expressed with the use of
nonlinear potential are powerful tools in the study of the local behaviou
r of the solutions. Not only we provide such estimates expressed in terms
of a potential of generalized Wolff type\, but also we investigate their r
egularity consequences. For scalar equations we do not need to impose any
structural conditions on the the operator and we study generalized $A$-har
monic functions being distributional solutions to problems with nonnegativ
e measure. Lower and upper estimates we provide are sharp in the sense tha
t the potential 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 solutions. For systems we impose typical assumptions of the Uhlenbeck
-type structure of the operator and separated variables\, whereas the meas
ure can be signed as another notion of very weak solutions is employed. In
this case the upper bound is shown with the same potential as in the scal
ar case and presented together with its precise consequences for the local
behaviour of solutions. The talk is based on joint works:(scalar) with F.
~Giannetti and A.~Zatorska-Goldstein [arXiv:2006.02172] and (vectorial) wi
th Y.~Youn and A.~Zatorska-Goldstein\, [arXiv:2102.09313]\, [arXiv:2106.11
639].\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Schwarzacher (Charles University)
DTSTART:20210907T134500Z
DTEND:20210907T143000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/7
DESCRIPTION:Title: Construction of a right inverse for the divergence in non-cyli
ndrical time dependent domains\nby Sebastian Schwarzacher (Charles Uni
versity) as part of BIRS workshop: Nonlinear Potential Theoretic Methods i
n Partial Differential Equations\n\n\nAbstract\nWe discuss the constructio
n of a stable right inverse for the divergence operator in non-cylindrical
domains in space-time. The domains are assumed to be Hölder regular in s
pace and evolve continuously in time. The inverse operator is of Bogovskij
type\, meaning that it attains zero boundary values. We provide estimates
in Sobolev spaces of positive and negative order with respect to both tim
e and space variables. The regularity estimates on the operator depend on
the assumed Hölder regularity of the domain. The results can naturally be
connected to the known theory for Lipschitz domains. As an application\,
we prove refined pressure estimates for weak and very weak solutions to Na
vier--Stokes equations in time dependent domains. This is a joint work wit
h Olli Saari.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuomo Kuusi (University of Helsinki)
DTSTART:20210907T143000Z
DTEND:20210907T151500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/8
DESCRIPTION:Title: Homogenization\, linearization and large-scale regularity for
nonlinear elliptic equations\nby Tuomo Kuusi (University of Helsinki)
as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial
Differential Equations\n\n\nAbstract\nWe will consider nonlinear\, unifor
mly elliptic equations with variational structure and random\, highly osci
llating coefficients and discuss the corresponding stochastic homogenizati
on theory. After recalling basic ideas on how to get quantitative rates of
homogenization for nonlinear uniformly convex problems\, we will discuss
our recent work\, jointly with S. Armstrong and S. Ferguson\, showing that
homogenization and linearization commute. This is in the sense that the l
inearized equation homogenizes to the linearization of the homogenized equ
ation (linearized around the corresponding solution of the homogenized equ
ation). This procedure can be iterated to show higher regularity of the ho
mogenized Lagrangian as well as large-scale regularity for minimizers.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Verena Bogelein (Paris-Lodron-University Salzburg)
DTSTART:20210907T154000Z
DTEND:20210907T162500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/9
DESCRIPTION:Title: Higher regularity in congested traffic dynamics\nby Verena
Bogelein (Paris-Lodron-University Salzburg) as part of BIRS workshop: Non
linear Potential Theoretic Methods in Partial Differential Equations\n\n\n
Abstract\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
degenerate problems. Continuity properties of the gradient have been inves
tigated in the scalar case by Santambrogio & Vespri and Colombo & Figalli.
