On Vladimir Goncharov
24 years ago I have been awarded a NATO
Research Fellowship to support the first
visit of Vladimir Goncharov to Évora in
June-December 1994.
I had met him in the Spring of 1993, at
SISSA, in Trieste, Italy, during my
first sabbatical semester visiting SISSA,
where I had completed my PhD at the end
of 1988, after 4 years of study and
research there, supervised by Arrigo
Cellina.
Vladimir was then invited by Arrigo, to
visit SISSA for the first time, a few
months, I believe, after completing his
PhD at Irkutsk University, near the
great Lake Baikal in Siberia, supervised
by Professor Tolstonogov, a friend of
Arrigo.
We talked a lot during those months at
SISSA, and a nice friendship began.
During that second half of 1994 we wrote
together in Évora our first joint
research paper, "On minima of a
functional of the gradient: a continuous
selection", published in 1996 in the
research journal Nonlinear Analysis.
This was a generalization of a famous
paper by Arrigo Cellina, "On minima of a
functional of the gradient: a sufficient
condition", also published in Nonlinear
Analysis, a few years before.
While Arrigo proved that a minimizer
exists — in the non convex case — if and
only if the boundary data of the
surfaces under competition — assumed
affine — have its slope belonging to a
full-dimensional face of the epigraph of
the autonomous Lagrangian, dependent
only on the gradients of competing
surfaces. (More precisely, belonging to
the vertical projection of such face.)
While Arrigo considered just one affine
boundary data, we have considered the
whole family of all possible affine
boundary data, in its dependence on the
slope-parameter, and proved existence of
a solution depending continuously on
this parameter, in the space of
continuous functions, at a.e. boundary
data.
In October 1995, at the end of my
seminar talk on this paper at SISSA,
Gianni DalMaso asked me why almost every
slope instead of every slope.
I explained that it had to do with the
method used for the proof, namely the
Vitali Covering Theorem together with
the Baire Category Method, both non-constructive.
So, besides the a.e. question, our paper
had the inconvenience of furnishing a
non-constructive proof only.
Thus we had proved that a solution
exists, but did not know how to
construct it explicitly.
Gianni told me:
"I believe I know how to construct
explicitly a solution to your problem,
continuously depending on the
slope-parameter. We can work together on
that, if you wish."
We agreed, and after some months Gianni
gave me a dozen sheets of paper where he
had handwritten two lemmas detailing
such construction.
I then wrote this new paper, using these
two lemmas. It was published in 1999,
again in Nonlinear Analysis, with the 3
co-authors, Vladimir, Gianni and Antonio.
I remember well that I was writing it
when I travelled to Irkutsk in August
1996. I continued writing it in Irkutsk,
at some point I was invited by several
colleagues of Vladimir for me to join
them in a canoe trip, rowing along the
shore of lake Baikal. So we had tents
and slept on shore, I would wake up
earlier than my rowing companions, to
continue writing the paper; then we
would row for several hours until
reaching a new place to put our tents
for the night. While the others started
a fire and cooked our dinner, I
continued writing the paper.
After one week of rowing and writing, we
finally took the train back to Irkutsk,
and I handed the completed manuscript to
Vladimir, for him to check it. Not only
our aim turned out to be completely
fulfilled, even better we have
constructed explicitly a minimizer
depending in a locally
Lipschitz-continuous way upon our
slope-parameter.
I was glad to write this paper myself,
because I was in debt towards Vladimir.
Indeed, for our first paper it had been
the other way round: it was Vladimir who
had the idea and wrote the paper; while
my role was to check it.
Later on we published together, me and
Vladimir, a third research paper, "On
minima of a functional of the gradient:
upper and lower solutions", published in
2006 again in Nonlinear Analysis.
For this paper again the roles were
reversed: Vladimir wrote the paper and I
have checked it. Thus I will remain
forever in debt towards Vladimir, since
he conceived and wrote himself two of
our 3 joint research papers — the first
and the third — in which I only checked;
while I conceived and wrote — helped by
the two crucial lemmas conceived by
Gianni DalMaso — our second research
paper, which he checked.
So I can look up to the sky and tell him:
“Vladimir, if you can hear me now, I owe
you one !”.
In our third research paper, we have
constructed — or rather Vladimir has
constructed — a minimizer, continuously
depending on the slope-parameter, which
is always above (or always bellow) all
the other minimizers. So it is the
highest (or the lowest) of all the
minimizers.
I was quite impressed by Vladimir's
capacity to conceive such an elaborate
construction, and to find all the
technical tools necessary to prove
validity in full detail of his
construction.
Let's go back to 1996 for a while. I
have then organized in Évora my first
international math research meeting, in
June 1996. I got then a fellowship to
support the second visit of Vladimir to
Évora: he was here in April-June 1996.
This meeting was quite interesting, I
believe, with many famous mathematicians
participating: Arrigo Cellina, Luc
Tartar, Irene Fonseca, Bernard Dacorogna,
Nicola Fusco, Marcellini, Mikhail Sychev,
Pablo Pedregal, Terry Rockafellar,
António Ornelas