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