\nIn this talk we establish the optimal regularity of weak solutions in t
he vectorial case for any $p>1$. This is joint work with F. Duzaar\, R. Gi
ova and A. Passarelli di Napoli.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristiana De Filippis (Università di Torino)
DTSTART:20210907T162500Z
DTEND:20210907T171000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/10
DESCRIPTION:Title: Perturbations beyond Schauder\nby Cristiana De Filippis (
Università di Torino) as part of BIRS workshop: Nonlinear Potential Theor
etic Methods in Partial Differential Equations\n\n\nAbstract\nSchauder est
imates hold in the nonuniformly elliptic setting. Specifically\, first der
ivatives of solutions to nonuniformly elliptic variational problems and el
liptic equations are locally H\\"older continuous\, provided coefficients
are locally H\\"older continuous. In this talk I will present new regulari
ty results for minima of nonuniformly elliptic functionals with emphasis o
n delicate borderline regulairty criteria. My talk is based on papers:\n-C
. De Filippis\, Quasiconvexity and partial regularity via nonlinear potent
ials. Preprint (2021)\;\n-C. De Filippis\, G. Mingione\, Lipschitz bounds
and nonautonomous integrals. Arch. Ration. Mech. Anal.\, to appear\; C. De
Filippis\, G. Mingione\, Nonuniformly elliptic Schauder estimates. Prepri
nt (2021).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ovidiu Savin (Columbia University)
DTSTART:20210907T173500Z
DTEND:20210907T182000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/11
DESCRIPTION:Title: The multiple membrane problem\nby Ovidiu Savin (Columbia
University) as part of BIRS workshop: Nonlinear Potential Theoretic Method
s in Partial Differential Equations\n\n\nAbstract\nFor a positive integer
$N$\, the $N$-membranes problem describes the equilibrium position of $N$
ordered elastic membranes subject to forcing and boundary conditions. If t
he heights 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
obstacle problems with interacting free boundaries which can cross each ot
her. When $N=2$ there is only one free boundary and the problem is equival
ent to the classical obstacle problem. I will discuss a work in collaborat
ion with Hui Yu about the regularity of the free boundaries in the two dim
ensional case.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniela De Silva (Barnard College - Columbia University)
DTSTART:20210907T182000Z
DTEND:20210907T190500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/12
DESCRIPTION:Title: Inhomogeneous global minimizers to the one-phase free boundar
y problem\nby Daniela De Silva (Barnard College - Columbia University)
as part of BIRS workshop: Nonlinear Potential Theoretic Methods in Partia
l Differential Equations\n\n\nAbstract\nGiven a global 1-homogeneous minim
izer $U_0$ to the Alt-Caffarelli energy functional\, with $sing(F(U_0)) =
\\{0\\}$\, we provide a foliation of the half-space $\\mathbb R^{n} \\time
s [0\,+\\infty)$ with dilations of graphs of global minimizers $\\underlin
e U \\leq U_0 \\leq \\bar U$ with analytic free boundaries at distance 1 f
rom the origin. This is a joint work with D. Jerison and H. Shahgholian.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rupert Frank (Caltech / University of Munich)
DTSTART:20210908T133000Z
DTEND:20210908T141500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/13
DESCRIPTION:Title: Which magnetic fields support a zero mode?\nby Rupert Fra
nk (Caltech / University of Munich) as part of BIRS workshop: Nonlinear Po
tential Theoretic Methods in Partial Differential Equations\n\n\nAbstract\
nMotivated by the question from mathematical physics about the size of mag
netic fields that support zero modes for the three dimensional Dirac equat
ion\, we study a certain conformally invariant spinor equation. We state s
ome conjectures and present results in their support. Those concern\, in p
articular\, two novel Sobolev inequalities for spinors and vector fields.
The talk is based on joint work with Michael Loss.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Kristensen (University of Oxford)
DTSTART:20210908T141500Z
DTEND:20210908T150000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/14
DESCRIPTION:Title: Decompositions of sequences of PDE constrained maps\nby J
an Kristensen (University of Oxford) as part of BIRS workshop: Nonlinear P
otential Theoretic Methods in Partial Differential Equations\n\n\nAbstract
\nIt is a convenient and well-known fact that for exponents p>1\, any Lp-w
eakly converging sequence of PDE constrained\nmaps admits a decomposition
into sequences of PDE constrained maps where one converges in measure (no
oscillation) and\nthe other is p-equi-integrable (no concentration). For p
=1 the relevant corresponding result concerns weakly* convergent sequences
\nof PDE constrained measures and is false: the oscillation and concentrat
ion cannot be separated while simultaneously satisfying\nthe PDE constrain
t. In this talk we explain how the concentration regardless of the failure
of a decomposition result retains its PDE\ncharacter. The presented resul
ts are parts of joint works with Andre Guerra and Bogdan Raita.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Salani (Università di Firenze)
DTSTART:20210908T152500Z
DTEND:20210908T161000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/15
DESCRIPTION:Title: The intimate relationship between log-concavity and heat flow
\nby Paolo Salani (Università di Firenze) as part of BIRS workshop: N
onlinear Potential Theoretic Methods in Partial Differential Equations\n\n
\nAbstract\nThe talk will be based on some papers in collaboration with Ka
zuhiro Ishige (The University of Tokyo) and Asuka Takatsu (Tokyo Metropoli
tan University) where we investigate the preservation of concavity propert
ies by heat flow. Surprisingly\, we have recently proved that there exist
concavities stronger than log-concavity that are preserved by the Dirichle
t heat flow\, however\, when we consider a suitable class of concavities\,
log-concavity remains the strongest possible. Moreover\, in our latest pa
per\, we prove that\, when starting with an initial datum which shares any
concavity weaker than log-concavity\, then the solution may lose immediat
ely any reminiscence of concavity. In this way we almost complete the stud
y of preservation of concavity by the Dirichlet heat flow\, started by Bra
scamp and Lieb in 1976.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jana Bjorn (Linkoeping University)
DTSTART:20210908T161000Z
DTEND:20210908T165500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/16
DESCRIPTION:Title: Fine potential theory via analysis on metric spaces\nby J
ana Bjorn (Linkoeping University) as part of BIRS workshop: Nonlinear Pote
ntial Theoretic Methods in Partial Differential Equations\n\n\nAbstract\nW
eshow how p-harmonic functions and Sobolev spaces on metric spaces\, base
d on upper gradients\, naturally lead to fine potential theory\, even in t
he setting of Euclidean spaces.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tobias Weth (Goethe-University Frankfurt am Main)
DTSTART:20210908T172000Z
DTEND:20210908T180500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/17
DESCRIPTION:Title: Morse index versus radial symmetry for fractional Dirichlet p
roblems\nby Tobias Weth (Goethe-University Frankfurt am Main) as part
of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial Differe
ntial Equations\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 M
orse index of radially symmetric sign changing solutions to\nsemilinear fr
actional Dirichlet\nproblems in the unit ball. In particular\, the result
applies to the\nDirichlet eigenvalue problem for the\nfractional Laplacian
and implies that eigenfunctions corresponding to\nthe second Dirichlet ei
genvalue\nare antisymmetric. This resolves a conjecture of Banuelos and Ku
lczycki.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Scott Armstrong (Scott Armstrong)
DTSTART:20210908T180500Z
DTEND:20210908T185000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/18
DESCRIPTION:Title: Optimal doubling inequalities for periodic elliptic equations
\nby Scott Armstrong (Scott Armstrong) as part of BIRS workshop: Nonli
near Potential Theoretic Methods in Partial Differential Equations\n\n\nAb
stract\nI will discuss recent work with T. Kuusi and C. Smart on quantitat
ive unique continuation for solutions of periodic elliptic equations on la
rge scales.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dominic Breit (Heriot-Watt University)
DTSTART:20210909T133000Z
DTEND:20210909T141500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/19
DESCRIPTION:Title: Global Besov regularity for nonlinear elliptic problems\n
by Dominic Breit (Heriot-Watt University) as part of BIRS workshop: Nonlin
ear Potential Theoretic Methods in Partial Differential Equations\n\n\nAbs
tract\nWe prove global Besov estimates for the p-Laplacian with right-hand
side in divergence form under optimal assumptions on the regularity of th
e boundary of the domain $\\Omega$. In particular\, we show that $B^s_{\\v
arrho\,q}(\\Omega)$-regularity transfers from the forcing $F$ to the non-l
inear flux $|\\nabla u|^{p-2}\\nabla u$ provided the boundary belongs to t
he class $B^{s+1-1/q}_{\\varrho\,q}$ and has a small Lipschitz constant. I
n the linear case $p=2$ this recovers a sharp result from Maz'ya-Shaposhni
kova.\nThis is a joint work with A. Balci and L. Diening.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frédéric Robert (Universite de Lorraine)
DTSTART:20210909T141500Z
DTEND:20210909T150000Z
DTSTAMP:20260404T041619Z
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://stable.researchseminars.org/talk/BIRS_21w5100/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Filippo Gazzola (Politecnico di Milano)
DTSTART:20210909T152500Z
DTEND:20210909T161000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/21
DESCRIPTION:Title: Long-time behavior of partially damped systems modeling degen
erate plates with piers\nby Filippo Gazzola (Politecnico di Milano) as
part of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial D
ifferential Equations\n\n\nAbstract\nWe consider a partially damped nonlin
ear beam-wave system of evolution PDE's modeling the dynamics of a degener
ate plate. The plate can move both vertically and torsionally and\, conseq
uently\, the solution has two components. We show that the component from
the damped beam equation always vanishes asymptotically while the componen
t from the (undamped) wave equation does not. In case of small energies we
show that the first component vanishes at exponential rate. Our results h
ighlight that partial damping is not enough to steer\nthe whole solution t
o rest and that the partially (controlled) damped system can be less stabl
e than the undamped system. Hence\, the model and the behavior of the solu
tion enter in the framework of the so-called indirect damping and destabil
ization paradox. These phenomena are valorized by a physical interpretatio
n leading to possible new explanations of the Tacoma Narrows Bridge collap
se and to possible damages due to the damping control parameter. This is j
oint work with A. Soufyane (Sharjah\, UAE)\, based on a previous model dev
eloped with M. Garrione (Milano\, Italy).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tadele Mengesha (The University of Tennessee\, Knoxville)
DTSTART:20210909T161000Z
DTEND:20210909T165500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/22
DESCRIPTION:Title: Calderon-Zygmund type estimates for nonlocal PDEs with Holder
continuous kernel\nby Tadele Mengesha (The University of Tennessee\,
Knoxville) as part of BIRS workshop: Nonlinear Potential Theoretic Methods
in Partial Differential Equations\n\n\nAbstract\nIn this talk I will pres
ent a result on Sobolev regularity of weak solutions to linear nonlocal eq
uations. The theory we develop is concerned with obtaining higher integrab
ility and differentiability of solutions of linear nonlocal equations. In
addition to the standard conditions on the coefficient symmetricity and el
lipticity\, if we assume uniformly Holder continuity of the coefficient\,
then weak solutions from the energy space that correspond to highly integr
able right hand side will have an improved Sobolev regularity\nalong the d
ifferentiability scale in addition to the expected integrability gain. Th
is result is consistent with self-improving properties of nonlocal equatio
ns that has been observed by other earlier works. To prove our result\, we
use a perturbation argument where optimal regularity of solutions of a si
mpler equation is systematically used to derive an improved regularity for
the solution of the nonlocal equation. This is a joint work with Armin Sc
hikorra and Sasikarn Yeepo.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lars Diening (Bielefeld University)
DTSTART:20210909T172000Z
DTEND:20210909T180500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/23
DESCRIPTION:Title: Elliptic Equations with Degenerate Weights\nby Lars Dieni
ng (Bielefeld University) as part of BIRS workshop: Nonlinear Potential Th
eoretic Methods in Partial Differential Equations\n\n\nAbstract\nWe obtain
new local Calderon-Zygmund estimates for elliptic equations with matrix-v
alued weights for linear as well as non-linear equations. We introduce a n
ovel $\\log-BMO$ condition on the weight. In particular\, we assume smalln
ess of the logarithm of the matrix-valued weight in $BMO$. This allows to
include degenerate\, discontinuous weights. The assumption on the smallnes
s parameter is sharp and linear in terms of the integrability exponent of
the gradient. This is a novelty even in the linear setting with non-degene
rate weights compared to previously known results\, where the dependency w
as exponential. We also consider regularity up to the boundary. The expone
nt of integrability depends again linearly on the smallness condition on t
he boundary.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert McOwen (Northeastern Univeristy)
DTSTART:20210909T180500Z
DTEND:20210909T185000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/24
DESCRIPTION:Title: Gilbarg-Serrin Equation and Lipschitz Regularity\nby Robe
rt McOwen (Northeastern Univeristy) as part of BIRS workshop: Nonlinear Po
tential Theoretic Methods in Partial Differential Equations\n\n\nAbstract\
nWe discuss conditions for Lipschitz and $C^1$ regularity for a uniformly
elliptic equation in divergence form with coefficients that were introduce
d by Gilbarg & Serrin. In particular\, we find cases where Lipschitz regul
arity holds but the coefficients are not Dini continuous\, or do not even
have Dini mean oscillation. The form of the coefficients also enables us t
o obtain specific conditions and examples for which there exists a weak so
lution that is not Lipschitz continuous. (This is joint work with V.G.Maz
’ya.)\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Florica Cîrstea (University of Sydney)
DTSTART:20210910T133000Z
DTEND:20210910T141500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/25
DESCRIPTION:Title: Anisotropic elliptic equations with gradient-dependent lower
order terms and L1 data\nby Florica Cîrstea (University of Sydney) as
part of BIRS workshop: Nonlinear Potential Theoretic Methods in Partial D
ifferential Equations\n\n\nAbstract\nFor every summable function $f$\, we
prove the existence of a weak solution for a general class of Dirichlet an
isotropic elliptic problems in a bounded open subset $\\Omega$ of $\\mathb
b R^N$. The principal part is a divergence-form nonlinear anisotropic oper
ator $\\mathcal A$\, the prototype of which is $$\\mathcal A u=-\\sum_{j=1
}^N \\partial_j(|\\partial_j u|^{p_j-2}\\partial_j u)$$ with $p_j>1$ for a
ll $1\\leq j\\leq N$ and $\\sum_{j=1}^N (1/p_j)>1$. As a novelty\, our low
er order terms involve a new class of operators $\\mathfrak B$ such that $
\\mathcal{A}-\\mathfrak{B}$ is bounded\, coercive and pseudo-monotone from
$W_0^{1\,\\overrightarrow{p}}(\\Omega)$ into its dual\, as well as a grad
ient-dependent nonlinearity with an ``anisotropic natural growth" in the g
radient and a good sign condition. This is joint work with Barbara Brandol
ini (Universita degli Studi di Palermo\, Italy).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bruno Premoselli (Université Libre de Bruxelles)
DTSTART:20210910T141500Z
DTEND:20210910T150000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/26
DESCRIPTION:Title: Towers of bubbles for critical stationary Schrodinger equatio
ns in large dimensions\nby Bruno Premoselli (Université Libre de Brux
elles) as part of BIRS workshop: Nonlinear Potential Theoretic Methods in
Partial Differential Equations\n\n\nAbstract\nIn this talk we consider per
turbations of critical stationary Schrodinger equations\, such as Yamabe-t
ype equations on manifolds or Brézis-Nirenberg-type equations on bounded
open sets. We are interested in the blow-up behavior of such equations\; i
n particular in how blowing-up solutions may develop « multi-bubble blow-
up »\, that is how several interacting concentrating peaks may appear.\n
In dimensions larger than 7\, on a locally conformally flat manifold\, we
construct positive blowing-up solutions of such equations that behave like
towers of bubbles concentrating at a critical point of the mass function.
The result does not assume any symmetry on the underlying manifold. The c
onstruction is performed by combining finite-dimensional reduction methods
with a linear bubble-tree analysis. Our approach works both in the positi
ve and sign-changing case: as a byproduct of our analysis we prove the exi
stence\, on a generic bounded open set of $\\mathbb{R}^n$\, of blowing-up
solutions of the Brézis-Nirenberg equation that behave like towers of bub
bles of alternating signs.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Van Schaftingen (Université catholique de Louvain)
DTSTART:20210910T152500Z
DTEND:20210910T161000Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/27
DESCRIPTION:Title: Ginzburg–Landau functionals on planar domains for a general
compact vacuum manifold\nby Jean Van Schaftingen (Université catholi
que de Louvain) as part of BIRS workshop: Nonlinear Potential Theoretic Me
thods in Partial Differential Equations\n\n\nAbstract\nGinzburg–Landau t
ype functionals provide a relaxation scheme to construct harmonic maps in
the presence of topological obstructions. They arise in superconductivity
models\, in liquid crystal models (Landau–de Gennes functional) and in t
he generation of cross-fields in meshing. For a general compact manifold t
arget space we describe the asymptotic number\, type and location of singu
larities that arise in minimizers. We cover in particular the case where t
he fundamental group of the vacuum manifold in nonabelian and hence the si
ngularities cannot be characterized univocally as elements of the fundamen
tal group. We obtain similar results for $p$–harmonic maps with $p<2$ go
ing to $2$. The results unify the existing theory and cover new situations
and problems.\nThis is a joint work with Antonin Monteil (Paris-Est Crét
eil\, France)\, Rémy Rodiac (Paris–Saclay\, France) and Benoît Van Vae
renbergh (UCLouvain).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Nitsch (Università di Napoli "Federico II")
DTSTART:20210910T161000Z
DTEND:20210910T165500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/28
DESCRIPTION:Title: Some optimization problems in thermal insulation\nby Carl
o Nitsch (Università di Napoli "Federico II") as part of BIRS workshop: N
onlinear Potential Theoretic Methods in Partial Differential Equations\n\n
\nAbstract\nOptimal insulation consists in finding the ``best" displacemen
t of a prescribed volume of insulating material around a given conductor.
According to circumstances\, the ``best" configuration can be the one whic
h minimizes the heat dispersion\, maximizes the heat content\, minimizes t
he heat rate loss etc. \nWe provide a flavor of the state of the art\, and
then we focus on the case of prescribed heat source (inside the conductor
)\, with convective heat transfer across the solid and the environment. Th
is corresponds to consider the stationary heat equation inside both conduc
tor & insulator together with Robin boundary conditions at the external bo
undary. We aim at maximizing the heat content (the $L^1$ norm of the solut
ion) among all the possible distributions of insulating material with fixe
d mass\, and we prove an optimal upper bound in terms of geometric quantit
ies alone. Eventually we prove a conjecture according to which the ball su
rrounded by a uniform distribution of insulating material maximizes the he
at content.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Beck (Augsburg University)
DTSTART:20210910T172000Z
DTEND:20210910T180500Z
DTSTAMP:20260404T041619Z
UID:BIRS_21w5100/29
DESCRIPTION:Title: Lipschitz bounds and non-uniform ellipticity\nby Lisa Bec
k (Augsburg University) as part of BIRS workshop: Nonlinear Potential Theo
retic Methods in Partial Differential Equations\n\n\nAbstract\nn this talk
we consider a large class of non-uniformly elliptic variational problems
and discuss optimal conditions guaranteeing the local Lipschitz regularity
of solutions in terms of the regularity of the data. The analysis covers
the main model cases of variational integrals of anisotropic growth\, but
also of fast growth of exponential type investigated in recent years. The
regularity criteria are established by potential theoretic arguments\, inv
olve natural limiting function spaces on the data\, and reproduce\, in thi
s very general context\, the classical and optimal ones known in the linea
r case for the Poisson equation. The results presented in this talk are pa
rt of a joined project with Giuseppe Mingione (Parma)\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Camillo De Lellis (Institute for Advanced Study)
DTSTART:20210910T180500Z
DTEND:20210910T185000Z
DTSTAMP:20260404T041619Z
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:
Nonlinear Potential Theoretic Methods in Partial Differential Equations\n
\n\nAbstract\nThe Onsager conjecture\, recently solved by Phil Isett\, sta
tes that\, below a certain threshold regularity\, Hoelder continuous solut
ions of the Euler equations might dissipate the kinetic energy. The origin
al work of Onsager was motivated by the phenomenon of anomalous dissipatio
n and a rigorous mathematical justification of the latter should show that
the energy dissipation in the Navier-Stokes equations is\, in a suitable
statistical sense\, independent of the viscosity. In particular it makes m
uch more sense to look for solutions of the Euler equations which\, beside
s dissipating the total kinetic energy\, satisfy as well a suitable form
of local energy inequality. Such solutions were first shown to exist by L
aszlo Szekelyhidi Jr. and myself. In this talk I will review the methods u
sed so far to approach their existence and the most recent results by Iset
t and by Hyunju Kwon and myself.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_21w5100/30/
END:VEVENT
END:VCALENDAR